Higher-Order Evidence : new essays. 9780198829775, 0198829779

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OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

Higher-Order Evidence

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

Higher-Order Evidence New Essays

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Mattias Skipper Asbjørn Steglich-Petersen

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Great Clarendon Street, Oxford,  , United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © the several contributors  The moral rights of the authors have been asserted First Edition published in  Impression:  All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press  Madison Avenue, New York, NY , United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number:  ISBN –––– DOI: ./oso/.. Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Contents Contributors Introduction Mattias Skipper and Asbjørn Steglich-Petersen

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1. Formulating Independence David Christensen

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2. Higher-Order Uncertainty Kevin Dorst

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3. Evidence of Evidence as Higher-Order Evidence Anna-Maria A. Eder and Peter Brössel

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4. Fragmentation and Higher-Order Evidence Daniel Greco

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5. Predictably Misleading Evidence Sophie Horowitz

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6. Escaping the Akratic Trilemma Klemens Kappel

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7. Higher-Order Defeat and Evincibility Maria Lasonen-Aarnio

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8. The Puzzles of Easy Knowledge and of Higher-Order Evidence: A Unified Solution Ram Neta

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9. Higher-Order Defeat and the Impossibility of Self-Misleading Evidence Mattias Skipper

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10. Higher-Order Defeat and Doxastic Resilience Asbjørn Steglich-Petersen

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11. Return to Reason Michael G. Titelbaum

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12. Whither Higher-Order Evidence? Daniel Whiting

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13. Evidence of Evidence in Epistemic Logic Timothy Williamson

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14. Can Your Total Evidence Mislead About Itself? Alex Worsnip

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Index of Names Index of Subjects

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Contributors Peter Brössel Junior-Professor of Philosophy Ruhr University Bochum

Ram Neta Professor of Philosophy University of North Carolina, Chapel Hill

David Christensen Professor of Philosophy Brown University

Mattias Skipper PhD Candidate in Philosophy Aarhus University

Kevin Dorst Fellow by Examination University of Oxford

Asbjørn Steglich-Petersen Professor of Philosophy Aarhus University

Anna-Maria A. Eder Assistant Professor of Philosophy University of Cologne

Michael G. Titelbaum Professor of Philosophy University of Wisconsin-Madison

Daniel Greco Assistant Professor of Philosophy Yale University

Daniel Whiting Professor of Philosophy University of Southampton

Sophie Horowitz Assistant Professor of Philosophy University of Massachusetts, Amherst

Timothy Williamson Wykeham Professor of Logic University of Oxford

Klemens Kappel Professor of Philosophy University of Copenhagen

Alex Worsnip Assistant Professor of Philosophy University of North Carolina, Chapel Hill

Maria Lasonen-Aarnio Associate Professor of Philosophy Helsinki University

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Introduction Mattias Skipper and Asbjørn Steglich-Petersen

Normally, when evidence speaks for or against believing some proposition, it does so by virtue of speaking for or against the truth of that proposition. If, for example, I look out the window and see that the sky is darkening, the evidence I have thereby acquired speaks in favor of believing that it will rain by virtue of indicating that it will, in fact, rain. Sometimes, however, evidence constrains our beliefs in a more indirect way than that. If I learn that my weather predictions have been systematically too optimistic in the past, I also seem to have gained some evidence that speaks in favor of believing that it will rain. But in this case, the evidence does not straightforwardly indicate whether it will rain or not. Rather, it favors the relevant belief by virtue of casting doubt on my ability to predict the weather in an accurate way. Over the past decade, an increasing number of epistemologists have relied on a distinction between “first-order evidence” and “higher-order evidence” to capture these two ways in which evidence can constrain our beliefs. The thought is that, in the example above, my observation that the sky is darkening is first-order evidence, because it is directly relevant to the question of whether it will rain. By contrast, the information about my overly optimistic weather predictions is higher-order evidence, because it does not concern the weather per se, but rather concerns my ability (or lack thereof) to accurately predict the weather. More generally, we might tentatively characterize higher-order evidence as evidence about what you should believe. Such evidence might manifest itself in a number of different ways: as evidence about what your evidence supports; as evidence about what evidence you possess; as evidence about which normative principles govern our beliefs; or, as in the example above, as evidence about your own ability to assess the available first-order evidence. This tentative characterization of higher-order evidence obviously calls out for further clarification. Indeed, one of the ambitions of the present volume is to get a clearer view of how, exactly, the intuitive distinction between first-order and higherorder evidence is best understood. If we can make progress on the question of what higher-order evidence is, we will likely be in a better position to assess its normative significance. Why is it important to understand the normative significance of higher-order evidence? Part of the motivation stems from the fact that higher-order evidence is a

Mattias Skipper and Asbjørn Steglich-Petersen, Introduction In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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pervasive feature of human life. As David Christensen () has pointed out, we all live our lives in states of epistemic imperfection, not only because we base our beliefs on limited evidence, but also because we do not always respond to the evidence we have in the most rational way. Accommodating one’s own rational imperfections is an important part of what it takes to live a responsible epistemic life. As such, higherorder evidence is a valuable source of epistemic self-improvement: it gives us an opportunity to rectify our own rational mistakes. Higher-order evidence also lies at the heart of a number of central epistemological debates, spanning from classical disputes between internalists and externalists about epistemic justification to more recent discussions about the normative significance of peer disagreement. It has become increasingly clear that many controversies within these debates stem, at least in part, from conflicting views about the normative significance of higher-order evidence. The hope is that the present volume will not only advance our understanding of higher-order evidence in general, but also help shed light on a wider array of interrelated topics in different areas of epistemology. The contributors to this volume address a diverse range of issues about higherorder evidence, and we cannot hope to do them full justice. In the rest of this introduction, we will try to highlight a few different points of entry into the debate about higher-order evidence, and situate the chapters along the way. A summary of each individual chapter can be found by the end of the introduction.

Main themes a) The normative significance of higher-order evidence How, if at all, does evidence about what you should believe influence what you should believe? In other words, what is the normative significance of higher-order evidence? This fundamental question runs through many of the debates discussed in this volume. The question is especially salient in cases of misleading higher-order evidence: that is, cases where an agent starts out with a fully rational belief, but then acquires some evidence, which indicates that the agent’s belief is not rational. Can such evidence defeat the agent’s belief? Or is rational belief not defeasible by misleading higher-order evidence? This is a question that any theory of higherorder evidence must answer. According to one prominent line of thought, due in large part to David Christensen, higher-order evidence must indeed place normative demands on our beliefs, since we would otherwise license a distinct kind of dogmatism or question-begging reasoning.¹ The thought is that, if I ignore or disregard a body of higher-order evidence, I must effectively take the higher-order evidence to be misleading. But the higher-order evidence is, ipso facto, only misleading if my initial belief was rational. Thus, in disregarding the higher-order evidence, I must assume that my initial belief was indeed rational. Yet, I thereby seem to beg the question in much the same way as someone who disregards a body of first-order evidence on the grounds ¹ See, e.g., Christensen (; ).

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that it opposes his or her prior opinion. To avoid falling prey to this sort of dogmatic or question-begging reasoning, I should give up (or at least lose confidence in) my initial belief. This is so even if my initial belief was in fact rational. While this line of reasoning may seem intuitively appealing, a number of authors have challenged the view that higher-order evidence has any influence over what it is rational to believe. The challenges have come from a number of different directions. For example, Titelbaum () has argued that higher-order defeat is incompatible with the view that epistemic akrasia is irrational (more on epistemic akrasia below); Lasonen-Aarnio () has argued that higher-order defeat is incompatible with the view that epistemic dilemmas are impossible; Whiting () has argued that there is no distinct normative role for higher-order evidence to play within a broadly reasons-based theory of epistemic rationality. In their contributions to this volume, all three authors develop these lines of argument in new and illuminating ways. In Chapter , Titelbaum defends his position against a number of challenges raised by Schoenfield (; ; ), Worsnip (), and Smithies (). In Chapter , Lasonen-Aarnio argues that proponents of higher-order defeat are committed to the implausible claim that one can acquire strong misleading evidence about the rational status of one’s beliefs, whereas one cannot have such evidence. In Chapter , Whiting argues that the only distinct role that higher-order evidence might play within a reasons-based framework is as a practical reason. He tentatively takes this to suggest that higher-order evidence does not, in fact, place any normative demands on belief. Furthermore, in Chapter , Sophie Horowitz raises doubts about the truth-conduciveness of higherorder evidence. As she points out, we usually assume that evidence, although occasionally misleading, is a reliable guide to the truth. But according to Horowitz, higher-order evidence is predictably misleading: it tends to point away from the truth, rather than towards it. Importantly, even if the challenges presented by these authors can ultimately be met, the question remains how, exactly, higher-order evidence acquires its normative significance. So far, the literature on higher-order evidence contains only some rather tentative answers to this question. One suggestion, due to Christensen (), is that higher-order evidence can require an agent to “put aside” or “bracket” the available first-order evidence, thereby preventing the agent from giving the first-order evidence its due. Another suggestion, due to Silva (), is that higher-order evidence can prevent an agent from basing his or her belief properly on the first-order evidence, thereby undermining the agent’s doxastic (rather than propositional) justification for that belief. This volume contains two novel suggestions for how we should understand the normative bearing of higher-order evidence. In Chapter , Mattias Skipper argues that higher-order evidence gets its normative significance by influencing which conditional beliefs it is rational for an agent to have. This proposal, he argues, has the additional benefit that it can help us understand a number of peculiar features of higher-order evidence that are otherwise hard to make sense of. In Chapter , Asbjørn Steglich-Petersen suggests that we understand higher-order evidence as something that primarily affects the resilience of a credence rather than its level. A desirable feature of this proposal, he argues, is that it can help solve two important

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puzzles about higher-order defeat: that it seems to defy being understood in terms of conditionalization, and that it sometimes seems to generate a dilemma between respecting all of one’s evidence and avoiding epistemic akrasia.

b) Epistemic akrasia Is there a connection between what you should believe and what you should believe about what you should believe? Or can any combination of first-order and higherorder beliefs be rationally held? Many philosophers have been attracted to the idea that akratic beliefs—beliefs of the form “it’s raining, but I shouldn’t believe that it’s raining”—can never be rational. Such beliefs display a kind of “mismatch” between one’s first-order and higher-order beliefs, which seem incoherent in much the same way as Moorean beliefs of the form “it’s raining, but I don’t believe that it’s raining” (Horowitz ; Greco ; Titelbaum ; Worsnip ). But despite the intuitive oddness of akratic beliefs, a number of authors have argued that such beliefs can sometimes be rational. For example, LasonenAarnio (forthcoming) has argued that anti-akratic principles do not sit well with an evidentialist norm of belief; Christensen () has offered accuracy-based reasons to think that agents who rationally believe themselves to be anti-reliable should sometimes be epistemically akratic; and Coates () and Worsnip (; this volume) have both argued that it is possible for one’s total evidence to support akratic beliefs, because such evidence can be self-misleading (more on self-misleading evidence below). Five chapters in this volume offer new contributions to the debate on epistemic akrasia. In Chapter , Kevin Dorst argues that epistemic akrasia is rational whenever it is rational to have higher-order uncertainty: uncertainty about what one should believe. And, he argues, rational higher-order uncertainty is pervasive: it lies at the foundations of the epistemology of disagreement. In Chapter , Klemens Kappel proposes a fine-grained view of epistemic rules, which he argues can accommodate higher-order defeat without generating cases of rational epistemic akrasia. In Chapter , Asbjørn Steglich-Petersen proposes a resilience-based view of higherorder evidence, which he argues can help explain why epistemic akrasia is irrational. In Chapter , Mattias Skipper proposes a view of higher-order evidence, which he argues can help to reconcile anti-akratic principles with an evidentialist norm of belief. And, in Chapter , Daniel Greco outlines a fragmentation-based view of evidence possession, which he uses to explain why epistemic akrasia is irrational as well as to escape several other puzzles about higher-order evidence.

c) Is evidence of evidence evidence? The slogan “evidence of evidence is evidence” has sparked a lively debate in the wake of Feldman’s () initial paper on the topic. While many authors have found the slogan intuitively plausible, it has been challenged on the grounds that evidential support is not in general transitive (Fitelson ; Tal & Comesaña ). What is clear is that the slogan itself is open to different interpretations. For example, some interpretations of slogan are interpersonal in nature, whereas others are

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intrapersonal. On an interpersonal interpretation, the question is (roughly) this: If I have evidence that you have evidence for p, do I thereby have evidence for p? By contrast, on an intrapersonal interpretation, the question is (roughly): if I have evidence that I have evidence for p, do I thereby have evidence for p? The present volume sheds new light on both interpersonal and intrapersonal interpretations of the “evidence of evidence is evidence” slogan. In Chapter , Peter Brössel and Anna-Maria A. Eder argue that interpersonal versions of the slogan are best formulated and assessed within a framework that treats “evidence of evidence” as a kind of higher-order evidence (rather than as a kind of first-order evidence). They then present a new Bayesian framework, Dyadic Bayesianism, which can treat “evidence of evidence” in this way. Finally, they use Dyadic Bayesianism to formulate and assess different interpersonal renderings of the “evidence of evidence is evidence” slogan. In Chapter , Timothy Williamson uses tools from probabilistic epistemic logic to formulate and assess several different intrapersonal renderings of the “evidence of evidence is evidence” slogan. He shows that these principles often imply versions of controversial iteration principles, such as positive and negative introspection (Williamson , pp. –; ). Williamson tentatively takes these results to count against the idea that evidence of evidence is necessarily evidence.

d) Self-misleading evidence Can your total evidence at once support believing p while supporting the belief that your total evidence does not support believing p? Or is such self-misleading evidence impossible? Different answers to this question each seem appealing at first sight. On the one hand, as Greco () points out, it is natural to think that one can be misled about virtually any subject matter Why should propositions about what one’s total evidence supports be an exception? On the other hand, self-misleading evidence is peculiar in that it supports akratic beliefs of the form “it’s raining, but my total evidence doesn’t support that it’s raining.” Thus, if self-misleading evidence is possible, we seem forced to choose between two otherwise attractive ideas: the idea that it is rational to believe what one’s evidence supports, and the idea that epistemic akrasia is irrational. Four chapters in the present volume offer new insights into the question of whether self-misleading evidence is possible. In Chapter , Alex Worsnip argues that such evidence is, indeed, possible. His contribution extends and refines the central argument developed in (Worsnip ), and provides illuminating probabilistic renderings of the key steps in that argument. In Chapter , Ram Neta offers a unified solution to what he calls the puzzles of “easy knowledge” and of “higher-order evidence.” His solution centers around on certain structural constraints on an agent’s evidence that can be understood as ruling out the possibility of self-misleading evidence, although Neta himself does not put matters in those very terms. In Chapter , Mattias Skipper defends a view of higher-order evidence, which, he argues, renders self-misleading impossible. Finally, in Chapter , Maria LasonenAarnio points out a tension between the claim that self-misleading evidence is impossible and the widely accepted view that one can acquire misleading higherorder evidence about the rational status of one’s beliefs.

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e) Peer disagreement Many epistemologists have agreed that peer disagreement can sometimes provide the disagreeing parties with higher-order evidence by giving them reason to doubt the rationality of their initial opinions about the disputed matter. But how, if at all, should such the parties to such disagreements revise their opinions? This question has generated an extensive literature over the past decade. It has become customary to divide the main views about the epistemic significance of peer disagreement into two broad camps. On the one hand, there are “conciliationist” views, according to which peer disagreement should typically lead you to substantially revise your opinion in the direction of your peer. Versions of this view have been prominently defended by, e.g., Christensen (; ) and Elga (). Central to Christensen’s view, in particular, is the idea that one’s assessment of one’s own reliability on the disputed matter should be suitably independent of the reasoning that one used to form one’s initial opinion. While many authors have found this sort of independence principle intuitively appealing, a suitably clear and precise formulation of the principle is still lacking. In his contribution to this volume, Christensen seeks to remedy this situation by formulating and assessing different candidate precisifications of the independence principle. Along the way he highlights a number of difficulties associated with the attempt to capture the idea that our beliefs should be sensitive to higher-order evidence about our own cognitive reliability. On the other hand, there are “steadfast” views, according to which you are typically permitted to maintain your initial opinion in the face of peer disagreement, at least if that belief was rational to begin with. A moderate version of this view has been defended by Kelly (), who accepts that peer disagreement can have some defeating force, but denies that the defeating force is as strong and pervasive as Christensen and Elga would have us think. A more radically steadfast view has been defended by Titelbaum (), who argues that peer disagreement (in fact, higher-order evidence in general) can never have any defeating force. In his contribution to this volume, Titelbaum defends his stance against various objections, but also exempts from his view certain cases of peer disagreement that do not provide the parties with any higher-order evidence in the first place.

Summaries of chapters Chapter . David Christensen: “Formulating Independence” We often get evidence that bears on the reliability of some of our own first-order reasoning. The rational response to such “higher-order” evidence would seem to depend on a rational assessment of how reliable we can expect that reasoning to be, in light of the higher-order evidence. “Independence” principles are intended to constrain this reliability-assessment, so as to prevent question-begging reliance on the very reasoning being assessed. However, extant formulations of Independence principles tend to be vague or ambiguous, and coming up with a tolerably precise

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formulation turns out to be tricky. This chapter aims to make some progress toward developing a more precise Independence principle, and also to highlight some of the difficulties the project entails. The hope is to take a step toward understanding how rational belief is informed by evidence bearing on agents’ cognitive reliability.

Chapter . Kevin Dorst: “Higher-Order Uncertainty” You have higher-order uncertainty if and only if you are uncertain of what opinions you should have. Dorst defends three claims about it. First, the higher-order evidence debate can be helpfully reframed in terms of higher-order uncertainty. The central question becomes how your first- and higher-order opinions should relate—a precise question that can be embedded within a general, tractable framework. Second, this question is nontrivial. Rational higher-order uncertainty is pervasive, and lies at the foundations of the epistemology of disagreement. Third, the answer is not obvious. The Enkratic Intuition—that your first-order opinions must “line up” with your higher-order opinions—is incorrect; epistemic akrasia can be rational. If all this is right, then it leaves us without answers—but with a clear picture of the question, and a fruitful strategy for pursuing it.

Chapter . Anna-Maria A. Eder and Peter Brössel: “Evidence of Evidence as Higher-Order Evidence” In everyday life and in science, we often change our epistemic states in response to “evidence of evidence.” An assumption underlying this practice is that the following “EEE Slogan” is correct: “evidence of evidence is evidence” (Feldman , p. ). In this chapter, Eder and Brössel suggest that evidence of evidence is best understood as higher-order evidence about the epistemic states of agents. In order to model evidence of evidence in this way, they introduce a new powerful framework for modelling epistemic states, Dyadic Bayesianism. Based on this framework, they discuss different characterizations of evidence of evidence and argue for one of them. Finally, they show that the tenability of the EEE Slogan depends on the specific kind of evidence of evidence.

Chapter . Daniel Greco: “Fragmentation and Higher-Order Evidence” The concept of higher-order evidence—roughly, evidence about what our evidence supports—promises epistemological riches; it has struck many philosophers as necessary for explaining how to rationally respond to disagreement in particular, and to evidence of our own fallibility more generally. But it also threatens paradox. Once we allow higher-order evidence to do non-trivial work—in particular, once we allow that people can be rationally ignorant of what their evidence supports—we seem to be committed to a host of puzzling or even absurd consequences. Greco’s aim in this chapter is to develop a way of reaping the riches without incurring the costs; he presents an independently motivated framework that, he argues, lets us mimic the particular case judgments of those who explain how to accommodate evidence of our fallibility by appeal to higher-order evidence, but without commitment to the absurd consequences.

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Chapter . Sophie Horowitz: “Predictably Misleading Evidence” Evidence can be misleading: it can rationalize raising one’s confidence in false propositions, and lowering one’s confidence in the truth. But can a rational agent know that her total evidence supports a (particular) falsehood? It seems not: if we could see ahead of time that our evidence supported a false belief, then we could avoid believing what our evidence supported, and hence avoid being misled. So, it seems, evidence cannot be predictably misleading. In this chapter Horowitz develops a new problem for higher-order evidence: it is predictably misleading. She begins by introducing and motivating the challenge. She then examines a radical strategy for explaining higher-order evidence, according to which is there are two distinct epistemic norms at work; she argues that this type of view is unsatisfactory because it fails to explain why both norms are epistemically significant. Finally, she suggests that mainstream accounts of higher-order evidence may be able to answer the challenge after all. To do so, they must distinguish between a rational agent’s total body of evidence, on the one hand, and the perspective from which she forms beliefs, on the other. They also must deny that epistemic rationality is a matter of believing what is likely given one’s evidence.

Chapter . Klemens Kappel: “Escaping the Akratic Trilemma” Much of the recent literature on higher-order evidence has revolved around the following three theses. First, one’s credence in any given proposition p should rationally reflect one’s evidence e bearing on the truth of p. Second, one’s credence in any given higher-order proposition p0 (concerning the evidential relation between e and p) should rationally reflect one’s evidence e0 bearing on the truth of p0 . Third, it is epistemically irrational to have a high credence in p based on e, while having a high credence that e does not support p, or that one’s processing of e is somehow faulty (The Non-Akrasia Requirement). All three theses are prima facie plausible, yet they jointly lead to inconsistencies. This is what might be called The Akratic Trilemma. This chapter assesses two recent responses to The Akratic Trilemma (Titelbaum ; Lasonen-Aarnio ), argues that both responses fail, and offers a novel way out of the Trilemma.

Chapter . Maria Lasonen-Aarnio: “Higher-Order Defeat and Evincibility” One of the ambitions of the past decades of epistemology has been to accommodate the view that higher-order evidence that a belief one holds is rationally flawed has a systematic kind of defeating force with respect to that belief. Such a view is committed to two claims. First, it is possible to acquire misleading evidence about the normative status of one’s doxastic states: even if one’s belief is perfectly rational, one might acquire deeply misleading evidence that it is irrational. Second, such evidence has defeating force with respect to the belief, zapping its rational status. In this chapter, Lasonen-Aarnio aims to do two things. First, to outline a view she calls normative evincibility, according to which one always has a kind of epistemic access—access that can come in different strengths—to the normative status of one’s doxastic states (intentions, actions, etc.). Lasonen-Aarnio shows how commitment to higher-order defeat in effect incurs a commitment to a form of normative

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evincibility. Second, she argues that the idea that it is possible to acquire misleading evidence about the normative status of one’s doxastic states is in tension with evincibility. In particular, those who like higher-order defeat are committed to the claim that though one can acquire even deeply misleading evidence about the rational status of one’s doxastic states, one cannot have evidence that is (too) misleading about the rational status of one’s doxastic states. Lasonen-Aarnio argues that these claims cannot be jointly accommodated.

Chapter . Ram Neta: “The Puzzles of Easy Knowledge and Higher-Order Evidence: A Unified Solution” In this chapter, Neta aims to provide a unified solution to two widely discussed epistemological puzzles. He begins by setting out each of these two puzzles. He then surveys some of the proposed solutions to each puzzle, none of which generalize. Finally, he argues that the two puzzles arise because of a widespread confusion concerning the relation of substantive and structural constraints of rationality. Neta argues that clearing up this confusion allows us to clear up both puzzles at once.

Chapter . Mattias Skipper: “Higher-Order Defeat and the Impossibility of Self-Misleading Evidence” Evidentialism is the thesis, roughly, that one’s beliefs should fit one’s evidence. The enkratic principle is the thesis, roughly, that one’s beliefs should “line up” with one’s beliefs about which beliefs one ought to have. While both theses have seemed attractive to many philosophers, they jointly entail the controversial thesis that self-misleading evidence is impossible. That is to say, if evidentialism and the enkratic principle are both true, one’s evidence cannot support certain false beliefs about which beliefs one’s evidence supports. Recently, a number of epistemologists have challenged the thesis that self-misleading evidence is impossible on the grounds that misleading higherorder evidence does not have the kind of strong and systematic defeating force that would be needed to rule out the possibility of self-misleading evidence. In this chapter, Skipper responds to this challenge by proposing an account of higher-order defeat that does, indeed, render self-misleading evidence impossible. Central to the proposal is the idea that higher-order evidence acquires its normative force by influencing which conditional beliefs it is rational to have. What emerges, Skipper argues, is an independently plausible view of higher-order evidence, which has the additional benefit that it allows us to reconcile evidentialism with the enkratic principle.

Chapter . Asbjørn Steglich-Petersen: “Higher-Order Defeat and Doxastic Resilience” It seems obvious that when higher-order evidence makes it rational for one to doubt that one’s own belief on some matter is rational, this can undermine the rationality of that belief. This is known as higher-order defeat. However, despite its intuitive plausibility, it has proved puzzling how higher-order defeat works, exactly. To highlight two prominent sources of puzzlement, higher-order defeat seems to defy being understood in terms of conditionalization; and higher-order defeat can sometimes place agents in

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    ø - what seem like epistemic dilemmas. In this chapter, Steglich-Petersen draws attention to an overlooked aspect of higher-order defeat, namely that it can undermine the resilience of one’s beliefs. The notion of resilience was originally devised to understand how one should reflect the “weight” of one’s evidence. But it can also be applied to understand how one should reflect one’s higher-order evidence. The idea is particularly useful for understanding cases where one’s higher-order evidence indicates that one has failed in correctly assessing the evidence, without indicating whether one has over- or underestimated the degree of evidential support for a proposition. But it is exactly in such cases that the puzzles of higher-order defeat seem most compelling.

Chapter . Michael G. Titelbaum: “Return to Reasons” In this chapter, Titelbaum discusses a number of responses to his “Rationality’s Fixed Point (or: In Defense of Right Reason)” (). First, he explains how he understands rationality, and why he takes akratic states to be rationally forbidden. Second, he reconstructs his argument from the irrationality of akrasia to the Fixed Point Thesis (that mistakes about rationality are mistakes of rationality). Third, he notes that that argument can’t be avoided by distinguishing ideal rationality from everyday rationality, rationality from reasonableness, or structural norms from substantive. He also considers rational dilemmas. Fourth, he shows that the Fixed Point Thesis and the Right Reasons position on peer disagreement that follow from it are compatible with both externalist and internalist conceptions of rationality, and survive concerns about what an agent is able to figure out. Fifth, he assesses Declan Smithies’s () view that disagreement defeats doxastic justification. Finally, he revises his Right Reasons position to address cases in which peer disagreement rationally affects an agent’s opinions without providing higher-order evidence.

Chapter . Daniel Whiting: “Whither Higher-Order Evidence?” First-order evidence is evidence which bears on whether a proposition is true. Higherorder evidence is evidence which bears on whether a person is able to assess her evidence for or against a proposition. A widespread view is that higher-order evidence makes a difference to whether it is rational for a person to believe a proposition. This chapter considers in what way higher-order evidence might do this. More specifically, it considers whether and how higher-order evidence plays a role in determining what it is rational to believe distinct from that which first-order evidence plays. To do this, Whiting turns to the theory of reasons and tries to situate higher-order evidence within it. The only place Whiting finds for it there, distinct from that which first-order evidence already occupies, is as a practical reason, that is, as a reason for desire or action. One might take this to show either that the theory of reasons is inadequate as it stands or that higher-order evidence makes no distinctive difference to what it is rational to believe. Whiting tentatively endorses the second option.

Chapter . Timothy Williamson: “Evidence of Evidence in Epistemic Logic” The slogan “Evidence of evidence is evidence” is obscure. It has been applied to connect evidence in the current situation to evidence in another situation. The link

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may be diachronic or interpersonal. Is present evidence of past or future evidence for p present evidence for p? Is evidence for me of evidence for you for p evidence for me for p? The chapter concerns putative intra-perspectival evidential links. Is present evidence for me of present evidence for me for p present evidence for me for p? Unless the connection holds between a perspective and itself, it is unlikely to hold between distinct perspectives. Evidence will be understood probabilistically. Formal models will be used from epistemic logic, which provides a natural integration of first- and second-level epistemic conditions. An integrated framework is needed to give a fair chance to the idea that evidence of evidence is evidence. We ask questions like this: if the probability on the evidence that the probability on the evidence of a hypothesis H is at least % is itself at least %, when does it follow that the probability on the evidence of H is indeed at least %, or at least more than %? These resemble synchronic analogues of probabilistic reflection principles. Bridge principles between first-level and higher-level epistemic conditions often imply versions of controversial principles, such as positive and negative introspection. Formalizations of intra-perspectival principles that evidence of evidence is evidence have similar connections.

Chapter . Alex Worsnip: “Can Your Total Evidence Mislead About Itself?” It’s fairly uncontroversial that you can sometimes get misleading higher-order evidence about what your first-order evidence supports. What is more controversial is whether this can ever result in a situation where your total evidence is misleading about what your total evidence supports: that is, where your total evidence is misleading about itself. It is hard to arbitrate on purely intuitive grounds whether any particular example of misleading higher-order evidence is, more than that, an example of misleading total evidence (about total evidence). This chapter tries to make some progress by, first, offering a simple mathematical model that suggests that higher-order evidence will tend to bear more strongly on higher-order propositions about what one’s evidence supports than it does on the corresponding first-order propositions; and then by arguing that given this, it is plausible that there will be some cases of misleading total evidence (about total evidence). In doing so, Worsnip follows a broadly similar strategy to one he pursued in a previous paper, but in what he hopes is a much more precise, detailed, and epistemologically sophisticated form.

References Christensen, D. (). “Epistemology of Disagreement: The Good News.” In: The Philosophical Review , pp. –. Christensen, D. (). “Higher-Order Evidence.” In: Philosophy and Phenomenological Research , pp. –. Christensen, D. (). “Disagreement, Question-Begging, and Epistemic Self-Criticism.” In: Philosophers’ Imprint . Christensen, D. (). “Disagreement, Drugs, Etc.: From Accuracy to Akrasia.” In: Episteme , pp. –.

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    ø - Coates, A. (). “Rational Epistemic Akrasia.” American Philosophical Quarterly (), pp. –. Elga, A. (). “Reflection and Disagreement.” In: Noûs (), pp. –. Feldman, R. (). “Reasonable Religious Disagreements.” In L. Antony (ed.), Philosophers Without God: Meditations on Atheism and the Secular Life, Oxford University Press, pp. –. Feldman, R. (). “Evidence of Evidence is Evidence.” In Matheson and Vitz (eds), The Ethics of Belief, Oxford University Press, pp. –. Fitelson, B. (). “Evidence of Evidence is not (Necessarily) Evidence.” In: Analysis , pp. –. Greco, D. (). “A Puzzle about Epistemic Akrasia.” In: Philosophical Studies , pp. –. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs , pp. –. Kelly, T. (). “Peer Disagreement and Higher Order Evidence”. In A. Goldman and D. Whitcomb (eds), Social Epistemology: Essential Readings, Oxford University Press, pp. –. Lasonen-Aarnio, M. (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research , pp. –. Lasonen-Aarnio, M. (forthcoming). “Enkrasia or Evidentialism? Learning to Love Mismatch.” Forthcoming in Philosophical Studies. Schoenfield, M. (). “Chilling out on Epistemic Rationality: A Defense of Imprecise Credences (and Other Imprecise Doxastic Attitudes).” In: Philosophical Studies , pp. –. Schoenfield, M. (). “Bridging Rationality and Accuracy.” In: The Journal of Philosophy , pp. –. Schoenfield, M. (). “An Accuracy Based Approach to Higher Order Evidence.” In: Philosophy and Phenomenological Research , pp. –. Silva, P. (). “How Doxastic Justification Helps Us Solve the Problem of Misleading Higher-Order Evidence.” In: Pacific Philosophical Quarterly (), pp. –. Smithies, D. (). “Ideal Rationality and Logical Omniscience.” In: Synthese , pp. –. Tal, E. and J. Comesaña (). “Is Evidence of Evidence Evidence?” In: Noûs , pp. –. Titelbaum, M. (). “Rationality’s Fixed Point (Or: In Defense of Right Reason).” In: T. Gendler and J. Hawthorne (eds), Oxford Studies in Epistemology , Oxford University Press, pp. –. Whiting, D. (). “Against Second-Order Reasons.” In: Noûs (), pp. –. Williamson, T. (). Knowledge and its Limits. Oxford: Oxford University Press. Williamson, T. (). “Very Improbable Knowing.” In: Erkenntnis , pp. –. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research (), pp. –.

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1 Formulating Independence David Christensen

. Introduction People often encounter evidence which bears directly on the reliability or expected accuracy of their thinking about some topic. This has come to be called “higher-order evidence.”¹ For example, suppose I have some evidence E, and come to have high confidence in hypothesis H on its basis. But then I get some evidence to the effect that I’m likely to do badly at assessing the way E bears on H. Perhaps E bears on H statistically, and I’m given evidence that I’m bad at statistical thinking. Or perhaps E is a set of CVs of male and female candidates, H is the hypothesis that a certain male candidate is a bit better than a certain female candidate, and I get evidence that I’m likely to overrate the CVs of males relative to those of females. Or perhaps E consists of gauge and dial readings in the small plane I’m flying over Alaska, H is the hypothesis that I have enough fuel to reach Sitka, and I realize that my altitude is over , feet, which I know means that my reasoning from E to H is likely affected by hypoxia. Or finally, perhaps E is a body of meteorological data that seems to me to support rain tomorrow, H is the hypothesis that it’ll rain tomorrow, and I learn that my friend, another reliable meteorologist with the same data E, has predicted that it won’t rain tomorrow. In cases like these, it seems that part of what determines how confident I should end up being in H is some kind of assessment of my own reliability, or expected accuracy, in forming my initial credence in H. So, in general, the credence I end up with should be informed by what I’ll call a “reliability-assessment.” The cases also illustrate the range of considerations that should inform this reliability assessment. They include factors that bear on my general competence, my psychological quirks, my current circumstances, and even, in the case of

¹ The term “higher-order evidence” has actually been given several non-equivalent understandings in the literature. Some take it as evidence bearing on evidential relations (which makes the term particularly appropriate). Others focus on evidence about the rationality of the person’s thinking, or (as I will here) on evidence about the reliability or accuracy of the person’s thinking. As Jonathan Vogel pointed out to me, it’s not clear that the latter understandings are naturally captured by “higher-order.” But the literature has focused on a common set of examples—examples of the sort I’ll concentrate on here. So I’ll go with the “higher-order” terminology. David Christensen, Formulating Independence In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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   disagreement, generic evidence that I’ve screwed up somehow or another today. But there’s one way of assessing my reliability that seems clearly irrational: I should not reason that since E strongly supports H, and I’ve come to believe H on the basis of E, that my thinking in the present instance is fine, despite my poor statistical skills/my implicit bias/my likely hypoxia/my friend’s disagreement. I should not assess my reliability in this way, even if my reasoning from E to H happens, in this case, to be perfectly correct. The relevant reliability assessment needs to be independent of this sort of reasoning. Nevertheless, it turns out to be difficult to formulate a plausible Independence principle. This chapter aims to make some progress on this project, and also to illustrate some of the difficulties that it entails. We might start with a highly contrived example, devised to make it maximally easy to see what should, and what should not, be allowed to inform the relevant independent reliability assessments. This will allow us to sketch an Independence principle, which can then be refined and tested by considering less straightforward cases. Logic on Drugs: highly reliable: (A)

Alicia is told two things by a source she rationally believes to be

Karla was born in May if and only if Kayla wasn’t; and

(B) Either Kayla, or Layla and Lola, the Lumpkin twins, were born in May. She rationally becomes extremely confident in A and B, thinks about their implications a bit, and rationally becomes highly confident—say, . confident—that: (P) Karla wasn’t born in May unless Layla Lumpkin was. Then Alicia learns that before she started to think about all these birthdays, someone slipped her a powerful drug. The drug distorts people’s complex truthfunctional reasoning about birthdays. It causes people who reach high credences from doing this sort of reasoning to favor incorrect conclusions in % of the time. Alicia has played with this drug before, at parties. When she forms high credences, she has a long history of forming them in wrong conclusions % of the time, in problems just like this one, even while feeling perfectly clear-headed. She reflects on all this, and becomes significantly less confident in P—say, she reduces her confidence to around .. I will assume that the credence she adopts is the one that’s most rational for her, and that the credence she’s most rational to adopt depends on an assessment of her likely reliability on this occasion. It’s clear that this reliability-assessment should take into account some facts about her present situation: in particular, that she has ingested the logic-disrupting drug. But just as clearly, the reliability-assessment relevant to Alicia’s final credence in P should not be informed by the following train of reasoning: . A . B So, . P . I came to believe P So, . The drug did not interfere with my reliability today!

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This train of reasoning would seem to beg the question in some sense: it would dismiss the worry about a bit of Alicia’s reasoning by relying on the very reasoning in question. Putting the point informally, it seems that the relevant reliability assessment must be independent of Alicia’s reasoning from A and B to P. But how might we make this informal statement more precise? In trying to make progress on this question, it will be useful to begin by restricting attention to a certain class of cases: ones where the agent begins by forming a rational initial credence in some proposition P based on a well-defined bit of first-order evidence, and then learns some higher-order evidence that bears on the reliability of the reasoning by which that first-order evidence led to her credence in P. I will also assume that the agents form rational independent reliability-assessments. It will also be useful to work with a particular model of what form the reliabilityassessments might take. I’ll work here with a model (defended elsewhere²) on which they take the form of hypothetical conditional credences. In Alicia’s case, we may think of her as focusing on the reliability of her own reasoning from her very high credence in A and B to her initial very high credence P. We can ask how likely Alicia should think it is that P is really true, conditional on the fact that that she, having ingested the drug, reached that very high credence in P on the basis of A and B. If this conditional credence is not to be affected by the sort of question-begging reasoning described above, it will have to be independent from Alicia’s reasoning from A and B to P. Intuitively, we can see Alicia as stepping back from her confidence in P, and considering herself as a kind of measuring device, whose high credence in P serves as an indicator of how likely P is to be true. I take it that Alicia’s independent hypothetical credence in P, given the information about her being drugged and about the results of her initial reasoning, should be about .. Here, then, is a very rough first pass at characterizing reliability estimates that are independent in the requisite way: Independence, preliminary sketch: When an agent has formed an initial credence c in P on the basis of first-order evidence E, and then gets some evidence that bears on the reliability of her reasoning from E to her credence in P, her final credence in P should reflect the Independent Hypothetical Credence (IHC) it would be rational for her to have in P: that is, the rational credence in P independent of her reasoning from E to her initial credence in P, but conditional on her having formed credence c in P on the basis of E, and on the reliability evidence the agent has about herself.³ In the present example, the agent reached very high initial credence, and the rational IRC for her was considerably lower. But it is important that this need not always be

² See Christensen (). ³ In this and subsequent formulations, “reflect” is intended to indicate that the agent’s final credence takes into account her IHC. On my favored way of understanding higher-order evidence, the agent’s final credence would simply match her IHC. But I think that even plausible views on which higher-order evidence is taken as less powerful will depend on an independent reliability-assessment (see Christensen , ). So I’ll use “reflect” to allow us to consider what that assessment should look like, without commitment to the exact way it would help determine the agent’s final credence.

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   the case. Suppose that an agent is hoping for a nice hike tomorrow, and her direct assessment of the meteorological evidence resulted in credence . in rain. Then she gets good evidence that she tends to reach overly-optimistic opinions about the weather. In such a case, the agent’s rational IHC in rain, given that she reached . initial credence in rain, and given the reliability evidence she now has about herself, would be higher than . (the amount by which it would be higher would depend on the specifics of the evidence about her optimism). So IHCs can be lower, or higher, or even equal to agents’ initial credences.⁴ With this framework in hand, let us turn to making the intuitive idea more precise, and then to consider some cases where applying Independence will be more difficult.

. Independent from “the agent’s reasoning”? The above description of Alicia’s reliability-assessment requires that it be “independent from Alicia’s reasoning from A and B to P.” But what this means is hardly clear. Similar formulations in the disagreement literature include “independent of the dispute” and “independent of the disagreement.” Less common formulations talk about independence from the initial belief, and independence from the first-order “evidence” on which the initial belief was based, or independence from “the reasons” for the initial belief (which seems ambiguous between independence from the evidence on which the belief was based, and independence from the reasoning from that evidence to the initial belief ).⁵ I think that the most promising approach will not exactly require independence from the relevant first-order evidence. For one thing, it is probably a mistake to divide items of evidence between the “first-order” and “higher-order” categories. Some items of evidence have both sorts of import, even with respect to the same belief. Consider a standard sort of example used to motivate conciliatory accounts of disagreement: I go to dinner with a friend and, after doing mental calculations based on the amount on the check, come to a confident conclusion about what our shares are. Then I learn that my friend disagrees. If we have long and equally good track records in share-calculation, this seems to call for significant loss of confidence on my part. And it would intuitively beg the question for me to use my initial calculation as the basis for dismissing my friend’s opinion as incorrect. But suppose we sought to prevent this question-begging by requiring my reliability-assessment to be independent of my “first-order evidence” (in this case, the figures on our check). And suppose further that I happen to know that my friend is particularly bad at calculations when the figure on the check has s in it, and that today’s check contains three s. Here, it seems that I should take this information into account in my reliability assessment, and not lose so much confidence in my original answer. But this would be prevented if my reliability assessment had to be independent of the evidence of the figures on the check.⁶ ⁴ Thanks to an editor of this volume for prompting me to clarify this point. ⁵ For a sampling of various formulations by defenders of such principles, see Elga (), Kornblith (), Christensen (), Vavova (), and Matheson (). ⁶ Thanks to Adam Elga for helpful discussion of this sort of case.

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This suggests that what needs to be “bracketed” by my reliability-assessment is not evidence E as such, but certain routes by which E might support P, and thereby support the claim that I was reliable on this occasion. Although it is often convenient to talk about “first-order evidence” vs “higher-order evidence,” we should—at least when we are being careful—distinguish instead between different ways that a piece of evidence can bear on the doxastic attitude it’s rational for an agent to take toward a given proposition. Roughly, evidence bears on the rationality of an agent’s attitude in a higher-order way insofar as it bears on that attitude indirectly, via bearing on how the agent should assess her reliability in assessing the direct bearing of her evidence on the relevant proposition.⁷ So Alicia’s information about being drugged bears on her rational credence in P in a higher-order way. In the restaurant-check example we just considered, the number on the check bears on my credence in my answer in both higher-order and first-order ways. I think that disagreement cases typically illustrate these dual roles that a single item of evidence can play: when I believe P on the basis of E, the disagreement of a friend who shares E serves both as first-order evidence for not-P (in the usual testimonial way), and as higher-order evidence that I’ve reasoned from E to P unreliably. This rough characterization is, no doubt, inadequate. But I think that the examples can at least serve to flesh out the intended idea, and that speaking in terms of different ways evidence can bear on an attitude, rather than speaking of different kinds of evidence, is a step in the right direction.⁸ Suppose, then, that our Independence principle should focus on the train of reasoning from E to P, not on E itself. This fits with informal formulations which require the assessment to be “independent of the reasoning behind the initial belief.” But this raises another question: what does it mean for, say, Alicia’s reliability assessment to be independent of “the reasoning behind” her initial belief? Does it just mean that the assessment must not depend on the particular episode of reasoning in which Alicia engaged at the beginning of the story? Or does it mean that the reliability assessment that would be rational for Alicia should be independent of the general fact that A and B support P—that is, independent of the first-order bearing of A and B on P? The former interpretation would see the relevant reliability assessment in terms of the conditional credence that would be rational to have in P, given that Alicia was drugged, and that she’d reached a high initial credence in P, and that A and B—which entail P—are very likely true. That is, we would be allowing the reliability assessment that was rational for Alicia to be informed by A and B’s actual first-order support for P. Since A and B entail P, that rational conditional credence would presumably be very high. But that’s not the result the account is after; it really defeats the purpose behind requiring Independence. So I think that the latter interpretation is closer to what we need.⁹ The reliability-assessment that would be ⁷ This differs from some common ways of characterizing higher-order evidence, which see it as directly bearing on the rationality, rather than the reliability, of the agent’s assessment of the evidence’s direct bearing on the relevant proposition. I suspect that our concern with rationality in these contexts derives from a more fundamental concern for accuracy; for more on this distinction, see Christensen (). ⁸ Having made this distinction, I will often revert to informal talk of first-order and higher-order evidence, to avoid cumbersome formulations. ⁹ This distinction is developed in van Wietmarschen (), which argues that only the former interpretation can be motivated, and that seeing this shows that conciliatory accounts of disagreement

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   rational for Alicia should be fully independent of A and B’s first-order bearing on P. In general terms, then, we might amend our rough characterization as follows: Independence, second sketch: When an agent has formed an initial credence c in P on the basis of the first-order bearing of evidence E, and then gets some evidence that bears on the reliability of her reasoning from E to her credence c in P, her final credence in P should reflect the Independent Hypothetical Credence (IHC) it would be rational for her to have in P: that is, the rational credence in P independent of E’s first-order bearing on P, but conditional on her having formed credence c in P on the basis of E, and on the reliability evidence the agent has about herself.¹⁰ This formulation avoids having to divide items of evidence into first-order and higherorder. It also allows for the possibility of agents using facts about their evidence or reasoning to inform their reliability-assessments. So, for example, in Alicia’s case, if the drug did not affect cognition about people whose names began with K or L, our formulation would allow Alicia to take that into account and raise her assessment of her reliability. This fits well with a more general point that has been made in the literature: “independent of the reasoning behind the agent’s initial belief” cannot mean that facts about this reasoning are precluded from informing the reliability-assessment. Agents should be allowed to take into account whether, for example, they felt clearheaded while doing the reasoning, whether the reasoning began with a number involving s, whether it involved statistical analysis, etc. Independence principles require a distinction between depending on facts about my reasoning, and depending on that reasoning itself in a way that relies on the reasoning’s cogency.¹¹ This feature of our formulation will help handle a type of example due to Andrew Moon (). Moon develops a series of counterexamples to the sort of Independence principles that have been invoked in the disagreement literature—ones which require that one’s assessment of the disagreer’s reliability be “independent of the disputed belief” or “independent of the reasoning behind the disputed belief.” Here is a representative example, adapted slightly: Reliable Source: Boris knows Cho to be a reliable source. Cho tells him two things: (P) Peggy is at the party. (Q) If Peggy is at the party, then Quinn is unreliable about Peggy’s whereabouts. Boris rationally becomes confident in both P and Q. But then, Boris meets Quinn, about whom he has no other special information. Quinn says that Peggy was not at the party. Boris puts P and Q together, dismisses Quinn’s testimony, and retains his belief that P.

apply only to the well-foundedness (doxastic justification) of beliefs, not to their evidential support (propositional justification). I resist these conclusions, and discuss the issue in greater depth, in Christensen (). See also Smithies () for a view on which higher-order evidence is relevant to doxastic, but not propositional, justification. ¹⁰ I say “bearing on” rather than “support for” to cover cases where first-order evidence tells against the relevant proposition, or where it would make rational a mid-range credence. ¹¹ For more on this issue, see Christensen (), Arsenault & Irving (), Kelly (), Matheson (), Christensen ().

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

Moon argues that Boris’s reaction of dismissing Quinn’s disagreement here is legitimate. But he argues that it relies crucially on Boris’s using his beliefs that P and Q to support Quinn’s unreliability, and thus dismiss Quinn’s disagreement. And P is the very belief disputed by Quinn. So formulations of Independence which forbid this sort of dependence must be wrong. He proposes building in exception clauses to Independence principles which would permit reliability-assessments to be based on the disputed belief (or the reasoning behind it), as long as certain conditions obtain. I think that there are reasons to worry about building in these exception clauses to our principle.¹² But it is worth noticing that if we understand Independence in the way we are examining, Reliable Source seems to be handled correctly. Boris’s IHC should be independent of his evidence’s first-order support for P, but will be based on the fact that he formed a high initial credence in P, and on the evidence he has that’s relevant to his reliability. And it seems to me that the IHC will indeed be pretty high here, because of considerations involving facts about Boris’s original reasoning: Boris formed a high credence in a proposition that was asserted by a source he knew to be reliable. The proposition was then denied by a source he knew nothing special about, and, in addition, the known-reliable source provided evidence that the other source was unreliable in cases where the proposition was true. It would seem that, in cases fitting this general description, the initial high credence is very likely to be accurate. But this reasoning does not depend on the claim that Peggy is at the party—it just relies on facts about the types of evidence Boris based his credence on, and on how these types of evidence interact.¹³ If our formulation handles this sort of example correctly, perhaps we need not add potentially troublesome exception clauses to our Independence principle.

. Independent of the first-order bearing of what evidence on P? One of the ways which Alicia’s case is particularly simple is that the first-order evidence bearing on P is cleanly isolated: as a first approximation, A and B bear on P, ¹² Here is an example of why I think we should worry. One of the exception clauses permits the agent to rely on premises behind her initial belief to reach conclusions about the epistemic credentials of her friend’s belief, as long as they concern the non-truth-entailing credentials of that belief. This is needed because Moon is sympathetic to the basic motivation behind the Independence principles in the literature. So he wants to disallow the sort of question-begging dismissal of disagreement that occurs in standard restaurant-checkstyle cases when I start out believing P, find out that my peer believes not-P, and reason that since P, her belief is false this time, and hence not warranted/known—so I needn’t lose confidence in mine. But consider a weather-forecasting peer disagreement case, where we build in that my peer and I each very occasionally become over- or under-confident in rain the next day. If I believe it’ll rain tomorrow, and she, on the basis of the same meteorological evidence, thinks it won’t, I should not be able to reason as follows: “Well, our data make rain highly probable, so she must have misinterpreted the data today, so I needn’t worry about her disagreement.” But the downgrading of my friend’s belief ’s credentials in this case is not based on any claim entailing the truth or falsity of the claim that it will rain. ¹³ I should note that Moon’s description of Boris’s reasoning to his conclusion about Quinn’s reliability— which explicitly moves from “P” and “If P, then Quinn is unreliable about Peggy’s whereabouts” to “Quinn is unreliable about Peggy’s whereabouts”—does seem quite natural, intuitively. But I think that this should not persuade us that Boris may really rationally rely on his belief that P to dismiss worries about his reliability in reasoning about P. See Christensen (forthcoming) for detailed discussion of this issue.

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   and nothing else does.¹⁴ So in thinking about her reliability estimate, it seemed obvious that it should simply be independent of the first-order bearing of A and B on P. But if we are to find a general description of the requisite sort of independence, we will have to think about more problematic cases. Consider, for example: Dinosaur Disagreement: Dora and Esa disagree about the extinction of the dinosaurs: Dora believes that it was caused by volcanoes, while Esa thinks it was caused by a meteor impact. Let us suppose that each of them is rationally quite confident that the other is her (at least rough) equal in intelligence, education, diligence, honesty, acquaintance with the relevant literature, and the other factors that they take to bear on likely reliability—each reasonably thinks that the other would be just as likely to get the right answer to this sort of scientific question as she herself is. Suppose we think that Dora should lose substantial confidence in her belief, given Esa’s disagreement. We would like Dora’s relevant Independent Hypothetical Credence in the volcano hypothesis—independent of her views on dinosaur extinction, but given what she has reason to believe about her own and Esa’s reliability—to be relatively modest. But in this case, there’s no simple way of isolating the first-order evidence E which bears on these hypotheses: clearly, all sorts of geological, biological, astronomical, and archaeological evidence are relevant. And similar points will apply to lots of disagreement cases—for example, philosophers’ disagreements about the nature of consciousness, or economists’ disagreements about the effects of minimum-wage laws. So simply specifying that Dora’s IHC is independent of the first-order bearing of “evidence E” on the volcano hypothesis looks problematic.¹⁵ Moreover, even when there is some clearly specifiable bit of relevant first-order evidence, there are sorts of cases where the proposed approach would not seem to fit. There are cases where the very same bit of evidence bears on the relevant proposition in more than one first-order way. In some such cases, the agent’s higher-order evidence will threaten her ability to respond to some of these ways, but not others. So will the right reliability-assessment be independent of the first-order bearing of this evidence, or not? Consider the following example: Two Diseases: Freny is a doctor whose patient has symptoms that might indicate either of two diseases D or D (though he may have neither). Freny orders a blood test, and examines the results. They eliminate D simply and decisively, and also eliminate D, but through a more complex type of statistical reasoning. Freny forms very low credences in both D and D. Then she’s informed that she’s been dosed with a drug that doesn’t affect the sort of simple thinking required to eliminate D on the basis of the blood test, but that does render more complex statistical thinking very unreliable. It seems that Freny should end up with high confidence that her patient is free of D, but not with high confidence that her patient is free of D.

¹⁴ This isn’t quite right. Alicia’s very high credence in A and B is itself based on the reliable source’s testimony. But I believe that this temporary simplification is harmless. ¹⁵ See Christensen (, pp. ff.) for an expression of this worry, whose importance was emphasized to me by Jennifer Lackey.

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

It’s hard to see how the difference between the way Freny’s evidence bears on D and the way it bears on D would be captured by separating this evidence into two chunks: it’s the same blood test result that grounds both chains of reasoning. But how else could the independent hypothetical credence that would be rational for her reflect one avenue of support but not the other? A clue to one approach can be found in thinking back to Alicia’s case. There, we saw that Alicia’s IHC in P was .. This was not, of course, the credence directly supported by her first-order evidence (being told that A and B) alone; that was the point of imposing an Independence requirement. But Alicia’s . hypothetical credence was also not unaffected by Alicia’s having that evidence. It was conditional on her having reached a very high initial credence in P—an initial credence she would not have reached had she not had A and B to go on. And it is only the fact that her initial credence was so high—combined with the information about the drug—that made her IHC ., rather than some much lower value. Moreover, if the drug had been much weaker, Alicia’s IHC would have been correspondingly closer to .. If the drug had had only a negligible expected effect on Alicia’s thinking, her IHC would have been extremely close to the . level made rational by the first-order bearing of her evidence—even though the IHC was independent of A and B’s first-order support for P. So: when a bit of first-order evidence is part of the evidence whose first-order bearing on the relevant proposition is “bracketed” by the IHC, the fact that the agent has that evidence may still affect the agent’s IHC, and thus affect the agent’s rational final credence.¹⁶ This suggests that one approach to the Two Diseases case would involve taking Freny’s IHC to be independent of the first-order bearing of the blood-test results entirely. The idea is that the fact that Freny reached certain credences on the basis of the drug test might still allow the blood-test data to inform her IHC in an intuitively reasonable way. So let us ask, first, what credence Freny should have that her patient is free of D, independent of the blood test’s first-order bearing on D, but conditional on Freny having formed low credence in D on the basis of statistical reasoning from the blood test, and on Freny being a generally reliable doctor who was drugged in a way likely to make that sort of reasoning unreliable. Here, it seems that Freny’s IHC that her patient is free of D will not be high, since her elimination of D is likely to have been based on faulty reasoning. This seems like the appropriate result. Next, let’s ask what credence Freny should have that her patient is free of D, independent of the blood test’s first-order bearing on D, but conditional on Freny having formed low credence in D on the basis of simple reasoning from the blood test, and on Freny being a generally reliable doctor who was not drugged in a way likely to make that sort of reasoning unreliable. Here, it seems that Freny’s IHC that her patient is free of D will be high, since her elimination of D is unlikely to have been based on a mistake. This is also the result we want.

¹⁶ Recall that we are working with two important simplifying assumptions: that the agent reached an initial credence on the basis of her first-order evidence before getting her higher-order evidence, and that that initial credence was rational. We will turn to relaxing these assumptions below.

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   So despite our stipulation that Freny’s IHC be independent of the first-order bearing of the blood-test evidence, the blood-test evidence did affect the IHC it was rational for her to have in D. The fact that she formed her initial low credence in D on the basis of that evidence, combined with the fact that her higher-order evidence indicated that she was in a good position to assess the bearing of that blood-test evidence on D, led to an IHC that reflected, indirectly, the blood-test’s elimination of D. On the other hand, the blood-test evidence did not similarly affect Freny’s IHC for D. While Freny did in fact appreciate its significance for D correctly, her higherorder evidence did not allow her to trust that initial low credence in D. So perhaps we can after all handle cases where a single piece of evidence bears on the agent’s beliefs in some first-order ways that are threatened by the agent’s higher-order evidence, and in other first-order ways that are not: we can take the relevant IHC to be independent of all the first-order bearing of that evidence on the relevant propositions. If this technique allows us to recover the effect that the evidence legitimately has when it bears on matters in ways that aren’t threatened by the agent’s higher-order evidence, the same technique might be extended to offer a neat solution to the problem posed by Dinosaur Disagreement. There, it looked like we’d need to find a way of delimiting exactly what first-order “evidence E” was relevant to Dora’s belief in the volcano hypothesis. But perhaps we don’t need to do that after all. Instead, we might make the relevant IHC independent of the first-order bearing of all evidence on the volcano hypothesis. Insofar as Dora’s higher-order evidence does not cast doubt on her ability to interpret certain parts of that evidence, her earlier appreciation of that evidence will affect the IHC it’s rational for her to have, in the same way as the blood test’s elimination of D ended up affecting Freny’s IHC in D. This suggests a further refinement of our rough formulation of Independence: Independence, third sketch: When an agent has formed an initial credence c in P on the basis of the first-order bearing of her evidence, and then gets some evidence that bears on the reliability of her reasoning from that evidence to her credence c in P, her final credence in P should reflect the Independent Hypothetical Credence (IHC) it would be rational for her to have in P: that is, the rational credence in P independent of her evidence’s first-order bearing on P, but conditional on her having formed credence c in P on the basis of the first-order bearing of her evidence, and on the reliability evidence the agent has about herself. This formulation has one other advantage worth noting. It allows us to accommodate a kind of case that could otherwise seem problematic: the sort of case that arguably does not involve any first-order evidence for P. Suppose, for example, that we vary Logic on Drugs so that Alicia does not learn A or B from a reliable source. Instead, she just decides to consider the material conditional “if A and B, then P.” Alicia, ever the good logician, recognizes this as a logical truth and becomes highly confident of it. Then she’s told she’s been drugged in a way that makes people like her misidentify truth-functionally complex claims about birthdays as logically true about % of the time. Alicia reduces her credence in the claim to around ..

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To get this result, we should see Alicia as having a rational IHC in the claim, conditional on her having reached a high credence in the claim, and on her having been drugged in a specific way, of around .. And one might worry that since the claim is a logical truth, the rational IHC will have to be , no matter what it’s conditional on. But it seems to me that we need not say this. We might see logical truths as supported in a first-order way by any set of premises (including, of course, the empty set). After all, the way that A and B support P in the original version of our example is via the logical relations among the claims: the logical relations guarantee that P is true if A and B are. But these are the same logical relations that guarantee the truth of the material conditional whether or not A and B are true. So it seems natural that if we make Alicia’s rational IHC in our original case independent of the first-order support A and B give to P, via those logical relations, we should consider Alicia’s rational IHC in the variant case to be independent of the bearing of these logical relations on the truth of the material conditional, no matter what Alicia’s first-order evidence may be.¹⁷

. What if there’s “no higher-order evidence”? So far, we’ve been thinking of cases where agents get clear evidence that some part of their thinking has been compromised. It might seem at first as if this is some fairly limited class of special cases that comprise the domain of theories of higher-order evidence. And the literature on higher-order evidence and disagreement often reads as if it’s a discussion of somewhat anomalous cases. But a little reflection should cast doubt on that thought. Consider some variations on Logic on Drugs. We might begin with a case where Alicia gets excellent evidence that she’s been given a drug that always makes everyone reach wrong conclusions about birthdays. We can then construct one kind of spectrum by reducing the strength of Alicia’s reasons for trusting the person who informs her about the drugs, until this evidence is extremely weak. Or we can consider cases where Alicia gets excellent evidence that she’s been given a drug which always causes those sensitive to it to make mistakes, but which only affects % of people it’s given to. Or one which only affects %, or even .%. Clearly, in each of these spectra, the last cases should be treated much like cases where Alicia gets no special information about being drugged. So cases involving clear evidence of impairment seem to shade smoothly into those involving no such evidence. We can also construct a spectrum of variant cases where Alicia’s evidence concerns a drug that affects everyone, but which induces % inaccuracy, or % inaccuracy or .% inaccuracy. This sort of spectrum can be continued to include cases where Alicia gets evidence that her reliability has been enhanced rather than compromised. (A realistic example of such a spectrum might involve beverages ranging from cognac

¹⁷ This makes especially clear that our IHCs will not be conditional probabilities. But it seems clear that taking higher-order evidence seriously will lead to non-probabilistic credences in general. See Christensen ().

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   to coffee.) So a general account of higher-order evidence should handle cases ranging from impairment to enhancement, presumably in a smooth way. It’s also worth noting that, insofar as Alicia is like an ordinary person, she will have some evidence relevant to her reliability, even apart from any drug information. This might come from how clear or fuzzy she felt in thinking about the birthday problem. It might come from her track record on very similar problems—or, less tellingly, from her vague impressions of her track record on less-similar problems. And so on. So if Alicia is an ordinary person, she likely will have some higher-order evidence bearing on her likely reliability. And this does not seem peculiar to logic problems—it applies to a vast range of cases where people draw conclusions from evidence. All of these considerations together suggest that our account of higher-order evidence should apply to most, if not all, of the beliefs we form on the basis of first-order evidence. If that’s right, then in a great many cases, our beliefs should be required to reflect our rational IHCs.¹⁸ However, it’s not obvious how the relevant rational IHCs would be arrived at in cases where reliability evidence is meager. And it is hard to say much about this without presupposing one or another general account of epistemic rationality. But having flagged the problem, I’ll sketch one approach that strikes me as attractive (though it no doubt incorporates assumptions that some would reject). Let us begin by considering the ordinary range of beliefs that ordinary adults typically have in ordinary matters—cases where agents aren’t parties to disagreement, or likely victims of cognition-degrading factors such as drugs, biases, or fatigue. It does seem reasonable—and I think it will probably be uncontroversial— that these people are usually rational in trusting their own cognition. One way of putting this, very roughly, is to say that, in most cases, agents would be rational in thinking that P is likely to be true (false), on the supposition that they believe that P (~P). Less roughly, agents are typically rational to have high (low) credence in P being true, conditional on their having a high (low) credence in P. Next, it seems that this sort of conditional credence would typically be rational, even independent of the agent’s particular attitude toward P. Someone may believe P, and also may be rational to be confident that P is likely true on the condition that she believes it, and the rationality of this conditional credence need not be just based on her confidence that P is true anyway. So, for example, if we describe the belief about P in general terms (“a belief about which of two letters comes first in the alphabet,” “a belief about what country a well-known city is in,” “a belief about whether a certain kind of animal is a mammal”), agents will typically be rational to be confident that P is true on the condition that they believe that P.¹⁹ If this is right, then even when we consider the ordinary beliefs of reasonable people—beliefs that are not subject to the ¹⁸ At this point, I’m still restricting our discussion to cases where an agent forms a belief on the basis of first-order considerations, and asking how she should take higher-order considerations into account. Thinking about ordinary cases makes salient how severe a limitation this is. Section . takes on this problem. ¹⁹ A more precise version would talk about credences rather than beliefs; this would require talking about cases where the agent has some middling credence in P. But the point here is just the intuitive one: that for most of us, and for many of the ordinary topics we have opinions about, we have reason for confidence in our opinions that is independent of those particular opinions themselves.

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sort of higher-order evidence that the literature has highlighted—we should typically see agents as rational in having Independent Hypothetical Credences that cohere with those ordinary beliefs. There are, of course, deep and difficult questions about how this sort of selfconfidence is ultimately grounded. One way of raising such questions begins by considering the theoretical limit case, where an agent has no higher-order evidence at all to work with. It’s quite possible that such cases never occur in adult human thinkers, but perhaps we can ask the theoretical question of what reliabilityassessment would be rational, completely independent of an agent’s higher-order evidence. It seems plausible that when we ask ourselves this question, the answer is that agents have some default entitlement to trust their own thinking. This is a familiar idea, made plausible in part by the apparent impossibility of rationally basing beliefs on deliverances of sources whose reliability one cannot rationally trust. On this sort of picture, an agent with no higher-order evidence at all would be rational to have some (defeasible, of course) confidence in the accuracy of her beliefs. It’s worth noting that this picture fits well with one kind of line in the disagreement literature: Some have worried that Independence principles will lead to skepticism in disagreements with, for example, paranoid conspiracy theorists who reject vast proportions of our beliefs. The worry is that if we put aside all the disputed issues, we’ll be left with no independent reason to think we’re more reliable than the conspiracy theorist. In response, it has been argued that conciliation is required only when one has strong independent reason to believe the other person to be reliable, and that putting aside most of our beliefs, we’re left without strong reason think this.²⁰ On the current suggestion, it’s natural to think that when there’s only very slim dispute-independent evidence to go on, it will be insufficiently robust to significantly undermine our default self-trust. So our default self-trust provides a graded mechanism whereby undermining the justification of our beliefs occurs only to the extent that we have substantial independent reason to trust those with whom we disagree. But perhaps we don’t need to settle this in order to say that there are IHCs that are rational for ordinary people who have ordinary beliefs. Of course, these will be highly sensitive to specific reliability information of the sort featured in standard discussions of higher-order evidence. And they clearly may be pushed in either direction by specific higher-order evidence, making room for evidence of enhanced reliability as well as the sorts of diminished reliability the literature has concentrated on.

. What if the agent has no “initial credence”? We’ve been simplifying the discussion by supposing that our agents have formed rational credences on the basis of their first-order evidence before they get the higherorder evidence. But surely this is not how matters typically work out. One typically forms opinions on controversial issues by looking at the first-order evidence while ²⁰ See, e.g., Elga (), Christensen (), and Vavova (); related issues are discussed in Worsnip () and Pittard ().

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   already aware of the fact that others have varying opinions. One may learn about implicit bias before looking at a stack of CVs, or learn about possible hypoxia before wondering whether one has enough fuel to reach Sitka. In thinking about this sort of case, we should pay attention to the distinction between propositional and doxastic rationality. Here, I’d like to concentrate on the propositional notion: When we think about agents who get higher-order evidence before getting the first-order evidence, what credence is propositionally rational for them? And I will be supposing that what credence is propositionally rational for agents depends not only on their first-order evidence, but on their higher-order evidence as well. Thinking about cases where the agent does not have an initial credence based only on first-order considerations will clearly complicate our discussion considerably. Our Independence principles so far have simply assumed that agents had already formed rational credences on their first-order evidence, and it was those credences that were the subject of the independent reliability-assessments. That clearly can’t work in cases where the agent has not formed any initial credence. So what could our reliability-assessments possibly apply to? Without delving into different possible approaches to this question, I’ll assume for present purposes a particular answer to it, in order to lay bare the issues that come up for formulating Independence. The basic idea is this: since we’re asking what final credence is rational given the agent’s total evidence, we can apply the agent’s higherorder-evidence-based reliability-assessment to the credence that would be rational for her on the basis of the first-order bearing of her evidence. In effect, we are imagining that the agent reacts rationally to her first-order evidence, and takes that reaction as the subject of the reliability-assessment. This would of course mesh nicely with our verdicts on the cases considered above, where we stipulated that the agent had already formed the credence that was rational given her first-order evidence. It also fits with the natural idea that an agent who gets higher-order evidence before first-order evidence should end up with the same credence as one who gets the same batches of evidence, but in the opposite order.²¹ Moving to applying reliability assessments to credences the agent hasn’t actually formed, though, introduces another complication. When we were concentrating on agents who had already formed an initial credence in the relevant proposition P, the agent’s IHC was conditional on the fact that she herself had formed that initial credence. But in the present framework, the agent’s evidence presumably includes her memory of not having formed any such credence. This does not necessarily mean that she would be rational to be absolutely certain that she did not form an initial credence—perhaps she is not. But she might well be rational to think that if she had formed such a credence, then given that she can’t remember any such thing, there must be something drastically wrong with her. This would obviously suggest that she was cognitively impaired in some way, and thus affect her IHC in ways that we want to avoid. Here is what seems to me to be the most promising way of avoiding this problem: Instead of considering an IHC conditional on the agent herself having formed the

²¹ See Christensen () for extended discussion of this idea.

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

initial credence, we might consider an IHC that’s focused on a relevantly similar hypothetical agent—that is, one who is similar to the agent with respect to those higher-order factors that affect expected reliability. (In this way, the relevant hypothetical credence is similar to what would be rational for a third party to form, upon learning that an agent of the relevant sort had formed the initial credence in question.) Putting these ideas together, we get something along the following lines: Independence, final sketch: Let c be the credence in P that would be rational given the first-order bearing of the agent’s evidence on P. Then the credence in P that would be rational for the agent, given all her evidence, should reflect the Independent Hypothetical Credence (IHC) it would be rational for her to have in P: that is, the rational credence in P independent of her evidence’s first-order bearing on P, but conditional on a relevantly similar agent adopting credence c in P on the basis of the first-order bearing of that agent’s evidence. There are, no doubt, further problems with this sketch—so it’s “final” only in representing the approach that looks most promising to me today. But before closing, in sections . and .. I’d like to discuss one apparent difficulty that this formulation poses, and another difficulty that becomes apparent when we move beyond the simplified examples we’ve been looking at so far.

. What if there are multiple first-order routes from E to P? Agents who have actually formed initial credences are, of course, aware of more than the fact that they formed a certain initial credence. They may well be aware, for instance, of the way in which they moved from their first-order evidence to their initial credence in P. For example, in a case like Logic on Drugs, Alicia may well remember which logical rules she actually used initially, in inferring P from A and B. In our original case, this information was not particularly relevant, since the drug we were imagining affected all complex truth-functional reasoning about birthdays. But designing our drugs a bit more narrowly can give us cases where this sort of information can be relevant. For example, suppose that Alicia knew that the logic-disrupting drug she’d been given only affected reasoning done by (attempted) Modus Ponens. In such a case, the confidence it would be rational for her to end up with would depend on whether she had reason to think she’d relied on MP in forming her high credence in P. This might not raise any difficulties in Alicia’s case, but we should also consider cases where the agent does not form their initial credence before getting the higher-order evidence. So consider the following case: Logic on MP-disrupting Drugs: Gabi is informed that they have been slipped a drug that degrades people’s truth-functional reasoning about birthdays. But this drug only degrades Modus Ponens reasoning: those who attempt to reason by MP make mistakes in applying the rule % of the time, while feeling perfectly clearheaded. The drug does not affect other reasoning at all. After learning about the drug, Gabi is told A and B, as Alicia was, and then asked whether P.

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   What should Gabi’s IHC be? If Gabi had inferred P from A and B before getting the information about the drug, they would have information about how they had done that—information relevant to the reliability of their inference (i.e., whether they had used MP in deriving P from A and B). But in the present case, Gabi never performed that initial inference. So in thinking about what credence in P is propositionally rational for Gabi, how should the information about the drug be brought to bear? We should first note that the credence c that’s rational on Gabi’s first-order evidence is very high, since A and B entail P. So our question then becomes: how likely is P to be true, given that an agent relevantly like Gabi reached a very high credence in P on the basis of first-order support from their evidence? The answer to this question obviously depends on how likely that agent would have been to reason via MP. If the agent was highly likely to have reasoned via MP, it will be close to % likely that they made a mistake, so the IHC would be significantly lower than c. On the other hand, if the agent would be unlikely to use MP, the probability of mistake would be much lower, and the rational IHC much closer to c. The next thing to notice is that the reliability evidence Gabi has does not only make it likely that Gabi has been drugged in a certain way. It also includes the fact that Gabi was told they had been drugged in that way. If Gabi is like most of us, the fact that they have good reason to think they would be likely to mess up in MP-based reasoning would make them much more likely to try reason about the problem without using MP. If it were reasonable, given Gabi’s evidence about this, to expect Gabi—or a relevantly similar agent—to be able to succeed in this task, then the IHC that would be rational on Gabi’s evidence would presumably be quite high. On the other hand, suppose that Gabi’s evidence suggests that they are so drawn to reasoning by MP that they would end up slipping and reasoning by attempting MP, without realizing it, even if they tried not to. Then the same would go for the relevantly similar agent. And of course the rational IHC for Gabi in this situation would be significantly lower. So it turns out that the IHC that would be rational for Gabi would depend on how likely it was (on their evidence) that they would end up reasoning by attempted MP. This might seem problematic at first. After all, one might think, if we’re asking what credence is propositionally rational for an agent, we should answer this by assuming that they react to their evidence rationally. And given Gabi’s evidence about having ingested the drug, the rational reaction would include avoiding MP. So despite what evidence Gabi has about their own propensity to use MP, their IHC in P would be very high. But intuitively, in cases where Gabi has good reason to believe they would reason unreliably, high credence in P would not be propositionally rational. But I think this worry would be misplaced. At bottom, it’s not really different from the general thought that higher-order evidence is always irrelevant to the question of what credence is propositionally rational, since a perfectly rational agent would react perfectly to her first-order evidence. Insofar as we acknowledge that agents are rationally required to take higher-order evidence seriously—in the sense that what credences are propositionally rational for them are sensitive to information they have about their own expected reliability—it makes sense that the credence in P that’s rational for Gabi will depend on how likely Gabi’s higher-order evidence makes it that they will reason unreliably. Gabi’s situation is not really different in kind from

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

the situations of agents who have evidence that they might be hypoxic, or biased. In all of these cases, the credence that’s propositionally rational for the agent depends on the agent’s evidence about their own reliability. So it seems to me that while moving to our third sketch of Independence does introduce complexities in thinking about cases involving multiple paths from E to P, the complexities simply reflect important complexities of the epistemic situations in question.

. What if the higher-order evidence is indiscriminate in its target? There is one more way in which the examples we’ve been looking at were chosen to simplify our discussion of Independence principles. In each case, the higher-order evidence targeted a fairly narrow part of the agent’s reasoning. This is particularly clear in our artificial cases involving designer drugs, but also applies in other cases. Sexism may distort my CV-assessments while leaving large parts of my cognition unaffected. Fatigue or hypoxia may degrade complex thinking, while leaving simpler thinking relatively unscathed. Disagreement about a particular topic may indicate that I’ve made a mistake in thinking about that topic, without necessarily indicating a more widespread cognitive problem. But it’s clear that not all cases will fit into this tidy mold. Sometimes, higher-order evidence may provide reason for the agent to doubt wide swaths of her thinking— including, in some cases, her thinking about how to accommodate higher-order evidence. This sort of case introduces a new dimension of difficulty in theorizing about higher-order evidence.²² For a simple example of this sort, we might think about more powerful drugs. Suppose that Hui is part of a research team studying a new reason-disrupting drug. The team has seen subjects like Alicia and Gabi reach reasonable conclusions about P—conclusions that reflect their higher-order reasons for doubting their ability to reason truth-functionally about birthdays while drugged. So they decide to kick the challenge up a notch. They design a drug that not only severely compromises ordinary truth-functional reasoning about birthdays, but also degrades the sort of thinking required to take higher-order evidence about drugs into account—in particular, drugged agents do not form the IHCs supported by their evidence. Hui volunteers to be a subject for the initial tests of the drug. She walks into the experiment room, and an associate brings her a pill and some water. Twenty minutes later, she is given some information, and asked to think about what follows. Suppose she is told A and B from our original story, and wonders whether P. It sure seems to her as if P must be true if A and B are—but . . . Well, what credence in P is rational for Hui in this situation? In thinking intuitively about this case, we might start by asking how a fully rational agent would react to Hui’s predicament. One might think that since Hui has good evidence ²² Sliwa and Horowitz (, §.) pose this problem, and make points related to some of the points made below.

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   that she can’t rationally take higher-order evidence into account, she should just go with her first impression, and be highly confident in P. But we might imagine that she’s been through several trials with this drug, and knows that this strategy very frequently leads to high confidence in false claims—after all, the drug degrades first-order thinking severely, so she can’t trust that thinking any more than she can trust her thinking about the implications of her higher-order evidence. So maybe she should take all of this into account, and ignore not only her higher-order evidence, but also ignore A and B. But would not that be simply a different instance of using higher-order evidence to compensate for the drug’s expected effects? (In fact, even going with her initial impression that P follows from A and B, on the grounds that she can’t trust herself to reliably form beliefs on the basis of the drug information, would also be an instance of taking the drug information into account in forming her belief.) At this point, I, at any rate, have no clear intuitive idea what maximallyrational Hui would believe, or what result the right account of higher-order evidence should give in this case. Does Independence help us out here, giving us a verdict in a case where intuitions are confused? I don’t think so. Independence—at least as it’s sketched above— invokes rational Independent Hypothetical Credences. In this case, we would have to ask how confident it’s rational for Hui to be in P, independent of P’s first-order support, but contingent on an agent relevantly similar to Hui reaching high credence in P on the basis of first-order bearing of that agent’s evidence. But this assessment is exactly the sort that’s targeted by Hui’s higher-order evidence. And so it’s not clear what IHC would be rational for Hui in this case. Should we expect the correct account of Independence to deliver a clear verdict about what credence is rational for Hui’s? I think it’s not at all obvious that we should. Independence feeds into an account of higher-order evidence which puts constraints on rational credences: it relates rational “all things considered” credences to certain rational hypothetical conditional credences. But this constraint need not tell us the whole story about what either of these two credences should be. Insofar as Hui’s case is one where it’s quite unclear intuitively what the most rational overall credence, or the rational IHC, would be, it is less worrisome that Independence yields no clear verdict. We might consider an extreme version of Hui’s case, in which a drug is so powerful that it completely messes up people’s thinking about everything. It seems that Hui’s team could observe the wacky beliefs their unwitting experimental subjects form under the drug’s influence—but it’s not at all clear that this leaves Hui with some epistemically rational reaction to the information that she’s just ingested that drug. A similar problem crops up in examples posing the anti-expertise paradox, where agents get excellent evidence that they’ll believe P just in case P is false: no doxastic position seems rationally stable.²³ It would seem unreasonable to expect Independence to provide a cure for this sort of epistemic malady. Nevertheless, I think that cases involving indiscriminate higher-order evidence do pose problems for formulating Independence. Suppose we decide that we need not worry about certain paradoxical cases, or cases involving near-global self-doubt.

²³ See Conee (), Sorensen (), Christensen ().

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

Still, cases of global self-doubt seem to be at one end of a spectrum, with cases of narrowly focused self-doubt at the other end. One might hope that a more fully developed account of higher-order evidence could give us some insight into the rationality of beliefs based on what we might think of as somewhat indiscriminate higher-order evidence. And some such examples are much less artificial than ones involving super-drugs, or the ones that standardly figure in discussions of the antiexpertise paradox. For example, consider Isaac’s attitude toward the proposition that the President of his country is a habitual liar. When Isaac considers the matter directly, it seems quite clear to him that his President is a habitual liar. But Isaac also notices that he’s highly disgusted by many aspects of the President’s conduct and character. He sees the President’s policies as cruel and vindictive; he sees the President as surrounding himself with corrupt, self-serving advisors; and he sees the President as playing to his compatriots’ worst selves. Overall, he thinks of the President as base and immoral, and is deeply embarrassed that his country is governed by such a man: even hearing the sound of the President’s voice on the radio, or seeing his picture in the newspaper, turns Isaac’s stomach. So the topic of the President is emotionally fraught for Isaac, and he’s aware that his intense loathing for the President may compromise his ability to assess the President’s truthfulness accurately. It seems clear that this information is relevant to how confident Isaac may rationally be that the President is indeed a habitual liar. On the approach we’ve been developing, Isaac’s rational credence depends on what IHC would be rational for him, given his evidence about his likely cognitive impairment. But should Isaac expect himself to be able to assess his likely reliability on this matter in a cool, accurate way? I don’t think so. We may suppose that Isaac is sophisticated enough to see that his opinion on the President’s mendacity and his opinion on his own reliability in assessing the President’s mendacity are tightly linked. Insofar as his emotional investment in thinking the President a liar is likely to affect one, it seems quite likely to affect the other. So the rational IHC for Isaac would have to take into account his evidence that he’s likely to count himself more reliable than his independent reliability-relevant evidence really indicates. A parallel point applies to disagreement evidence. Suppose Isaac also knows that a significant number of his compatriots think that the President is not a habitual liar. How strong evidence does their disagreement provide for the claim that Isaac’s direct thinking on the matter is unreliable? Well, as usual, that depends on how reliable Isaac should take those who disagree to be—setting aside Isaac’s opinions on the President’s truthfulness, and on related matters on which he disagrees with them. So, for example, Isaac might want to consider whether he has good independent reason to think that his opponents are being irrationally manipulated by rhetoric, or whether he has good independent reason to suspect that their views are formed in response to racism, or xenophobia. But of course there’s a problem here. Since Isaac is aware that his opponents’ expected reliability is a threat to his own convictions, his emotional investment in those convictions would seem likely to warp his reliability-assessments of them (which of course feed directly into how reliable Isaac should expect his own contrary opinion to be).

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   Insofar as Isaac realizes all of this, it seems that that the rational IHC for him would be lower than it would be if Isaac did not have evidence that his emotions were likely to inflate his self-assessment. So this is not a case where we have no intuitive idea of how the agent’s higher-order evidence should affect his credence. For that reason, it seems plausible that a fully satisfactory account of Independence should explain how an agent’s evidence can bear in a higher-order way not only on what credences in ordinary propositions are rational for the agent, but on what IHCs are rational for the agent. It’s not clear to me how this will work out. When we ask how much Isaac’s evidence about his emotions should lower his IHC, it seems that this question should depend on how reliable Isaac should expect himself to be at arriving at the rational IHC. This might naturally seem to involve some sort of meta-IHC. But it’s not clear to me how this should be formulated. And it’s not clear where, or if, Isaac’s reasons for self-doubt will stop ramifying.²⁴ What should we make of this problem? We should not, of course, expect an Independence principle to give us an entire epistemology. We should not expect it to yield clear verdicts in every case, if there are cases in which there are no clearly correct answers. But it’s not unreasonable to hope that our account of Independence will help us understand how higher-order evidence affects rational IHCs—that it will give us some insight into how ramified self-doubt affects rational credences, when those credences do seem to exist. I’m at this point unsure of how difficult this work will be. But at a minimum, there is more work to be done.

. Conclusion Independence principles seem to be required in order to cope with a peculiarity that characterizes higher-order evidence: roughly, this sort of evidence targets the reliability of an agent’s thinking; but in assessing the import of this evidence, the agent must do so from within an epistemic perspective that’s constituted by her own thinking. She must act both as judger and as judged. The awkwardness that this involves comes out, in various ways, in the difficulties we’ve seen in formulating Independence principles. While I think that there are promising avenues for meeting some of these difficulties—as embodied in the Independence-sketches above—I would not claim to have a clear, clean, and precise way of meeting them all. One reaction one might have to the difficulties is to give up on the project—say, by holding that higher-order evidence simply does not bear on rational belief. But the gain in theoretical simplicity would, I think, be more than offset by the loss of explanatory power. When we consider the high-flying pilot, or the sleepy medical resident, or the person likely to be affected by sexist bias, it seems clearly irrational for them to maintain high confidence in their original judgments. ²⁴ See Lasonen-Aarnio () for an extended development of a related sort of worry, and Schechter () for a possible line of response.

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

Like it or not, we’re often cast in the role of judging our own epistemic performance. Acting as judge in one’s own case is, of course, famously problematic. But since epistemic agents cannot recuse themselves from this predicament, we epistemologists should do our best to understand how it is most rationally negotiated.

Acknowledgements Thanks to audiences at NYU and the University of Copenhagen, where some of the arguments forming the basis of this chapter were presented. Particular thanks to Zach Barnett, Cian Dorr, Adam Elga, Kit Fine, Ruth Horowitz, Hélène Landemore, Jim Pryor, Josh Schechter, and Jonathan Vogel for helpful discussions of these issues. And thanks to an editor of this volume for useful comments on an earlier draft.

References Arsenault, M. and Z. C. Irving (). “Aha! Trick Questions, Independence, and the Epistemology of Disagreement.” In: Thought (), pp. –. Christensen, D. (). “Does Murphy’s Law Apply in Epistemology? Self-Doubt and Rational Ideals.” In: T. S. Gendler (ed.), Oxford Studies in Epistemology II, Oxford University Press, pp. –. Christensen, D. (). “Higher-Order Evidence.” In: Philosophy and Phenomenological Research (), pp. –. Christensen, D. (). “Disagreement, Question-Begging and Epistemic Self-Criticism.” In: Philosophers Imprint (), pp. –. Christensen, D. (). “Conciliation, Uniqueness and Rational Toxicity.” In: Noûs , pp. –. Christensen, D. (). “Disagreement, Drugs, etc.: from Accuracy to Akrasia.” In: Episteme, (), pp. –. Christensen, D. (). “On Acting as Judge in One’s Own (Epistemic) Case (Marc Sanders Lecture).” In: Proceedings and Addresses of the American Philosophical Association. Conee, E. (). “Evident, but Rationally Unacceptable.” In: Australasian Journal of Philosophy , pp. –. Elga, A. (). “Reflection and Disagreement.” In: Noûs , pp. –. Foley, R. (). Intellectual Trust in Oneself and Others, Cambridge University Press. Kelly, T. (). “Disagreement and the Burdens of Judgment.” In D. Christensen and J. Lackey (eds), The Epistemology of Disagreement: New Essays, Oxford University Press. Kornblith, H. (). “Belief in the Face of Controversy.” In R. Feldman and T. Warfield (eds), Disagreement, Oxford University Press. Lasonen-Aarnio, M. (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research (), pp. –. Matheson, J. (). The Epistemic Significance of Disagreement, Palgrave Macmillan. Moon, A. (). “Disagreement and New Ways to Remain Steadfast in the Face of Disagreement.” In: Episteme (), pp. –. Pittard, J. (). “Disagreement, Reliability and Resilience.” In: Synthese , pp. –. Schechter, J. (). “Rational Self-Doubt and the Failure of Closure.” In: Philosophical Studies (), pp. –. Sliwa, P. and S. Horowitz (). “Respecting all the Evidence.” In: Philosophical Studies (), pp. –.

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   Smithies, D. (). “Ideal Rationality and Logical Omniscience.” In: Synthese (), pp. –. Sorensen, R. (). “Anti-Expertise, Instability, and Rational Choice.” In: Australasian Journal of Philosophy , pp. –. Van Wietmarschen, H. (). “Peer Disagreement, Evidence, and Well-Groundedness.” In: The Philosophical Review (), pp. –. Vavova, K. (). “Moral Disagreement and Moral Skepticism.” In: Philosophical Perspectives (), pp. –. Worsnip, A. (). “Disagreement about Disagreement? What Disagreement about Disagreement?” In: Philosophers Imprint (), pp. –.

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2 Higher-Order Uncertainty Kevin Dorst

Here is one of my main claims: Thesis:

Epistemic akrasia can be rational.

(Don’t worry, just yet, about what it means.) I am confident of Thesis, for I have a variety of arguments that I take to be good evidence for it. But—now that I think about it—whenever I sit down to write a paper, I’m confident of that paper’s thesis. In fact, that confidence usually has a similar basis: I have a variety of arguments that I take to be good evidence for it. And yet I’ve later found—all too often—that my arguments weren’t so good after all; that I’ve been overconfident in my past theses.¹ Having meditated on these facts, I’m still confident of Thesis, for I still think that I have good arguments for it. However, here’s another proposition that I now consider possible: Doubt: I should not be confident of Thesis. I’m not confident of Doubt—but nor do I rule it out: I leave open that maybe I shouldn’t be confident of Thesis. Question: how should my attitudes toward Thesis and Doubt relate? Thesis is a claim about some subject-matter. Doubt is a claim about what opinion I ought to have about that subject-matter. Let’s call my opinion about Doubt a higher-order opinion—an opinion about what opinion I should have. Since I am uncertain about what opinions I should have, I have higher-order uncertainty. Let’s call my opinion toward Thesis a first-order opinion—an opinion about something other than what opinions I should have. Generalizing our question: how should my first-order and higher-order opinions relate? For example: if I become more confident that I shouldn’t be confident of Thesis, should that lead me to be less confident of Thesis? Or: if I have a lot of higher-order uncertainty about how confident I should be in Thesis, can I nevertheless be fairly confident of it? I will not give a full answer to such questions—but I will take three steps toward one.

¹ “Overconfident” here—as in natural language—means being more confident than you should be; not to having some confidence in something that’s false. If you’re – that this fair coin that I’m about to toss will land heads, then you’re not overconfident—even if, in fact, it will land tails. Kevin Dorst, Higher-Order Uncertainty In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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   First step. Many have asked similar questions. But they have often framed it as a question of how one body of evidence—your first-order evidence—interacts with another body of evidence—your higher-order evidence.² My first claim: R: We should reframe the question: Given your total evidence, how should your first- and higher-order opinions relate? I defend R by showing how to build a general framework for studying the relationship between first- and higher-order uncertainty (section .), and then putting it to work (sections .–.). Second step. So reframed, our question is nontrivial: M T: Your total evidence often warrants being uncertain what opinions your total evidence warrants, and (hence) being modest: uncertain whether you’re rational.³ I defend the M T by arguing that rational modesty—i.e. rational higherorder uncertainty—is needed to account for the epistemic force of disagreement (section .). Third step. Many have pointed out that it seems irrational to believe that my Thesis is true, but I shouldn’t believe it. The inferred explanation has been that your first-order opinions must “line up” with your higher-order opinions. Call this the Enkratic Intuition. Many theories defend (or presuppose) it as the answer to our question.⁴ My final claim—my Thesis—is that the Enkratic Intuition is wrong: A: If modesty is rational, so too is epistemic akrasia. I defend A by using the above framework to precisify the Enkratic Intuition and show that it is inconsistent with higher-order uncertainty (section .; cf. Titelbaum ). That is the plan. Here is the picture. Higher-order uncertainty is pervasive and important. There is a general, tractable framework for studying it. Many open questions remain.

. Two problems Recall that I’m confident of: Thesis: Epistemic akrasia can be rational.

² E.g., Feldman (); Christensen (a, ); Horowitz (); Schoenfield (, ); Sliwa and Horowitz (). ³ I will assume a single normative notion that privileges certain opinions. I’ll call them the opinions that “you should have,” that “your (total) evidence warrants,” or that “are rational.” If you think these normative notions come apart, please replace my expressions with your preferred, univocal one. ⁴ E.g., Feldman (); Gibbons (); Christensen (b); Huemer (); Smithies (; ); Greco (); Horowitz (); Titelbaum (); Sliwa and Horowitz (); Littlejohn (); Worsnip (); Salow ().

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- 



But I also suspect that: Doubt: I should not be confident of Thesis. In thinking about cases like this, the standard operating procedure is to make (something like) the following distinction: () My first-order evidence about Thesis is the evidence that bears directly on it. (Example: my current arguments.) () My higher-order evidence about Thesis is the evidence that bears indirectly on it by bearing directly on claims like Doubt. (Example: the flaws in my past arguments.) Authors making this distinction often presuppose that we can meaningfully speak of two distinct bodies of evidence—my first-order evidence and my higher-order evidence.⁵ Distinction made, the standard question goes something like this. Given my first-order evidence, I should have some opinion about Thesis. Now add my higher-order evidence. Should my opinion in Thesis change? If so, how? Since we are to imagine two interacting bodies of evidence, call this the Two-Body Problem. My first claim is: R: We should reframe the question: Given your total evidence, how should your first- and higher-order opinions relate? In other words: instead of two interacting bodies of evidence, we have two interacting levels of opinions warranted by a single, total body of evidence. We have a Two-Level Problem, not a Two-Body one. I have no short, knock-down argument for R. Instead, what I have to offer is (my own) confusion generated by the Two-Body Problem, and clarity generated by the Two-Level one. Perhaps you will share them. Confusion first. One question: what exactly does it mean for a bit of evidence to bear “indirectly” on Thesis? In some sense, the claim that Jones has constructed an argument for Thesis does so. But this is not the sense that has been meant in the higher-order evidence discussion, which focuses on the possibility of rational errors (Christensen a)—sleep deprivation, hypoxia, irrationality pills, and the like. So perhaps evidence bears indirectly on Thesis when it bears on whether I’ve made a rational error in forming my opinion about Thesis? But suppose that I haven’t yet formed any opinion about Thesis, and then the oracle informs me that Doubt is true. Surely this is still higher-order evidence, even though it says nothing about a rational error on my part. Another question: how do these two bodies of evidence agglomerate? Suppose F is first-order evidence for q and H is higher-order evidence for q; what is the conjunction F ∧ H? It clearly bears directly on q, so it seems that it should be first-order evidence. But this means that when I go to base my beliefs about q on my first-order evidence, I will thereby base them on F ∧ H, bringing in the higher-order information H. So maybe

⁵ E.g., Feldman (); Christensen (a, ); Horowitz (); Schoenfield (, ); Sliwa and Horowitz ().

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   those beliefs should be based on my purely first-order evidence. But what does it mean for a bit of evidence to be purely, directly about q? Consider q itself—surely this proposition is purely, directly about q if anything is. But consider the proposition: Not-Known:

My first-order evidence does not put me in a position to know :q.

Not-Known is a paradigm case of the sort of proposition that higher-order evidence about q works “through”—if a proposition p bears on Not-Known, it bears indirectly on q. But q implies Not-Known. Thus even q itself does not bear purely directly on q! This is not meant to be clarifying. Nor is it meant to be a precise argument against the Two-Body Problem. What it is meant to be is an illustration of how easy it is to find oneself confused with this problem. My goal in the rest of the chapter is to argue that the Two-Level Problem leads to a clearer framing of the questions and their potential answers.

.. Framing the debate Assume that you have a single total body of evidence that determines what opinion you should have in any given proposition. Some of these propositions will be like Doubt—claims about what opinions you should have, that is, about what opinions your single, total body of evidence warrants having. Your opinions about such propositions are higher-order opinions. Other propositions will be like Thesis—claims that aren’t about what opinions you should have. Your opinions about such propositions are first-order opinions. Here is an interesting question: how do the first-order opinions warranted by your total evidence relate to the higher-order opinions warranted by your total evidence?⁶ For example, how confident of Doubt can my evidence warrant being before it necessarily warrants being less than confident of Thesis? Or: if we minimally change my evidence so that it warrants more confidence in Doubt, will it thereby warrant being less confident of Thesis? If so, how much? We can state things more precisely. Let C be a definite description for my actual degrees of belief—whatever they are. [C(q) = t] is the proposition that I’m t-confident of q—it’s true at some worlds, false at others. Let P be a definite description for the credences I should have, given my (total) evidence. For simplicity, assume unique precision: my evidence always warrants a unique, precise probability function P.⁷ [P(q) = t] is the proposition that my (current, total) evidence warrants being t-confident of q. So at any given world w, there’s a particular probability function that I ought to have—let Pw be a rigid designator for the function initialized by w. (Unlike the definite descriptions P and C, Pw refers to a particular probability function whose values are fixed and known.) Since I can (rationally) be unsure which world I’m in, I can (rationally) be unsure which probability function my credences should match: if the open possibilities are w₁, w₂, . . . then I can leave open whether [P = Pw₁] (the rational credence function is Pw₁) or [P = Pw₂] (the rational credence function is Pw₂), or . . .

⁶ I’m certainly not the first to approach the issue in this way—see Williamson (, , this volume); Christensen (b); Elga (); Lasonen-Aarnio (), and Salow (). ⁷ It would be fairly straightforward to generalize the framework to drop this assumption. It’s also worth noting that the models I use only presuppose intrapersonal uniqueness: there is a uniquely rational credence function for each agent, given their information and standards of reasoning. For the (de)merits of these assumptions, see White (, ); Joyce (); Schoenfield (), and Schultheis ().

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- 



With this notation in hand, here’s how we can regiment my attitudes toward Thesis and Doubt. For simplicity, suppose I’m confident of q iff my credence in q is at least ., and I leave open q iff my credence in q is nonzero. We can treat Thesis as a primitive proposition. On the other hand, Doubt is the proposition that I should not be confident of Thesis, that is, that the rational credence in Thesis is less than .. So Doubt = [P(Thesis) < .]. Thus my attitudes: I’m confident of Thesis: [C(Thesis)  .], and I leave open that this confidence is rational: [C(P(Thesis)  .) > ]. I leave open that I shouldn’t be confident of Thesis: [C(P(Thesis) < .) > ]. What’s distinctive about my epistemic situation is that I’m unsure which opinions my (total) evidence warrants: I think maybe it warrants having a credence of at least ., and maybe it warrants having a credence below .. If we further assume that I’m sure of my actual credences—so [C(C(Thesis)  .) = ]—it follows that I am unsure whether I’m rational; or, as I will say, I am modest.⁸ For since I’m certain that I’m confident of Thesis and I leave open that I shouldn’t be, I thereby leave open that I’m not rational: [C(C(Thesis) 6¼ P(Thesis)) > ]. We might expect that if such higher-order doubts are warranted, then they should constrain my confidence in Thesis. The Two-Level Problem is whether, why, and to what extent this is so: how are rational opinions constrained by rational opinions about what opinions you should have? Notice that this is a question about how my higher-order doubts should affect my first-order opinions: it is a question about P (the credences I should have), not about C (my actual credences). Stating the question more precisely: how is the value of P(Thesis) modulated by the varying values of P(P(Thesis) = t) for various t? Using only the resources already specified, here are a host of natural answers that we could give to this Two-Level Problem: A I: [P(q) = t] ! [P(P(q) = t) = ] If you should be t-confident of q, you should be certain that you should be tconfident of q. G A: [P(q)  t] ! [P(P(q)  t)  t] If you should be at least t-confident of q, you should be at least t-confident that you should be at least t-confident of q. D JJ: [P(P(q)  t)  s] ! [P(q)  t·s] If you should be at least s-confident that you should be at least t-confident of q, you should be at least t·s-confident that q. R: P(q|P(q) = t) = t Conditional on the rational credence in q being exactly t, you should adopt credence exactly t in q. S T: P(q|P(q)  t)  t Conditional on the rational credence in q being at least t, you should adopt a credence of at least t in q. ⁸ I follow Elga () in the “modesty” terminology; note that it is orthogonal to the sense of “immodesty” used in the epistemic utility theory literature (Lewis ).

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   The goal of the Two-Level Problem is to assess principles like these for plausibility and tenability. Do they allow rational modesty? If so, do they nevertheless enforce plausible connections between first- and higher-order attitudes—or do they let such attitudes split radically apart? The answers are often surprising. A I obviously rules out higher-order uncertainty, and implies each of the other principles (by trivializing them). Surprisingly, R implies that you must always be certain of A I (as we will see in section .). G A rules out most cases of higher-order uncertainty (see Williamson, this volume). On the other hand, Simple Trust implies D JJ, and both of these principles allow massive amounts of higher-order uncertainty—meaning that S T is much weaker than R. It is not my aim here to explain or justify these particular assessments of these particular principles (cf. Dorst ). I mention them to give a sense of the terrain— for my aim is to explain why the Two-Level Problem is a fruitful and tractable strategy for exploring the notion of higher-order evidence. To do that, I need to do three things. First, I need to address the foundational questions of how to model and interpret rational higher-order uncertainty (section ..). Second, I need to argue that the solution to the Two-Level Problem is not the trivial one given by A I—that higher-order uncertainty is often rational (section .). Finally, I need to argue that the solution to the Two-Level Problem is not the obvious one given by the Enkratic Intuition or R (section .).

.. Modeling it We want to model a particular agent (say, me) at a particular time (say, now) who’s uncertain about a particular subject matter (say, Thesis). I should be uncertain about Thesis. How do we model that? By saying that I should match my opinions to a probability function that’s uncertain of which world it’s in— it assigns positive probability to Thesis-worlds and positive probability to :Thesisworlds. I should also be uncertain about whether I should be confident of Thesis. How do we model that? The same way. By saying that I should match my opinions to a probability function that’s uncertain which world it’s in—it assigns positive probability to I-should-be-confident-in-Thesis-worlds, and positive probability to I-shouldnot-be-confident-in-Thesis-worlds. That is, just as Thesis expresses a proposition, so too I should be confident of Thesis expresses a proposition. What we need is a systematic way to represent such propositions. Here’s how.⁹ Let W be a (finite) set of epistemic possibilities that capture the distinctions relevant to my scenario. Propositions are modeled as subsets of W. Truth is modeled as membership, so q is true at w iff w 2 q. Logical relations are modeled as set-theoretic ones, so: :q = W – q; q ∧ r = q \ r; etc.

⁹ I’m drawing on the probabilistic epistemic logic literature, though it usually assumes you know your own probabilities (cf. van Ditmarsch et al. ); some exceptions: Samet (); Williamson (, , ); Lasonen-Aarnio (); Salow ().

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- 



C is a definite description for my actual degrees of confidence—whatever they are. It can be modeled as a function from worlds w to credence functions Cw—for simplicity, suppose Cw is always a probability function over W. (Note that while C is a definite description that picks out different functions at different worlds, Cw is a rigid designator for the credence function I have at world w.) Using it we can define propositions (subsets of W) about what I actually think. For any proposition q  W and t 2 [; ], let [C(q) = t] be the proposition that I’m actually t-confident of q: [C(q) = t] =df {w|Cw(q) = t}.¹⁰ P is a definite description for the degrees of confidence I should have—whatever they are. It too can be modeled as a function from worlds w to probability functions Pw over W, thought of as the credences I ought to have at w. What’s crucial for modeling higher-order uncertainty is that we can use P to define propositions about what I should think. For any proposition q  W and t 2 [, ], [P(q) = t] is the proposition that I should be t-confident of q: [P(q) = t] =df {w|Pw(q) = t}. Since we have identified facts about rational credences as propositions (sets of worlds), your (rational) credences are thereby defined for any higher-order claim about what credences you should have—that is, (rational) higher-order opinions fall right out of the model. In sum, we can model my epistemic situation with a credal-probability frame ⟨W, C, P⟩ capturing the relevant possibilities (W), what I actually think in those various possibilities (C), and what I should think in those various possibilities (P). I know that I should have credences that match P. I also—perhaps—know what my actual credences are. Higher-order uncertainty slips in because I may not know whether what I actually think (C) lines up with what I should think (P). If we assume that rational agents know their actual credences, such higher-order uncertainty can be rational iff there can be agents who are in fact rational—[C = P]—but who are modest: they are not certain that they are rational—[C(C = P) < ]. To get a grip on how this machinery works, let’s construct a toy model of my case: I’m confident of Thesis, but I am uncertain whether I’m rational to be confident of Thesis. Suppose I know that I should either be . or . confident of Thesis. Letting T abbreviate Thesis, here is what we would like to say about my case: () I should be sure that I should either be . or . confident of Thesis: [P([P(T) = .] ∨ [P(T) = .]) = ]. () In fact I should be . confident of Thesis: [P(T) = .] () I should leave open that I should be ., but also leave open that I should be .: [P(P(T) = .) > ] and [P(P(T) = .) > ]. () I’m in fact . confident of Thesis, and I should be certain that I am: [C(T) = .] and [P(C(T) = .) = ]. () My credences are in fact warranted by my evidence: [C = P]. Figure . is a credal-probability frame that makes ()–() true at worlds a and c.

¹⁰ Similar definitions apply to other claims about my confidence, e.g., that I’m more confident in q than r: [C(q) > C(r)] =df {w|Cw(q) > Cw(r)}.

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  

Thesis

〈.6, .1, .15, .15〉 a

〈.4, .2, .1, .3〉 b

c 〈.6, .1, .15, .15〉

d 〈.4, .2, .1, .3〉

[C = Pa]

Figure . Thesis Uncertainty

There are four relevant epistemic possibilities: W = {a,b,c,d}. Thesis is true at worlds a and b, so Thesis = {a,b} (hence :Thesis = {c,d}). Since I know my actual credences, C is a constant function: at each world w, Cw matches the credences that are rational at world a; Cw = Pa for all w. (Indicated by the label for the shaded region covering all worlds.) The sequences next to each world w indicate the credences I should have in the various possibilities, in alphabetical order. So the “⟨., ., ., .⟩” next to a indicates that Pa(a) = ., Pa(b) = ., Pa(c) = ., and Pa(d) = .. This in turn specifies the rational credences to have in any proposition by summing across worlds: Pa(T) = Pa({a,b}) = Pa(a) + Pa(b) = .. By our definitions, [P(T) = .] = {w| Pw(T) = .} = {a,c}—at a and c I should be . confident in my thesis—while [P(T) = .] = {b,d}—at b and d I should be .. Here’s the crucial point. At worlds a and c I should be . confident of Thesis. Yet at those worlds I should also assign positive credence to b and d—where I should instead be . confident of Thesis. This means I should have higher-order uncertainty: I should be . confident of Thesis, but I should leave open that I should instead be . confident of it. Precisely, ()–() are true at worlds a and c for the following reasons: () [P(T) = .] = {a,c} and [P(T) = .] = {b,d}, so ([P(T) = .] ∨ [P(T) = .]) = W. So [P([P(T) = .] ∨ [P(T) = .]) = ] = [P(W) = ] = W. () [P(T) = .] = {a,c}. () Every world w is such that Pw({a,c}) >  and Pw({b,d}) > . So [P(P(T) = .) > ] and [P(P(T) = .) > ] are true everywhere. () [C(T) = .] is true everywhere since [C = Pa] = W and Pa(T) = .. Since [C(T) = .] = W, [P(C(T) = .) = ] = [P(W) = ] = W.¹¹ () [C = P] is true at {a,c} since [C = Pa] is true everywhere and [P = Pa] = {a,c}. ¹¹ Here is a very subtle point. At world b, the credence in Thesis warranted by my evidence is .. Nevertheless, at b the credence warranted in the claim that my actual credence in Thesis is . is . That is, at b: [P(T) = .] and [P(C(T) = .) = ]. Though puzzling, this is correct. For in b I can tell what my actual credences are—I have overwhelming evidence that my credence in Thesis is in fact ., thus my evidence warrants being certain of this claim. In b those credences are not warranted by my evidence—instead I should have . credence in Thesis. What this shows is that we can’t understand Pb as giving the credences that a rational agent at b would have. Arguably, no rational agent could have credence . in Thesis while having my same (overwhelming) evidence that she has credence . in it—in adopting credence . in Thesis she would make it so that she had different evidence than I have (cf. Salow ). In short, Pb captures the opinions that are warranted (rational) given my evidence—not necessarily the opinions that would be warranted if I were to conform to my evidence, since in so conforming I may change my evidence (e.g., my evidence about my beliefs).

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- 



This is the framework within which I’m proposing we study higher-order evidence: the framework of higher-order uncertainty, modeled using (credal-)probability frames. It provides the formal backbone to . Most of my argument will consist of putting it to work. Before doing so, two final notes. First, it does not presuppose anything about how the warranted credences at various worlds are related to each other (for instance, they do not have to be recoverable by conditioning from a common prior). Thus it is consistent with the view that higher-order evidence should lead you to “bracket” some of your information (Christensen a)—or in some other way provides counterexamples to conditionalization. Second, W is a set of epistemic possibilities. Thus there is no formal problem with using such a framework to model (higherorder) uncertainty about logic. If you are unsure whether an argument is valid, we can simply add epistemic possibilities where it is (isn’t)—so long as we treat the claim that the argument is valid as an atomic proposition, no formal problems will arise. Difficult interpretive questions will arise, of course—but those exist for all approaches to modeling logically non-omniscient agents. For an initial application, let me illustrate how we can use this sort of “totalevidence” framework to define notions that correspond fairly well to the intuitive ideas of what’s warranted by your first- and higher-order evidence. Intuitively, the opinions warranted by my first-order evidence (my arguments) are simply the opinions that someone who was fully informed about evidential matters—who has no doubts about what was warranted by my evidence—would think. The reason I’m unsure what my first-order evidence warrants (and, therefore, what my total evidence warrants) is that I do have such higher-order doubts about evidential matters. Thus to determine what my first-order evidence warrants, we can ask: what opinions would I be warranted in having if all my higher-order doubts were removed? That is, if I were to learn what the rational credence function was (i.e., what it was before I learned what it was), what would be the rational reaction to this information?¹² Let’s apply this thought to Figure .. Notice that the credence function warranted at world a assigns . probability to Thesis—but it does so, in part, because it has higher-order uncertainty: it assigns . credence to being at b or at d, where a different credence in Thesis is rational. In other words, the rational . credence is modulated by higher-order doubts. At a, what would the rational opinions be if my higherorder doubts were removed? Let P^ capture these opinions: P^w () = df Pw(|P = Pw), and [P^ (q) = t] =df {w|P^w(q) = t} (cf. Stalnaker ms). Since P^ captures what the rational credences would be if higher-order doubts were removed, it can plausibly be understood as what my first-order evidence warrants. In Figure ., [P = Pa] = {a,c}, so ∧ ½P ¼ Pa Þ a ðaÞ :6 ^ P^a(T) = Pa(T|P = Pa) = Pa ðT = PaPðfa;cgÞ = :75 = .. Hence [P(Thesis) = .] is Pa ðP ¼ Pa Þ ^ true at a and c, while a similar calculation shows that [P(Thesis) = .] is true at b and d. So at a and c the first-order evidence strongly supports Thesis (my arguments are good), while at b and d it actually tells against Thesis (my arguments are bad).

¹² Careful here. If I learn the values of P, and P had higher-order uncertainty, then I learn something that P didn’t know. Thus, as we’re about to see, the rational reaction to learning the values of P may be different from P itself (Elga ; Hall , ).

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   Yet both of these opinions are modulated by higher-order doubts to the more moderate opinions of . and .. We have the opinions warranted by your total evidence (P) and those warranted by ^ what about the opinions warranted by your higheryour first-order evidence (P); order evidence? Intuitively, the opinions warranted by your total evidence should be “factorable” into the various possibilities you leave open for what your first-order evidence warrants, and your higher-order opinions about how likely those possibilities are to be actual. To see this, consider how we might alternatively represent Figure .. There are two possibilities for what the first-order evidence warrants—{a,c} and {b,d}. In this frame, each world agrees on the probability distribution within such cells: conditioned on {a,c} or {b,d}, every Pw has the same distribution. The differences between the Pw are due to their distributions across such cells: the worlds in {a,c} are split – between {a,c} and {b,d}, while those in {b,d} are split –. Thus this frame can be equivalently represented using numbers within cells to indicate the firstorder support there, and labeled arrows between cells to indicate the probability that the (total) evidence gives to being in each cell. That yields Figure .. In this picture, the credence in Thesis warranted by the total evidence at world a can be calculated by averaging the support of the two first-order-evidence cells, with weights determined by how confident Pa is of each cell: Pa(T) = .  . + .  . = .. Similarly, Pb(T) = .  . +   . = .. In fact, the reason we can redraw Figure . as Figure . is precisely because in Figure . this equality holds generally: the rational credence in q equals the rational expectation of the credence in q warranted by the first-order evidence. What do I mean? The rational expectation EP ½X of a quantity X is a weighted average of the various possible values of X, with weights determined by how confident you should be in each. If you should be 13 confident I have  hats and 23 confident that I have  hats, then your rational expectation of my number of hats (the 1 2 number you P most expect to be near correct) is 3 ð3Þ þ 3 ð6Þ ¼ 5. Formally, EP ½X ¼ the quantity we’re estimating is the t ðPðX ¼ tÞtÞ. In the case at hand, P ^ ^ ^ first-order support for q, PðqÞ, so EP ½PðqÞ ¼ t ðPðPðqÞ ¼ tÞtÞ. Given this, the principle that allows us to redraw Figure 2.1 as Figure 2.2 is: ^ HF: PðqÞ ¼ EP ½PðqÞ The rational credence in q (P(q)) equals the rational expectation of: the credence in ^ q that’s warranted by the first-order evidence ðEP ½PðqÞÞ. .75

Thesis

.25

.8 a

b .4

.2 c

d .6

[Pˆ = Pˆa]

[Pˆ = Pˆb] .5

.5

Figure . First- and Higher-Order Support

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- 



I call this principle HF because it captures the idea that the opinions warranted by your total evidence are factorable into your Higher-order expectations of your First-order evidence. In particular, when this principle holds it is natural to identify the opinions warranted by your higher-order evidence as simply the distribution across first-order-evidence cells that is warranted by your total evidence, i.e. PðP^ ¼ P^w Þ for various w. (This distribution is represented in the labeled arrows between cells in Figure ..) The upshot of this discussion is that for probability frames that validate HF, we can define well-behaved precisifications of the idea that some opinions are warranted by your “first-order evidence,” and others are warranted by you “higher-order evidence.”¹³ Which frames do so? HF turns out to be equivalent to the N R principle proposed by Adam Elga (). N R starts with the observation that if your evidence warrants being uncertain of what your evidence warrants, then if you learn¹⁴ what your evidence warrants, you have gained new information that was not already entailed by your evidence. (If P(P = Pw) < , then P(·|P = Pw) is more informed than P(·).) So what should you do when you learn what opinions your evidence warrants? Elga says: adopt the opinions that your evidence would warrant if it were to be updated with what you’ve just learned. Precisely: N R: P(·|P = Pw) = Pw(·|P = Pw) Upon learning the opinions warranted by the evidence, react to this information in the way that you (now) know the evidence would warrant. N R sounds truistic. It is one way of making precise the idea that your opinions should be guided by your opinions about what your evidence warrants. And it is what allows us to “factor” your total evidence into first- and higher-order components: Fact . A probability frame ⟨W, P⟩ validates HF iff it validates N R.¹⁵ Nevertheless, there are objections to N R (Lasonen-Aarnio ). It is not my goal here to defend the principle, but instead merely to show that it represents a choice-point in our ability to vindicate a version of the first/higher-order evidence distinction. (And to argue—below in section ..—that N R and HF should not be seen as the solution to the problem of higher-order evidence.) This concludes my proposal for how to think about higher-order evidence—the details behind R. The rest of the chapter applies it. Section . defends the M T that higher-order uncertainty is often rational, while section .

¹³ Admittedly, I have not told you what it means for a proposition to be first- or higher-order evidence. I have no idea how (or whether) that can be done. ¹⁴ N R is strictly about conditional beliefs, not learning. For ease of exposition I’ll switch between talk of the two—but we could reformulate everything in terms of conditional beliefs. ¹⁵ Proof in the Appendix; cf. Stalnaker (ms). A probability frame is a credal-probability frame without C. A frame validates a principle iff it makes the principle true at all worlds for all instantiations on which it is well defined.

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   argues that the obvious principles for connecting first- and higher-order opinions do not succeed.

. Rational modesty In this section I argue for: M T: Your total evidence often warrants being uncertain what opinions your total evidence warrants, and (hence) being modest: uncertain whether you’re rational. Your evidence warrants higher-order uncertainty iff for some q and all t: P(P(q) = t) < . So long as you know what your actual opinions are, you have higher-order uncertainty iff you are modest, so I will treat modesty and higher-order uncertainty together. Isn’t it obvious that we often do—and should—have such self-doubts? Intuitive cases abound. Bias: I’m inclined to think that Kim’s job talk wasn’t great; but, knowing the literature, I have good reason to suspect that I have implicit bias against her—I’m probably underappreciating her talk. Impairment: the answer to the test’s “challenge problem” seems obvious; but I’m running on four hours of sleeping—I’m probably missing something. Disagreement: I thought the evidence supported the defendant’s innocence; but you thought it supported his guilt—perhaps I’ve misassessed it. And so on. Clean cases can also be found (Christensen a; Elga ; Schoenfield ):  Flying your plane, you’ve done some reasoning and become confident that , feet is a Safe altitude (Safe). Then over the radio you’re told there’s a good chance you’re hypoxic, in which case your opinions may be slightly irrational. You know, given all this information, that you should be either somewhat or fairly confident of Safe. In fact you are fairly confident of Safe. Isn’t it obvious that in H you should be uncertain whether () your fair confidence is rational, or () you should instead be only somewhat confident in Safe? That is, isn’t it obvious that you shouldn’t be certain of what is warranted by your total evidence (including the radio announcement)? From one perspective, it certainly seems so. Being rational is hard. Very often we don’t live up to the challenge. We know this about ourselves. So very often we should think that maybe right now we’re not living up to the challenge—we should be unsure what it’s rational to think. But from another perspective, to admit such rational higher-order uncertainty is to give up the game. For—the thought goes—getting to the truth is the hard part, and the job of epistemology is to provide us with in-principle-accessible rules that ensure we do the best we can. If we allow higher-order uncertainty, we will have to deny a form of this “in-principle-accessible” claim: A I: [P(q) = t] ! [P(P(q) = t) = ] If you should be t-confident of q, you should be certain that you should be t-confident of q.

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- 

Safe

〈.6, .1, .15, .15〉 a

〈.4, .2, .1, .3〉 b

c 〈.6, .1, .15, .15〉

d 〈.4, .2, .1, .3〉



[C = Pa]

Figure . Hypoxic Uncertainty

To deny this principle is to say that sometimes you are required to have an opinion even though you can’t be sure that you are so required. This can seem unacceptable: failing to live up to requirements is grounds for criticism; how could you be legitimately criticized if you couldn’t tell what was required of you? Those attracted to this line of thought will want a different way to think about our cases. Let’s focus on H. Suppose that in this context you’re fairly confident iff your credence is ., and you’re somewhat confident iff your credence is .. Then the natural reading of the case is that you should be uncertain whether the rational credence in Safe is . or .: [P(P(Safe) = .) > ] and [P(P(Safe) = .) > ]. We can use the same model of me wondering about Thesis (from section ..) to model you wondering about Safe (Figure .). All of the above discussion applies equally well to this H case—you’re . confident of Safe and should be sure that you are, you are (and should) be unsure whether you should instead be ., etc. Question: is there a recipe for generating an internalist-friendly reading of cases like this? The main strategy I know of goes as follows.¹⁶ It’s intuitive to say, of a case like H, that “You should be uncertain of what you should think.” If we interpret both those “should”s in the same way, then this says you should have higher-order uncertainty. But we needn’t interpret them that way. Instead, we can interpret them as picking out different normative notions: there’s () what you should think given your cognitive imperfections, and () what you “should” think in the sense of what an ideal agent (with your evidence) would think. Thus the true reading of the sentence is: “You should (given your imperfections) be uncertain of what you should (ideally) think.” Moreover, you should (ideally) know what you should (ideally) think; and you should (given your imperfections) know what you should (given your imperfections) think. Instead of higher-order uncertainty within a normative notion, these cases reveal first-order uncertainty across normative notions. So far, so fair. But one more bit of explanation is needed: Why does each normative notion have no higher-order uncertainty? It’s not too hard to get a sense for why you should ideally be certain of what you should ideally think—ideal agents are special, after all. But what explains why you should given your imperfections be certain of what you should given your imperfections think? The line of reasoning from above is

¹⁶ The strategy is inspired by Stalnaker (ms)—though he may not agree with my formulation of it.

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   just as intuitive. Properly accounting for our imperfections is hard. Very often we don’t live up to the challenge. We know this about ourselves. So very often we should (given our imperfections) think that maybe right now we’re not properly accounting for our imperfections—we should (given our imperfections) be unsure what we should (given our imperfections) think. We should be modest. An internalist may reply as follows. When we are uncertain of what we should (ideally) think, what we should (given our imperfections) do is to match our opinions to our expectation of what we should (ideally) think. Since we know this, we do know what we should (given our imperfections) think. This strategy faces a dilemma. To illustrate, interpret “you should (ideally) have credence t” as “your first-order evidence warrants credence t,” as defined in section ^ ... On that definition, the first-order evidence warrants t-confidence in q ðPðqÞ ¼ tÞ iff the rational credence to have once your higher-order doubts are removed is t. What ^ It goes as follows: happens if we run the internalist reasoning using P and P? () You should (given your imperfections) be uncertain of what you should ^ (ideally) think about Safe: PðPðSafeÞ ¼ tÞ ]

Third: since you know that Judy can only get to the truth via her evidence, if you learn¹⁹ what credence she should have in Liable, then further learning what credence she actually has doesn’t provide any evidence for or against Liable. So learning which credence Judy should have in Liable should screen off her actual credence: Screening:

For all t,s: Py(L|[Pj(L) = t] ∧ [Cj(L) = s]) = Py(L|Pj(L) = t)

Finally: if you should have a given credence t, then upon learning that Judy’s credence is lower than that, you shouldn’t simply ignore her opinion. Budge:

For all t: if Py(L) = t, then Py(L|Cj(L) < t) 6¼ Py(L)

These premises jointly imply that you should have higher-order uncertainty: Fact . If Same, Disagree, Screening, and Budge are true at a world in a probability frame, so too are [Py(Py(L) = t) > ] and [Py(Py(L) 6¼ t) > ] for some t.²⁰ Upshot: to respect the epistemic force of peer disagreement in cases where you know that you and your peer share evidence and have no special access to the truth, higher-order uncertainty must be rational. If this is right, the principled internalist stand is shattered. Internalists may respond by biting the bullet: grant that it is intuitive that in this case you shouldn’t ignore Judy’s disagreement, but insist that in fact you should. They may tell a debunking story: in most cases of peer disagreement, other dynamics

¹⁹ Again, read my talk of “learning” as shorthand for conditional beliefs. ²⁰ Proof: Given a probability frame ⟨W, Py⟩, suppose Same, Disagree, Screening, and Budge are true at w. (I won’t add functions Pj and Cj; assume that the relevant propositions obey the expected logical relations.) For reductio, suppose the consequent of Fact  is false; so for some t0 , [Py(Py(L) = t0 ) = ] is true. Recalling that probability frames are finite, by finite additivity, Disagree implies that there are values si < t such that Py(Cj(L) = si) > . By total probability, Py(L|Cj(L) < t) is a weighted average of the values of Py(L|Cj(L) = si) (with some weights possibly ); so to establish that Py(L|Cj(L) < t) = Py(L) it will suffice to show that Py(L| Cj(L) = si) = Py(L) for each si. Since Py(Cj(L) = si) > , by our hypothesis that Py(Py(L) = t0 ) = , we have Py(L|Cj(L) = si) = Py(L|[Py(L) = t’] ∧ [Cj(L) = si]). By Same, this equals Py(L|[Pj(L) = t0 ] ∧ [Cj(L) = si]). By Screening, this in turn equals Py(L|Pj(L) = t0 ). By Same again, this equals Py(L|Py(L) = t0 ), which by hypothesis equals Py(L). It follows that Py(L|Cj(L) < t) = Py(L), contradicting Budge.

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   are present which lead to rational conciliation—yet in the highly circumscribed cases described, these dynamics are gone. I do not think such a response can work. The cases cannot be contained: to get the correct verdicts in usual cases of disagreement, higher-order uncertainty must be rational. This is my second argument. Granted, rarely should you be certain that the credence you ought to have in q is identical to the credence your peer ought to have. However, almost always you should consider it possible that you and your peer will disagree in a way that implies that one of you was irrational. Maybe your opinions will be very far apart (too far apart); maybe the manner in which you disagree will reveal that you were thinking very differently (too differently) about a piece of evidence; maybe something else. Suppose you are on a jury with an equally smart, equally informed peer Pete. All the evidence has been presented, but you have not yet convened to share your opinions. Let g be the proposition that the defendant is guilty. Let “P y ” and “P p ” be definite descriptions for the opinions you and pete should have (before convening). Let “C y ” and “C p ” be definite descriptions for the opinions that you and pete actually have, before convening. Let Disagree be the proposition that one of your opinions (before convening) was irrational, i.e. Disagree =df ([Cp(g) 6¼ Pp(g)] ∨ [Cy(g) 6¼ Py(g)]). Three premises: First: Before you convene with Pete, you should leave open that you two will disagree. (After all, you can’t be certain that Pete is rational.) Open:

P y(Disagree) > 

Second: Since we are responding to A I, we may safely assume that you should be certain of your actual credence that the defendant is guilty. Actual:

[C y(g) = t] ! [P y(C y(g) = t) = ]

Third: What should you think if you learn that you and Pete do disagree? You two are equally smart—hence, initially, equally likely to be (ir)rational. If you disagree, it follows that one of you was irrational. It would be arbitrary (and immodest) to assume that it must have been him. So upon learning Disagree, you should not be y certain that Pete was irrational, nor certain that you were. Let “PD ” be a definite description for the rational credences you should have upon learning that you and y Pete disagree, i.e. PD ðÞ = df P y(|Disagree). Then: Uncertain:

y

y

PD ðC p ðgÞ 6¼ P p ðgÞÞ < 1 and PD ðC y ðgÞ 6¼ P y ðgÞÞ < 1.

If Open, Actual, and Uncertain correctly describe your scenario, it follows that before you convene with Pete you should have higher-order uncertainty: Fact . If Open, Actual, and Uncertain are true at a world in a credal-probability frame, so too are [P y(P y(g) = t) > ] and [P y(P y(g) 6¼ t) > ] for some t.²¹ ²¹ Proof: Given a credal-probability frame ⟨W, C y, P y⟩, suppose Open, Actual, and Uncertain are all true at some world. (Again, I won’t formalize C p and P p; assume they obey the expected logical relations.) There will be some t for which [C y(g) = t] is true. We first show that P y(P y(g) = t) > . Notice that P y(Disagree ∧ y [P y(g) = C y(g)]) = P y(Disagree) PD ðPy ðgÞ ¼ Cy ðgÞÞ. By Open, the first multiplicand is > . The second y multiplicand equals   PD ðPy ðgÞ 6¼ Cy ðgÞÞ > 0. Since by Uncertain the subtracted term is < , it follows

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- 



If I am right that these premises correctly describe your scenario in typical cases of peer disagreement, it follows that such cases involve higher-order uncertainty. I believe the only way to resist this argument is to deny Uncertain—claiming that in cases where you and a peer share similar evidence, if you learn that your peer disagrees with you (i.e. that one of you was irrational), then you must be certain that you were rational and they were not. This is a desperate move. For one, we can make the case more extreme. Suppose that we re-run the scenario thousands of times and that in the cases where you Disagree, each one of you is (ir)rational equally often. Now we run the experiment again, and you discover that you disagree. Should you be certain that this time it was your peer who was irrational? That seems absurd. For two, unlike my first argument, the scenario described by this second argument is pervasive—it happens every time you should think you might fundamentally disagree with a peer about morality, religion, or philosophy. If that’s right, the internalist stand is shattered. Rational higher-order uncertainty is possible—in fact, pervasive.

. Enkrasia So far I have defended R (that we should think of higher-order evidence in terms of higher-order uncertainty—as a Two-Level Problem) and a M T (the Two-Level Problem is nontrivial—higher-order uncertainty can be rational). If this is right, the Two-Level Problem is well-formed and nontrivial. But does it have a simple solution? Many have seemed to suggest so. They point out that the following states seem to be irrational: () believing that my Thesis is true, but I shouldn’t believe it; () being confident that my Thesis is true, but I shouldn’t be confident of it; and () believing Thesis while being agnostic on whether that belief is rational. The inferred explanation has standardly been that rationality requires your first-order opinions to “line up” with your higher-order ones—that your first-order opinions must be sanctioned (or, at least, not disavowed) by your higher-order opinions. Call this the Enkratic Intuition. Many theories of higher-order evidence have been built on top of it.²² The Enkratic Intuition can be given a precise characterization within the higher-order-uncertainty framework. So if it is correct, our Two-Level Problem admits of a simple solution. But it is not correct. Here I defend (cf. Titelbaum ): A:

If modesty is rational, so too is epistemic akrasia.

y

that PD ðP y ðgÞ ¼ Cy ðgÞÞ>0 as well. Combined, we have that P y(Disagree ∧ [P y(g) = C y(g)]) > , and so P y(P y(g) = C y(g)) > . Since by Actual and our supposition that [C y(g) = t] we have P y(C y(g) = t) = , it follows that [P y(P y(g) = t) > ] is true. Next we show that [P y(P y(g) 6¼ t) > ] is also true. By parallel reasoning (through Open and Uncertain), we have that P y(Disagree ∧ [P p(g) = C p(g)]) > . Note that every world in which Disagree ∧ [P p(g) = C p(g)] is true is one in which [P y(g) 6¼ C y(g)]; thus P y(P y(g) 6¼ C y(g)) > . Since P y(C y(g) = t) = , we’ve established that ½Py ðPy ðgÞ 6¼ tÞ 6¼ 0, as desired. ²² Including Feldman (); Gibbons (); Christensen (b); Huemer (); Smithies (, ); Greco (); Horowitz (); Sliwa and Horowitz (); Titelbaum (); Worsnip (); Littlejohn (); Rasmussen et al. (); Salow (), and perhaps Vavova ().

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   My strategy is as follows. I’ll first argue that the Enkratic Intuition has a precise consequence for the relationship between your credence in q and your opinions about the rational credence in q. Then in section .. I’ll show that this consequence is inconsistent leaving open the rationality of higher-order uncertainty. Since we often should leave open that higher-order uncertainty is rational, we should often be akratic. This does not show that ()–() can be rational, of course. What it shows is that if they can’t be, a different explanation is needed. What does the Enkratic Intuition imply about the relationship between first- and higher-order credences? Suppose you’re . confident that it’ll rain tomorrow, yet you’re symmetrically uncertain whether this credence is overconfident, just right, or underconfident: C(Rain) = ., while: C(P(Rain) = .) = . C(P(Rain) = .) = . C(P(Rain) = .) = . Then your . credence seems perfectly well sanctioned by your higher-order opinion—the “pressure” from your higher-order beliefs to change your first-order credence is balanced. Why is that? Here’s a helpful metaphor (cf. Bertsekas and Tsitsiklis , p. ). The Enkratic Intuition suggests that to the degree you think the rational credence is higher than yours, that should “pull” your credence upward; and to the degree you think the rational credence is lower than yours, that should “pull” your credence downward. So imagine a bar labeled “” at one end and “” at the other is resting on a fulcrum: 0

1

Now imagine attaching a block on each spot t 2 [, ] along the bar with weight proportional to C(P(q) = t)—your credence that the rational credence in q is t. This will tip the scale: 1

0

Question: where would you have to put the fulcrum to balance the scale, so that it’s not leaning left or right? (Where would you have to put your credence to balance the pull of your higher-order doubts?) Answer: you must place it at the center of gravity c:

0

1

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- 



This is the point at which the sum of the torques from the weights to the left of the fulcrum (t < c) is equal to the sum of the torques from the weights to the right (c < t). The torque of a block weighing C(P(q) = t) at a distance |tc| is simply the weight timesPthe distance: C(P(q) = t)  |tc|. A bit of algebra shows that this means that c = t(C(P(q) = t)  t). This formula should be familiar: it is the mathematical expectation of P(q), calculated relative to C. So our metaphor leads to the plausible conclusion that your credence in q is enkratic only if it equals your expectation of the P rational credence in q. Recalling (from section .) that EC ½X ¼ df t(C(X = t)t) is your actual expectation of X, we have: E:

Your credence in q is enkratic only if C(q) = EC ½PðqÞ

I claim that E captures what the Enkratic Intuition requires of your degrees of belief.²³ Of course, E is a precisification of a intuitive principle that has primarily been motivated by appeal to the irrationality of certain outright beliefs, so there is a risk of a terminological impasse here. My point is simply that the intuitive motivation for the claim that believing p but I shouldn’t believe it is irrational is equally motivation for E. Thus if E must be rejected—as I’ll argue it must—then we cannot look to the Enkratic Intuition to solve the Two-Level Problem. Suppose E is correct. Then enkrasia is a rational norm only if the rational credence in q (P(q)) equals the rational expectation of the rational credence in q ðEP ½PðqÞÞ: R E: P(q) = ðEP ½PðqÞÞ Notice that this is not HF, for we are here calculating the expectation of the rational credence—not the expectation of the credence warranted by the first-order evidence. (More on this in section ...) R E is a consequence of the simple R principle discussed above: that upon learning that the rational credence in q is t, you should adopt credence t in q; P(q|P(q) = t) = t. So what do R E and R require?

.. Enkratic? Immodest They require that you be certain that you should be immodest. If R E is true, that’s because it is a structural requirement that helps constrain the rational response to higher-order doubts. So if enkrasia is a rational requirement and modesty is rational, it should be possible to know that enkrasia is a rational requirement while being modest. (We shouldn’t endorse a view on which the only times you can be modest are when you are uncertain whether you should be enkratic.) Problem: it turns out that if an agent knows that they should be enkratic, they must be certain that they should be immodest. Recall that a principle is valid on a probability frame iff it is true at all worlds for all instantiations of its free variables.

²³ This conclusion is not original to me (Christensen b; Sliwa and Horowitz ; Rasmussen et al. ; Salow ).

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   If you know R E, it must be valid on the probability frame that represents your epistemic situation. What do such frames look like? Letting Sq be the proposition that you should be Sure of q, that is, Sq =df [P(q) = ]: Fact  (Samet ). If a probability frame validates R E, then it validates S([P = π] $ S[P = π]).²⁴ [P = π] $ S[P = π] is true for all instantiations of π at a world w iff whatever the rational credence function is at w, you should be certain that it’s the rational credence function—iff you should be immodest. So to say the frame validates S([P = π] $ S[P = π]) is to say that you should be certain that you should be immodest. So if R E is correct, anyone who knows that it is must be certain that they should be immodest. But they shouldn’t be certain that they should be immodest. So R E is incorrect. Why does R E rule out higher-order uncertainty? The basic reason was discussed in conjunction with Elga’s () N R principle above. When higher-order doubts are rational, then learning what the rational credences are provides new evidence, and so changes the rational credences (Elga ). Metaphorically, R E enjoins you to aim at a moving target. To see why, consider the following case. You and Selena share evidence but disagree: Selena is self-confident—she’s . confident that she’s rational—while you are modest—you’re – on whether she or you is rational. Schematically, if s is the set of possibilities where Selena is rational, y is the set where you are, and C y is a definite description for your actual credences, we have Figure .. This frame illustrates why higher-order uncertainty is incompatible R E. If you’re rational, then the rational credence that Selena is rational (P(s)) is .. This is equivalent to the rational expectation of the truth-value T(s) of s ( or ): at y, P y(s) = EPy ½TðsÞ = ·P y(s) + ·P y(y) = .. In contrast, the rational expectation of the rational credence in s is: EPy ½PðsÞ = . · P y(P(s) = .) + . · P y(P(s) = .) = . · P y(s) + . · P y(y) = .. This expectation is higher than your expectation of the truth-value of s due to the fact that the rational credence is affected by higher-order doubts. You think it’s . likely that Selena is rational—in which case the rational credence of . is slightly below the truth-value of ; but you also think it’s . likely that you’re rational—in which case the rational credence of . is well above the truth-value of . These two divergences are asymmetric, so they do not cancel out—which is why your credence of . that Selena is rational is below your expectation of . for the rational credence. The crucial point that given higher-order uncertainty, you should not be s 〈.9, .1〉

y 〈.5, .5〉 [C y = Py]

Figure . Simple modesty ²⁴ This result is due to Samet () (although he has a different intended interpretation of P). In the Appendix I give a less mathematically involved proof than his.

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- 



trying to get your credence as close to the rational credence as you can—for sometimes the rational credence is modulated by higher-order doubts. This is subtle, but can become intuitive. What is not intuitive—but what Fact  shows—is that the example generalizes completely: there is no way to “balance” your higher-order doubts to respect the requirements of  , short of the trivial case of higher-order certainty. I do not know of a way of making this result seem intuitively obvious. And I take that fact to be evidence that higher-order uncertainty is subtle, and that we do well to explicitly test our principles in a model theory like that of probability frames.

.. A reply? Perceptive readers sympathetic to the Enkratic Intuition may wonder: if the problem with “trying” to get close to the rational credence is that it’s plagued with higherorder doubts, why don’t we reformulate the enkratic requirement to aim at what your first-order evidence supports? Doing so would yield HF: ^ HF: PðqÞ ¼ EP ½PðqÞ The rational credence in q (P(q)) equals the rational expectation of: the credence in ^ q that’s warranted by the first-order evidence ðEP ½PðqÞÞ. As discussed above, this principle permits higher-order uncertainty. Why isn’t it the proper precisification of the Enkratic Intuition? Because it does not explain the cases that motivate that intuition. Granted, it does explain why attitudes like being confident that it’ll rain, but my first-order evidence doesn’t support that are irrational. But it doesn’t explain why the following attitudes are irrational: () being confident that it’ll rain, but my total evidence doesn’t support that or () being very confident that it’ll rain, but my total evidence warrants being very confident it won’t. If any attitude is epistemically akratic, these ones are. Yet HF allows them. Example: two sycophants, Sybil and Phan, are each confident that the other person is the rational one. Sybil is . confident that Phan is rational, while Phan is . confident that Sybil is. If these opinions could be rational, we’d have a probability frame like Figure . (s is the possibility where Sybil is rational, and p is the possibility where Phan is). Since both Ps and Pp agree on everything when they update on the claim that one of them is rational, this frame validates N R and HF. Yet it gives rise to a paradigm case of akrasia: Ps ðp ∧ ½Pð:pÞ  :9Þ  :9 At s they should be very confident that Phan is rational but we should be very confident that he’s not. Such an attitude looks akratic, if anything does. Any principle that allows it—like HF—cannot capture the Enkratic Intuition. s 〈.1, .9〉

p 〈.9, .1〉

Figure . Sycophants

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   The upshot is clear: the Enkratic Intuition cannot form the foundation of a theory of higher-order uncertainty. We need some other principle to explain the irrationality of such radical splits between first- and higher-order attitudes.

. Proposal I’ve defended three claims. First is R: the problem of higher-order evidence is best formulated as one of higher-order uncertainty—as a Two-Level Problem about how first- and higher-order opinions should relate. Second is M T: such higher-order uncertainty is often rational—the solution to the Two-Level Problem is not trivial. Third is A: the Enkratic Intuition is incorrect—the solution to the Two-Level Problem is not straightforward. Where does this leave us? There are many reasons to want a general, strong theory of higher-order uncertainty. Such a theory would provide a foundation to the epistemologies of disagreement, debunking, and self-doubt. It would do so by formulating law-like principles connecting rational first- and higher-order opinions. It would use a framework like ours to both show that such principles are tenable and illustrate their consequences. I have proposed such a theory elsewhere (Dorst ). But the framework proposed here is compatible with many, many alternatives. I hope to have done enough to show that it provides fruitful terrain—terrain that is well worth exploring.

Appendix Fact . A probability frame ⟨W, P⟩ validates H iff it validates N R. Proof. We prove a Lemma: in any frame, [P = Px] = ½P^ ¼ P^x . If y 2 [P = Px], then Py = Px, so P^y = Py(|P = Py) = Px(|P = Px) = P^x , so y 2 ½P^ ¼ P^x . If y 2 = [P = Px], then since P^y (P = Py) = , P^y (P = Px) =  while P^x (P = Px) = , so P^y 6¼ P^x , so y 2 = ½P ¼ P^x . Now suppose that N R is valid. Taking an arbitrary world w and proposition q, since {[P = Px]} forms a partition: X P ðP ¼ Px Þ  Pw ðqjP ¼ Px Þ ðtotal probabilityÞ Pw ðqÞ ¼ P w Xx ¼ P ðP ¼ Px Þ  Px ðqjP ¼ Px Þ ðNEW REFLECTIONÞ P w Xx ¼ P ðP ¼ Px Þ  P^x ðqÞ ðdefinitionÞ P w Xx ^ ¼ P ðP^ ¼ P^x Þ  P^x ðqÞ ¼ EPw ½PðqÞ ðLemma; definitionÞ P w x

So HF is valid. For the converse, suppose that N R is not valid, so there is a w, q, x such that Pw(q|P = Px) 6¼ Px(q|P = Px). Consider the proposition q ∧ [P = Px]. Pw(q ∧ [P = Px]) = Pw(P = Px)  Pw(q ∧ [P = Px]|P = Px) =

Meanwhile, EPw

Pw ðP ¼ Px ÞPw ðqjP ¼ Px Þ ^ ∧ [P = Px])] = Pw(P = Px)P^w(q ∧ [P = Px]| = [P(q Pw ðP ¼ Px ÞPx ðqjP ¼ Px Þ

ð1Þ ð2Þ

Since by hypothesis Pw(q|P = Px) 6¼ Px(q|P = Px), it follows that () 6¼ (), and hence that ^ ∧ [P = Px])], so HF is not valid. Pw(q ∧ [P = Px]) 6¼ EPw [P(q ☐

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- 



Fact  (Samet ). If a probability frame validates R E, then it validates S([P = π] $ S[P = π]). P I will write Ew [P(q)] =df x2W(Pw(x)Px(q)) for the values of the expectation of P(q) at w. Given a probability frame ⟨W, P⟩, there is an induced binary relation R such that wRx iff Pw(x) >  and Rw = {x|Pw(x) > }. Lemma .. If ⟨W, P⟩ validates R E, then R is transitive: if wRx and xRy then wRy. Proof. Suppose wRx and xRy but :(wRy). Since xRy, Px(y) > . Since wRx, Pw(x) > . Therefore Ew [P(y)]  Pw(x)Px(y) > . Since by hypothesis :(wRy), Pw(y) = , contradicting R E at w. ☐ Lemma .. If ⟨W, P⟩ validates R E, then R is shift-reflexive: if wRy, then yRy. Proof. Suppose the frame validates R E, so by Lemma . R is transitive. Suppose for reductio that wRy but :(yRy). Since R is transitive, for all zi 2 Ry, Pzi(Ry) = . And since :(yRy) but Py(Ry) =  and Px(y) > , we have Ew ½PðRy Þ  ¼

X z 2RZ

Xi

zi 2Ry

ðPw ðzi ÞPzi ðRy ÞÞ þ Pw ðyÞPy ðRy Þ X Pw ðzi Þ1 þ Pw ðyÞ1 > P ðz Þ ¼ Pw ðRy Þ z 2R w i i

Z

☐

Contradicting R E at w.

Lemma .. If ⟨W, P⟩ validates R E, then R is shift-symmetric: if wRy and yRz, then zRy. Proof. Suppose the frame validates R E, and so by Lemmas . and . R is transitive and shift-reflexive. Suppose for reductio that wRy and yRz but :(zRy). By transitivity, all zi 2 Rz are such that Pzi(Rz) = . By shift-reflexivity, Py(y) >  and z 2 Rz, so Py(Rz) > . Finally, y 2 = Rz. Combining these facts: Ey ½PðRz Þ  ¼

X z 2RZ

Xi

zi 2RZ

ðPy ðzi ÞPzi ðRz Þ þ Py ðyÞPy ðRz ÞÞ X Py ðzi Þ1 þ Py ðyÞPy ðRz Þ> z 2R Py ðzi Þ ¼ Py ðRz Þ i

Z

☐

Contradicting R E at y. Lemma .. If ⟨W, P⟩ validates R E, then for all w 2 W: if wRy then Py(P = Py) = .

Proof. Suppose the frame validates R E, so by Lemmas . and ., R is transitive and shift-symmetric. Suppose for reductio that wRy but Py(P = Py) < . By transitivity, if yRz and zRx, then yRx; equivalently: if yRz then Rz  Ry. By shift-symmetry, if yRz, then zRy; so by transitivity Ry  Rz as well. Combined: if yRz, then Ry = Rz. Since Py(P = Py) < , there must be a proposition q such that Py(P(q) = t) <  for all t. Since W is finite, there is a set T = {t₁, . . . , tn} such that for all ti, Py(P(q) = ti) > , with at least two distinct ti 6¼ tj in T. Relabel so that t₁ < t₂ < . . . < tn. There must be some z 2 Ry such that Pz(q) = tn. By the above reasoning, Rz = Ry, meaning T is also the set of values s such that Pz(P(q) = s) > . Then: Ez ½PðqÞ ¼ ¼

P

ti 2T ðPz ðPðqÞ

P

ti . (i.e. Pr(p|e)
t >. ). The understanding of evidential support as incremental support and the understanding as absolute support are the two main understandings of evidential support in the literature. ² The presentation of his example is adapted here.

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p₂ (since p₁ supports p₂), and that Manuel knows p₂ supports p₃. It is still not true that p₁ supports p₃. (Adapted from Fitelson , pp. –) Formally, Fitelson’s counterexample against the EEE Slogan makes use of the fact that evidential support is not transitive: that p₁ supports Manuel knows p₂ and Manuel knows p₂ supports p₃ does not guarantee that p₁ supports p₃. By suggesting the right kind of interpretation of these propositions, Fitelsons turns this observation into a counterexample. In particular, formally the proposition Manuel knows p₂ is treated as if it were a first-order proposition p₂ with no special content. However, it is interpreted as a proposition about which evidence is available to an agent. Fitelson also discusses other specifications of the EEE Slogan that seem plausible but can be refuted by similar strategies and depend on the fact that, formally, evidence of evidence is treated as first-order evidence and that evidential support is not transitive. Since Fitelson published his counterexamples, epistemologists have focused on specifying conditions under which evidential support is transitive. This is motivated by the assumption that the EEE Slogan would be subject to the same conditions. For example, Roche () suggests—based on results from Bayesian confirmation theory that trace back to Keynes  and Shogenji —that evidence of evidence is evidence under screening-off. He makes this suggestion based on the fact that evidential support is transitive given such a condition.³ Clearly, if evidential support were transitive, we could not provide counterexamples such as Another Card Example. Tal and Comesaña () argue against Roche’s approach and suggest that the correct way of specifying the EEE Slogan is in terms of defeaters. Based on this idea, Tal and Comesaña provide one of the most precise and sophisticated accounts for hedging the EEE Slogan. Despite the sophisticated nature of their account, it too exemplifies what is going wrong in the present debate about the EEE Slogan: they too treat evidence of evidence as if it were first-order evidence. They do so at least in their formal reconstruction of cases involving evidence of evidence. In the following we study their account closely and learn from their mistakes.

.. Tal and Comesaña’s specifications of the EEE Slogan Tal and Comesaña start their approach with the following observation: [W]e believe that the correct way of fixing EEE principles is in terms [ . . . ] of defeaters. Notice a very interesting thing: when e is evidence for p which is evidence for q but e is at the same time a defeater for the support that p provides for q, the positive screeningoff condition will not be satisfied. That e is a defeater for the support that p provides for q means that e ∧ p is not evidence for q, whereas the positive screening-off condition requires that the support that e ∧ p gives to q not be lower than the support that p alone provides to q. If e ∧ p provides no support at all to q, then of course that support will be lower than the one that p alone provides given that p provides any positive support, as is required in all EEEs. We have therefore found a partial explanation of the evidential relevance of the positive screening-off condition: it rules out cases where e is itself a ³ For literature on the transitivity of evidential support given screening-off conditions see, for instance, Keynes , Roche , and Shogenji .

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 - .    ö defeater for the support that p provides to q. But given that the positive screeningoff condition is overkill, we should replace it with a no-defeaters condition. (Tal and Comesaña , p. )

The idea is this: if the agent’s evidence e supports the proposition p, that there is a proposition e* that is true and that supports q, and if e is not a defeater for the support that p provides to q, then e itself supports q to some positive degree. Based on this idea, they propose the following specification of the EEE Slogan: Tal and Comesaña’s EEE Suppose ‘S(p,q,+α)’ says that proposition p supports proposition q to some positive degree α and ‘T(p)’ says that p is true, then the following holds: S(e,9e*[T(e*)∧S(e*,p,+α)],+ β)∧S(e ∧9e*[T(e*) ∧S(e*,p,+α)],p,+ γ)! 9δS(e,p,+ δ) We shall argue that Tal and Comesaña’s EEE is incorrect and that their understanding of evidence of evidence is not helpful. Let us discuss Tal and Comesaña’s EEE by means of the following example:⁴ New Card Example Let p₁ be the proposition that c is a card in the suit of spades and p₂ be the proposition that card c is an ace. Now suppose there are two agents, Elisabeth and Andreas, who start with the same a priori credence function Pr. Then Elisabeth but not Andreas receives a piece of evidence e₁ saying that Vincent claims to know that c is an ace of spades, that is, Vincent claims to know p₁ ∧ p₂. Then, in a second step, Elisabeth and Andreas receive the proposition p₁ as evidence. Note that p₁ taken by itself is irrelevant for whether or not p₂. Before discussing the example in a formal way let us discuss it informally. First, intuitively, Elisabeth receives evidence of evidence for p₂ whereas Andreas does not. Second, intuitively, Elisabeth’s total evidence (i.e., e₁ ∧ p₁) is evidence for p₂, since it is evidence of evidence for p₂ (for this case we would like to employ the EEE Slogan); Andreas’s total evidence (i.e., p₁) is irrelevant for p₂ (the EEE Slogan is not applicable here). The reasoning behind both statements is the following. Elisabeth’s total evidence includes e₁: that Vincent claims to know that p₁ ∧ p₂ and the evidence p₁. These pieces of evidence together support that Vincent indeed has evidence for p₁ ∧ p₂ and that Vincent really knows p₁ ∧ p₂. The intuitions behind the EEE Slogan lead us to believe that Elisabeth possesses evidence of evidence for p₂ as well. Andreas only received evidence p₁ which is irrelevant for p₂ and the intuitions behind the EEE Slogan seem not to support that the obtained evidence is evidence of evidence. Now let us discuss this example in more detail, in terms of the Bayesian theory of evidential support. (We assume that the agents involved are rational from a Bayesian perspective.) We can make two observations here: first, note that both agents already know a priori that there is a proposition e* that support p₂ to some positive degree +α, simply because it implies the latter proposition. In particular, propositions such as p₁ ∧ p₂ and also Vincent knows (p₁ ∧ p₂) imply p₂ and, thus, support p₂ to some

⁴ Our discussion takes up an argument by Moretti (), which is worked out in more detail here.

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

positive degree +α. Let us use the log-likelihood measure⁵ to measure evidential support, then α = log[Pr(e|p₂)/Pr(e|p₂)] = 1, if e ⊨ p₂ (see Fitelson ). Thus, according to Elisabeth’s and also Andreas’s a priori credence function Pr: Pr(S(p₁ ∧ p₂, p₂, 1)) = , Pr(S(Vincent knows (p₁ ∧ p₂), p₂, 1)) =  and Pr(9e*[S(e*, p₂, 1)]) = . The only thing Elisabeth and Andreas do not know a priori is whether any of these propositions that support p₂ to degree 1 are also true. Second, note that the evidence that both Elisabeth and Andreas receive (i.e. p₁) provides evidential support for the truth of propositions that support p₂ to degree 1. In particular, p₁ (which we assumed to be irrelevant for p₂), supports p₁ ∧ p₂ (and T(p₁ ∧ p₂)), which implies that p₂. Thus, both agents receive evidence p₁ which supports the conjunction T(p₁ ∧ p₂) ∧ S(p₁ ∧ p₂, p₂, 1) and the corresponding existential statement 9e*[T(e*) ∧ S(e*, p₂, 1)]. For both agents, the evidence p₁ also supports something else. In particular, p₁ supports that Vincent knows that p₁ ∧ p₂. In particular Vincent knows (p₁ ∧ p₂) implies p₂ and p₁. Therefore, p₁ supports Vincent knows (p₁ ∧ p₂) and Vincent knows (p₁ ∧ p₂) supports p₂. (Arguably, however, Elisabeth receives much stronger evidence in favour of Vincent knows (p₁ ∧ p₂), after all Elisabeth’s total evidence includes the evidence e₁ that Vincent claims to know p₁ ∧ p₂.) In the following we present two problems that Tal and Comesaña have to face. The first problem for Tal and Comesaña’s EEE is this: intuitively, only Elisabeth receives evidence of evidence for p₂ and, thus, evidence for p₂. Andreas’s evidence should not be understood as evidence of evidence and the evidence received by Andreas should be irrelevant for p₂. According to Tal and Comesaña’s EEE, the opposite is true. Both agents receive evidence of evidence. Elisabeth receives the evidence e₁ ∧ p₁ and Andreas the evidence p₁. Both pieces of evidence support the existential claim 9e* [T(e*) ∧ S(e*, p₂, 1)]. Supporting such an existential claim is what distinguishes ordinary evidence from evidence of evidence, according to Tal and Comesaña’s EEE. This trivializes the characterization of evidence of evidence: all evidence is evidence of evidence for arbitrary p. This trivializes the notion of evidence of evidence for p. The second problem for Tal and Comesaña’s EEE is that they can hardly explain why Elisabeth’s evidence e₁ ∧ p₁ supports p₂, while Andreas’s evidence p₁ does not. We saw that both agents’ evidence supports the existential claim 9e*[T(e*) ∧ S(e*, p₂, 1)] (and they do so for the same propositions p₁ ∧ p₂ and Vincent knows (p₁ ∧ p₂)). Thus, they would need to argue that Andreas’s evidence is a defeater for the evidential support p₂ receives from the existential claim 9e*[T(e*) ∧ S(e*, p₂, 1)]. However, this is impossible: that there is a true proposition that implies p₂, implies that p₂ is true. Logical implication provides support that cannot be defeated (at least not in the present framework).⁶ Thus, according to Tal and Comesaña’s specification

⁵ The log-likelihood measure is one of the most prominent Bayesian confirmation measures in the literature. This is due to its flexibility, the possibility to factor out independent pieces of evidence, and its close relationship to the Bayes factor. See Brössel  and especially Fitelson ,  for a discussion of this measure. ⁶ We do not think that the original evidence p₁ functions in this case as higher-order evidence, à la Christensen , that calls our logical abilities into question, and thus would defeat the support p₂ receives from the existential claim.

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 - .    ö of the EEE Slogan we would expect that both agents receive evidence that supports p₂—this is contrary to what we have been arguing. Taken together, the above two problems show that Tal and Comesaña’s EEE is inadequate as a specification of our intuitive EEE Slogan. Although the discussion in Tal and Comesaña  enriches the debate tremendously by bringing to the foreground propositions that speak about evidential support, we still have nothing to help us explain the intuitive appeal of the EEE Slogan. It seems to us that the problem here is that we have been operating with an inadequate characterization of evidence of evidence. According to the characterization with which we have been operating, evidence e is evidence of evidence for p if e supports that there exists a true proposition e* which supports p. We believe the mistake is to allow the proposition e, the evidence of evidence for p, to be first-order evidence about the world, as long as e supports the existence of further true propositions that support p. But we have just seen that every proposition e supports the existence of true propositions that support p. Based on previous observations we propose a new characterization of evidence of evidence. Instead of characterizing evidence of evidence as first-order evidence and running the risk that one can show—yet again—that specifications of the EEE Slogan can be trivialized or shown to be inadequate, we propose to characterize evidence of evidence as higher-order evidence.

. Evidence of evidence as higher-order evidence In section ., we proposed to understand evidence of evidence as higher-order evidence. Now we need to present our detailed account of how to understand such higher-order evidence. In the present debate, we propose to understand evidence of evidence as higher-order evidence about the epistemic state of an agent. Thus, in a first step, we say more about how to model the epistemic states of agents. In a second step, we introduce candidates for characterizations of evidence of evidence.

.. Dyadic Bayesianism, higher-order evidence, and two assumptions ...   In Brössel and Eder  we argued that for social epistemological purposes it does not suffice to identify the epistemic state of an agent with the agent’s doxastic attitudes, like credences. To understand the dynamics of rational reasoning in social settings, we argued, we need a more finegrained framework for modelling the epistemic states of agents. We proposed such a framework, which we here refer to as Dyadic Bayesianism.⁷ This framework is also apt for present purposes. We introduce it first informally, and then in a formal way. According to Dyadic Bayesianism the following holds. First, the (rational) epistemic state of an agent is best modelled by a dyad consisting of (i) the agent’s (rational) reasoning, or confirmation,⁸ commitments—which reflect the justificatory ⁷ In Brössel and Eder  we refer to it as Pluralistic Bayesianism. ⁸ In Brössel and Eder  we use the term confirmational commitments. However, now we think that it is more apt to use the term ‘reasoning commitments’.

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import of the evidence and capture how the agent justifies her credences—and (ii) the agent’s total evidence. We refer to the latter component as the evidential state of the agent. Second, the agent’s (rational) credences equal her (rational) reasoning commitments conditional on the total evidence—which captures the idea that the agent uses her reasoning commitments to form her credences by conditionalizing on the evidence available to her. Finally, third, (rational) credences and (rational) reasoning commitments obey the probability calculus. This implies that the agents that we are considering are Bayesian agents, who have ideal epistemic states. (For the appropriate normative interpretation of these probabilities see Eder forthcoming.) Brössel (forthcoming) shows that the framework of Dyadic Bayesianism is fruitful for providing a satisfactory account of confirmation. In Brössel and Eder , we show that on the basis of Dyadic Bayesianism we can provide a powerful account for finding agreement among agents that initially disagree with each other. We show that with the account one can satisfy many requirements which alternative accounts that deal with peer disagreement do not satisfy. In this chapter we want to employ Dyadic Bayesianism to provide an account of evidence of evidence. Putting it in a more precise and formal way, Dyadic Bayesianism says the following: Dyadic Bayesianism An agent s’s (rational) epistemic state is () () ()

a dyad/ordered-pair ESs = hPrRs,tevsi consisting of (i) s’s reasoning commitments, PrRs, and (ii) s’s total evidence, tevs, such that s’s credences are as follows: PrCrs(p) = PrRs(p|tevs), and both PrCrs and PrRs obey the probability calculus.

... -  In this chapter we understand evidence of evidence as higher-order evidence. We characterize higher-order evidence as follows: Higher-Order Evidence A proposition e is higher-order evidence for some agent s if and only if there is some agent s* such that e describes, or evaluates, or guides the (formation of the) epistemic state of s*. Note first that we assume that judgements of the (ir-)rationality of epistemic states are evaluative or guiding in the sense relevant here. Furthermore, a consequence of the conception of higher-order evidence employed here is that propositions stating logical relations or evidential support relations are not higher-order evidence (provided the latter is understood as being independent of the agent’s reasoning commitments). Such propositions may become logically equivalent to higher-order evidence in our sense if we add certain presuppositions about the normativity or rationality of logic and evidential support. Without such presuppositions, however, they are not higher-order evidence because they neither describe, nor evaluate, nor guide an agent’s epistemic state.⁹ ⁹ Our Higher-Order Evidence corresponds to a narrow conception of higher-order evidence. There might be a broader conception according to which, for instance, evidence about logical or evidential support relations counts as higher-order evidence, independently of any additional presuppositions. In any

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 - .    ö Before we proceed and characterize evidence of evidence, we emphasize that it is one strength of Dyadic Bayesianism that we can distinguish different kinds of higher-order evidence that play an important role in epistemology. First, we can distinguish between higher-order evidence about evidence and higher-order evidence about reasoning commitments. The former describes, or evaluates, or guides the evidential state of an agent, the latter describes, or evaluates, or guides the reasoning commitments of the agent. And then there is higher-order evidence about evidence and reasoning commitments. In particular, higher-order evidence about the credences of an agent is such evidence. In our framework, the credences of an agent are (by definition) formed by updating her reasoning commitments on her available evidence. Information about that agent’s credences, thus, informs us about both her reasoning commitments and her evidence. Second, we can distinguish between intra- and interpersonal higher-order evidence. Intrapersonal higherorder evidence describes, or evaluates, or guides the epistemic state of oneself. Interpersonal higher-order evidence describes, or evaluates, or guides the epistemic state of another agent. In individual epistemology, intrapersonal higher-order evidence about one’s evidence plays an important role when we study the epistemic role of, for example, perception and memory. This kind of higher-order evidence is crucial for determining the input for our own reasoning. What it describes, or evaluates, or guides is not how we reason, but on what we (should or are permitted to) base our reasoning. Intrapersonal higher-order evidence about reasoning commitments in turn plays an important role when we want to study inferential reasoning. This kind of higherorder evidence describes, or evaluates, or guides how we reason, and thus is relevant for determining what credences or beliefs we (should or are permitted to) hold on the basis of our evidence. Interpersonal higher-order evidence is crucial in social epistemology. In Brössel and Eder  we discuss in detail how to revise one’s credence in the light of interpersonal higher-order evidence that describes the reasoning commitments of another agent and displays disagreement with the agent. In this chapter we place interpersonal higher-order evidence about the evidential state of another agent at the centre of our debate about evidence of evidence. We discuss this kind of higher-order evidence in detail in the following.

...   To isolate the question of how to deal with evidence of evidence from other issues in epistemology, we want to factor out other questions that are closely related. To this end we introduce two idealizing assumptions. First, we add an assumption about the evidence available to the agents. Following other authors working on the EEE Slogan (e.g., Tal and Comesaña ), we assume

case, most, if not all, of what we argue for would hold if we adopted a wider conception of higher-order evidence or rephrased Higher-Order Evidence in such a way that the right side of the biconditional expresses only a sufficient condition for higher-order evidence. We prefer to work with this more informative biconditional.

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

that the total evidence of agents does not include false propositions. This assumption simplifies our task considerably. Among others, we can ignore questions concerning whether one is justified in relying on the evidence available to other agents or to oneself. In addition, it ensures that the evidential states of two agents can always be combined to obtain a more comprehensive evidential state that is consistent. Thus, the assumption allows us to set aside the epistemological problems of perception and testimony.¹⁰ An obvious consequence of the possibility of combination is that the question of how to deal with specified evidence of evidence can be answered outright: if an agent s receives the higher-order evidence e that the total evidence of another agent s*, tevs*, contains the specified proposition e*, then s should update her evidential state by including e* in her total evidence. The remaining question is how to deal with evidence of unspecified evidence, namely evidence that states that some other agent has some evidence for p without specifying the exact evidential proposition that the other agent has. In the following we focus on evidence of unspecified evidence. Second, we make the idealizing assumption that both agents—the first agent s, who receives higher-order evidence of the second agent s*’s evidence, as well as s*, who has first-order evidence in support of a proposition—have the same reasoning commitments (i.e., PrRs = PrRs*) and that both of them are aware of this. For this reason, for the remainder of this chapter we drop the subscripts for the agents when we speak of reasoning commitments. A consequence of this assumption and Dyadic Bayesianism is that both agents agree in their credences when they share the same body of total evidence. (Remember: according to Dyadic Bayesianism an agent’s credence in a proposition equals the agent’s reasoning commitments with respect to the proposition conditional on the total evidence of the agent.) This allows us to ignore questions concerning whether one is justified in relying on some other agent’s reasoning capacities or rather on one’s own reasoning capacities, and whether it is possible that agents have different but equally rational reasoning commitments. It also allows us to ignore questions concerning how to deal with cases of peer disagreement, namely cases in which equally rational agents who share the same body of evidence assign different credences to the same proposition in light of the same body of evidence—questions such as: Should one be steadfast or change one’s credences in such a case? And if so, how should one revise one’s credences? We want to avoid these debates here, but at the same time acknowledge that a comprehensive theory of evidence of evidence—which we cannot present in this chapter—would engage with those questions. (We have already answered the latter question on how to change one’s credence when one is required to find agreement in Brössel and Eder .)

¹⁰ Perception is the primary source of information about the world. Testimony is the primary source of information about the evidence and beliefs of other agents. Given our assumptions we can ignore the question of whether and how strongly we are justified in accepting such perceptual and testimonial evidence. See, e.g., Brössel , Lyons , and Pryor  for a discussion of the epistemological problems of perception. See Adler  and Lackey and Sosa  for a discussion of the epistemological problems of testimony.

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 - .    ö

.. Evidence of evidence as higher-order evidence Now we are in a position to discuss possible characterizations of evidence of evidence as higher-order evidence that render the EEE Slogan correct.¹¹

...         With the above account of epistemic states to hand, we are in a position to introduce characterizations of evidence of evidence for some proposition p. Before we concentrate on evidence of evidence for some proposition p, let us consider what makes evidence into evidence of evidence. We submit the following characterization: A Characterization of EE A proposition e is (higher-order) evidence of evidence for some agent s if and only if there is some agent s* with epistemic state ESs* = hPrR,tevs*isuch that either: () there is some (specified) proposition e* such that e states that e* is included in the total evidence of s* (i.e. tevs* ⊨ e*) [evidence of specified evidence], or () e states that there is some (unspecified) proposition e* that is included in the total evidence of s* (i.e. tevs* ⊨ e*) [evidence of unspecified evidence]. Now we can turn to the question of when such evidence of evidence is evidence of evidence for some proposition p. The main idea is this: we obtain evidence e of evidence e* for a proposition p just in case e says that some agent s* possesses some piece of evidence e* that supports p. We are interested in when and why we are permitted or even ought to change our credence when we have evidence that some agent possesses a piece of evidence. Often, we receive such evidence about other agents and not about ourselves. Thus, we focus on evidence of evidence that is interpersonal higher-order evidence of evidence. Strictly speaking, evidence of evidence can be intrapersonal evidence of evidence too. For example, we might obtain higher-order evidence saying that we forgot an unspecified proposition e* that at an earlier time was implied by our evidence and that supported and still supports p. Arguably, this is evidence of evidence too. Nevertheless, in this chapter we restrict ourselves to interpersonal higher-order evidence about evidential states and ignore this possibility. (If one feels this restriction is too severe, then one can simply treat agents at different time points as if they were different agents.) As a first pass at the notion of evidence of evidence for p let us consider the following characterization. A Characterization of EE for p A proposition e is (higher-order) evidence of evidence for a proposition p for some agent s with epistemic state ESs = hPrR,tevsiif and only if there is some (other)¹² agent s* with epistemic state ESs* = hPrR,tevs*isuch that

¹¹ That we focus on this kind of evidence does not exclude that we are open to the possibility of there being different adequate characterizations of evidence of evidence. However, in this chapter we are interested in particular characterizations that make the EEE Slogan correct. For an alternative approach see Dorst and Fitelson ms. ¹² We add ‘(other)’ to emphasize our focus on interpersonal higher-order evidence.

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  



. there is some (specified) proposition e* such that e states that e* is included in the total evidence of s* (i.e. tevs* ⊨ e*) and that e* supports p (i.e. PrR(p|e*) > PrR(p)). [evidence of specified evidence for p], or . e states that there is some (unspecified) proposition e* that is included in the total evidence of s* (i.e. tevs* ⊨ e*) and that this e* supports p (i.e. PrR(p|e*) > PrR(p)). [evidence of unspecified evidence for p]. This seems to be an apt characterization of evidence of evidence for some proposition p if we want to characterize it in a very general manner. It is important to note, however, that A Characterization of EE for p certainly does not render the EEE Slogan correct. There are various reasons for this: some concern misleading evidence, others shared evidence, and some concern the fact that it is easy to acquire evidence that supports a proposition. We explain this in the following. According to A Characterization of EE for p, evidence of evidence that is known to be misleading evidence for a proposition is still evidence for the proposition. However, (higher-order) evidence of evidence for p that is known to be misleading evidence for p is certainly not evidence for that proposition. Consider the following example: Party Example Suppose you learn e₁, which states that Ann’s total evidence includes some unspecified evidence e* that supports the proposition p₁ that Ben will attend your party. However, Ben just called you to cancel because his partner has surprised him with concert tickets. In this case you do not take e₁ to be evidence for p₁. Instead you would take e₁ as what it is: higher-order evidence of Ann’s misleading evidence for p₁. Evidence of evidence for a proposition that is known to be misleading is certainly not evidence for the proposition: so the EEE Slogan is incorrect, given A Characterization of EE for p. There are also other cases that make clear that the EEE Slogan is incorrect when evidence of evidence for p is understood as suggested by A Characterization of EE for p. Consider the following case, which involves evidence of shared evidence: Another Party Example Suppose your total evidence supports the proposition p₁ that Ben will attend your party. You then obtain higher-order evidence e₁ that Ann has evidence that Ben will attend the party and that Ann and you share exactly the same evidence (including this higher-order evidence). You receive (higher-order) evidence of evidence for the proposition that Ben will attend the party. But this would not change your credences in the proposition. This higher-order evidence does not support the proposition and, intuitively, it should not increase your credence in the proposition. Hence, the EEE Slogan would be obviously incorrect when evidence of evidence for p is understood as suggested by A Characterization of EE for p. There is another way to see that the EEE Slogan cannot be correct when evidence of evidence for p is understood as suggested by A Characterization of EE for p. This is so because it is (too) easy to obtain (higher-order) evidence of evidence for a proposition p. In particular, when we receive evidence of evidence, almost always we receive evidence of evidence for p. Suppose you receive evidence that another

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 - .    ö agent s*’s total evidence tevs* implies a proposition e*. Then you also obtain evidence that this agent’s total evidence implies (e* ∨ p), which almost always supports p. ((e* ∨ p) is a logical consequence of p and logical consequences of some proposition always support it.) So almost always, evidence of evidence comes with some evidence of evidence for p, and we certainly do not want to say that all of this evidence is evidence for p, whatever p is. In addition, a characterization of evidence of the evidence of another agent should not be one where insignificant parts of this agent’s entire body of total evidence are taken into consideration, it should be evidence of evidence that is determinant for the agent’s current credence in that proposition. More specifically, given this evidence, the rest of the agent’s total evidence should be (probabilistically) irrelevant for the agent’s credence in the proposition in question.

...       As emphasized several times, the entire investigation here is devoted to answering when and why we may or should revise our credences in the light of evidence of evidence. For this reason, we are interested in higher-order evidence of new or unknown evidence for p, that is to say evidence of evidence for p that is part of the total evidence of some (other) agent, but that is unknown to us. For this reason we are also interested in evidence of evidence that is sufficiently significant for our credence in p, and ideally sufficiently relevant to determine our credence in p. Only when the unspecified evidence is sufficiently significant for p for the agent who possesses it can it be significant for the agent who merely receives higher-order evidence of the existence of such evidence for p. In the following we rephrase A Characterization of EE for p in line with the above considerations. The new characterization of evidence of evidence understands evidence of evidence as (higher-order) evidence which states that there exists an unknown piece of evidence for a proposition that is possessed by another agent. (Remember that A Characterization of EE for p does not require that the evidence is unknown to the agent who receives the evidence of evidence for p.) The other agent’s credence in the proposition given the agent’s total evidence is higher than one’s credence in it given one’s total evidence. This reflects that the other agent’s total evidence increases the support of the proposition in comparison to one’s total evidence. In accordance with this, we propose to revise the above characterization as follows: Our Characterization of EE for p A proposition e is (higher-order) evidence of unknown evidence for a proposition p for some agent s with epistemic state ESs = hPrR,tevsi if and only if there is some (other) agent s* with epistemic state ESs* = hPrR,tevs*i such that: . there is some (specified) proposition e* such that e states that proposition e* is included in the total evidence of s* (i.e. tevs* ⊨ e*) but not in the total evidence of s (i.e. tevs ⊭ e*) such that PrCrs*(p) = PrR(p|e*) > PrR(p|tevs) = PrCrs(p). [evidence of unknown specified evidence for p], or . e states that there is some (unspecified) proposition e* included in the total evidence of s* (i.e. tevs* ⊨ e*) but not in the total evidence of s (i.e., tevs ⊭ e*) such that PrCrs*(p) = PrR(p|e*) > PrR(p|tevs) = PrCrs(p). [evidence of unknown unspecified evidence for p].

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  



With this characterization to hand, we can study which specification(s) of the EEE Slogan can be turned into a tenable principle. Before doing so, let us explain why Our Characterization of EE for p captures one important and intuitive conception of evidence of evidence correctly. First, not just any evidence about the evidential state of (another) agent is evidence of evidence in the above sense. Given the above characterization, it is required that the higher-order evidence e claims that there is some (specified or unspecified) evidence e* included in the evidential state of s* that is unknown to the agent s. Since this evidence is unknown to s it might change s’s credence in p. Second, the evidence (specified or unspecified) e* is determinant for the agent s*’s credence, this being ensured by the assumption that PrCrs*(p) = PrR(p|e*); given e*, the rest of s*’s total evidence is probabilistically irrelevant to p. Our Characterization of EE for p is a very strong conception of evidence of evidence for p. For many possible usages, A Characterization of EE for p might be more fruitful, especially if we want to distinguish various forms of evidence of evidence, independently of the question whether these notions underwrite a version of the EEE Slogan. However, Our Characterization of EE for p is a very promising one for finding a correct specification of the EEE Slogan.

. New specifications of the EEE Slogan Given Our Characterization of EE for p, we know in advance that evidence of evidence for p informs the agent s that the total evidence of the second agent s* contains some proposition not already contained in the evidence of s. This assumption amounts to: tevs ⊭ tevs*. Now we need to distinguish two cases. In the first case, the second agent s* possesses all the evidence that the first agent s already possesses and more, that is: tevs* ⊨ tevs. In this case the agent s receives—what we refer to as—evidence of more comprehensive evidence. In the second case, s herself possesses evidence that the second agent s* does not possess, that is: tevs* ⊭ tevs. In this second case s receives—what we refer to as—evidence of complementary evidence.

.. Evidence of more comprehensive evidence Case 

tevs* ⊨ tevs (and tevs ⊭ tevs*)

Suppose agent s obtains the higher-order evidence e which states that s* possesses all the evidence of agent s and some additional unspecified proposition e* that is not included in the total evidence of s and that PrCrs*(p) = PrR(p|e*) > PrR(p|tevs) = PrCrs(p). This is a case in which the second agent s* is better informed than the first agent s and has a higher credence in p. s* possesses more evidence and, after receiving the higher-order evidence e, s knows this. Should agent s increase her credence in p in the light of the higher-order evidence e? Without our idealizing assumption that both agents have identical reasoning commitments, we do not think that the answer should be an unequivocal ‘Yes’. If, for whatever reason, s mistrusts the reasoning commitments of agent s*, then s should not increase her credence in p. Obtaining evidence of evidence that s* takes to be relevant for p is not evidence for p if we have reasons to believe that s* is

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 - .    ö mistaken about whether her more comprehensive evidence is probabilistically relevant for p. For example, very often patients have more evidence about their illnesses or injury than physicians; the patient knows where it hurts and how much, how badly her ankle bent when she landed, etc. But even then, physicians rarely feel they should defer in their judgement to the credences of the patient. Instead they convince patients to share their evidence so that they can apply their (medical experts) reasoning commitments to it (and if necessary commission further inquiries and tests). The reason for this is simple: physicians mistrust the reasoning commitments of their patients, and more often than not they are right in doing so. Given our simplifying assumption that both agents have identical reasoning commitments, we would answer the above question (i.e., should the agent s increase her credence in p in the light of the higher-order evidence e?) with an unequivocal ‘Yes’. The reason is that if both agents employ the same reasoning commitments, then the EEE Slogan has a similar spirit as van Fraassen’s () reflection principle,¹³ which we want to rephrase here in our terminology as a coherence principle for reasoning commitments. Reflection for Reasoning Commitments An agent s’s reasoning commitments PrR commit her to a credence r with respect to a proposition p conditional on the following evidence: there is a proposition e* such that s’s reasoning commitments concerning s’s credence in p given e* say that it should be r and that e* is her future total evidence. More formally: PrR(p|9e*[(PrR(p|e*) = r ∧ e* = tevsFuture ∧ e* ⊨ tevsCurrent]) = r, if s is certain that her future self is not misinformed (i.e., has no false evidence). This reflection principle is obviously relevant for the EEE Slogan: we only need to assume that it is irrelevant whose evidence e is—that is, whether it is agent s’s future evidence or agent s*’s current evidence. In this chapter we set aside questions concerning the reliability and trustworthiness of agents, and have assumed that all ¹³ The standard formulation of this principle is in terms of credences and reads as follows: Reflection Agent s’s current credence in p on the condition that her future credence in p is r, should be r. Formally, PrCurrent ðpjPrFuture Crs Crs ðpÞ ¼ rÞ ¼ r. if the following three conditions are satisfied: . s is certain that her future self is not misinformed (no false evidence). . s is certain that her future self has more information than she has now. . s believes that her future credence is obtained by rational updating. In the literature this principle is often interpreted as a coherence principle between your current credences and your anticipated future credences (Huttegger , Titelbaum forthcoming). Our Reflection for Reasoning Commitments below relies on the idea that if the agent’s future credences are different, this must be due to the fact that the agent will receive some unspecified evidence and that she will change her credences in response to this evidence as required by Dyadic Bayesianism, namely by forming her credences by conditionalizing her reasoning commitments on her total evidence including this unspecified future evidence. The latter assumption is the reason why we do not need to assume that one’s future credences are obtained by rational updating. Instead, we can be explicit about the assumption that one’s future evidence implies one’s old evidence (and is thus more informative). Our Reflection for Reasoning Commitments also makes it obvious that this is a coherence principle for one’s reasoning commitments. It requires the commitment to reason in accordance with your reasoning commitments whatever unspecified evidence will come along in the future. A related discussion on the connection between the reflection principle and disagreement can be found in Elga ().

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  



evidence is true. Given this assumption, it should be irrelevant whose evidence it is as long as it is logically stronger and, thus, more informative.¹⁴ In such cases, it seems advisable to defer to the other agent’s credences; they are better informed than we are and they are using the same reasoning commitments to form their credences. In the light of these assumptions, we can slightly reformulate the principle to obtain a tenable specification of the EEE Slogan: Evidence of More Comprehensive Evidence is Evidence An agent s’s reasoning commitments PrR commit her to a credence r with respect to a proposition p conditional on the following evidence: there exists a proposition e* such that s’s reasoning commitments concerning s’s credence in p given e* say that it should be r and that e* is the more comprehensive total evidence of another agent s*. More formally: PrR(p|9s*9e*[PrR(p|e*) = r ∧ e* = tevs* ∧ e* ⊨ tevsCurrent]) = r. This specification of the EEE Slogan seems to us as plausible as the above Reflection for Reasoning Commitments (and van Fraassen’s reflection principle discussed in n. ), and it is subject to similar qualifications and restrictions. In addition, we see how evidence of more comprehensive evidence for p can provide you with relevant information to increase your credence in p. Suppose e is higher-order evidence of more comprehensive evidence for p. Then e claims that there is some (specified or unspecified) proposition e* included in the total evidence of s* (i.e. tevs* ⊨ e*) but not in the total evidence of s (i.e. tevs ⊭ e*) and that r = PrCrs*(p) = PrR(p|tevs*) > PrR(p| tevs) = PrCrs(p). If we now add our assumption that tevs* ⊨ tevs, e implies the proposition that we need to apply the above principle: 9s*9e*[PrR(p|e*) = r ∧ e* = tevs* ∧ e* ⊨ tevsCurrent]. And when we apply the principle we defer to the agent s* with more evidence. Thus, higher-order evidence of (specified or unspecified) more comprehensive evidence for p is evidence for p.

.. Evidence of complementary evidence Let us consider cases in which both agents possess evidence that the other agent does not possess; here both agents possess complementary evidence. Case 

tevs* ⊭ tevs (and tevs ⊭ tevs*)

The majority of instances in which we would like to refer to the EEE Slogan to determine whether we should revise our credences are such that both agents have complementary pieces of evidence. We have already discussed some of these instances: first, recall Feldman’s Criminal Case Example, where he asked us to consider ‘two suspects in a criminal case, Lefty and Righty’ and the ‘two detectives investigating the case, one who has the evidence about Lefty and one who has the evidence incriminating Righty’ (Feldman , p. ). Feldman then notes that,

¹⁴ Cartesians in the sense of Barnett (forthcoming) might uphold this or a similar principle even if they give up the assumption that all evidence is true, since they defend perceptual impartiality. Dogmatists like Pryor () presumably will ultimately reject it, because they believe we should give some kind of priority to our own perceptual experiences.

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 - .    ö upon finding out that the other detective has evidence incriminating the other suspect, each detective is required to suspend judgement. Second, consider the example of Hanna observing Carina studying the content of the fridge. Presumably, Carina does not have all the evidence that Hanna, her partner, has. When Carina returns from the fridge she possesses an important piece of evidence that Hanna is missing. In the following we discuss whether one should revise one’s credences in the light of (higher-order) evidence of unknown (specified or unspecified) complementary evidence. Before we do so, let us consider how a Bayesian agent forms her credences in the light of her total evidence. Assume again that we measure evidential support with the help of the loglikelihood measure, l, of confirmation or evidential support (Fitelson ), which we introduced before. It is known that the agent’s credence in p in the light of the agent’s total evidence tevs is a function from her reasoning commitments PrR(p) (which correspond to some extent to her a priori credence in p) and the evidential support p receives from tevs, l(p, tevs).¹⁵ The higher the support l(p, tevs), the higher the resulting credence. Let us first study how we could incorporate evidence from other agents if we receive evidence of unknown specified evidence for p. (This will later also help us to better understand how to revise one’s credence in the light of evidence of unknown, unspecified, complementary evidence.) In particular, let this evidence say that the specified proposition e* is part of the total evidence of the second agent s* and suppose agent s’s total evidence is tevs. Remember, we assume that all evidence is true and that one can fully rely on the pieces of evidence included in someone’s evidential state. Thus, agent s would simply add e* to her current total body of evidence and update her credences accordingly. Then one can prove the following for the loglikelihood measure of evidential support (Brössel ):

...  2 lðp; tevs ∧e*Þ ¼ lðp; tevs Þ þ lðp; e*Þ þ log4

PrR ðtevs ∧e* jpÞ PrR ðtevs jpÞPrR ðe* jpÞ

3 5

PrR ðtevs ∧e* jpÞ PrR ðtevs jpÞPrR ðe* jpÞ

This theorem shows that the effect of adding e* to s’s total evidence concerning the proposition p depends on three factors: . the degree to which tevs supports p:   PrR ðtevs jpÞ lðp; tevs Þ ¼ log PrR ðtevs j  pÞ

¹⁵ The exact expression is not required in the main part of the chapter, and would hinder readability: e2lðp;tevs Þ PrR ðpÞ 1 PrR ðpÞ þ e2lðp;tevs Þ PrR ð pÞ 1

PrCrs ðpÞ¼ PrR ðpjtevs Þ ¼

1lðp;tev Þ s 2

e

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  



. the degree to which e* supports p:

  PrR ðe*jpÞ lðp; e*Þ ¼ log PrR ðe*j  pÞ

. the degree to which the total evidence of s and the new evidence e* cohere with each other in the light of p and in the light of p (in the sense of probabilistic coherence introduced in Shogenji ): 2 Pr ðtev ∧e* jpÞ 3 R s Pr ðtev jpÞPrR ðe* jpÞ 5 log4 R s PrR ðtevs ∧e* jpÞ PrR ðtevs jpÞPrR ðe* jpÞ If the sum of the latter two factors is greater than 0, then adding e to s’s total evidence would result in obtaining evidential support for p, and thus to increasing s’s credences in p (when compared with s’s credences in p given tevs alone). If the sum is negative, then this would result in counter-support, and thus in a decrease of s’s credence (when compared with s’s credences in p given tevs alone). This shows that even if we assume Our Characterization of EE for p and restrict the application of the EEE Slogan to evidence e of unknown specified evidence e* for p, the EEE Slogan is not correct in general. If the proposition e* does not cohere with the rest of one’s total evidence in the light of p, then this additional evidence will not increase our credence. And since the evidence e of evidence reports to the agent s which specified proposition e* is included in the other agent s*’s evidence, s herself needs to determine what her credence in p should be in the light of e* ∧ tevs. Let us ask whether one should revise one’s credence in p even if the evidence of specified evidence of the other agent for p is replaced by evidence of unspecified evidence. Perhaps the EEE Slogan holds, if it is restricted to evidence of unspecified evidence for p. We would want to include this unspecified evidence, since we assumed (for the purpose of the chapter) that the other agent’s evidence is true and that this agent uses the same reasoning commitments as we do. How should we revise our credence in p? The answer depends on the third factor: how well do the pieces of evidence cohere with each other in the light of the proposition p, and how well do they cohere with each other in the light of p. If we had reasons to assume that the third factor is neutral or even positive for the unspecified complementary evidence, then we would be justified in assuming that overall this complementary evidence would increase our credence in p. If we had reasons to assume that it is negative, however, we would not be justified in assuming that the unspecified complementary evidence would increase our credence. The important insight is this: even (higher-order) evidence of unknown, unspecified, complementary evidence is not always evidence. Even assuming Our Characterization of EE for p, the EEE Slogan is not correct in general. Indeed, with this observation as a guide, it is easy to construct examples that demonstrate how higher-order evidence of unknown unspecified complementary evidence for some proposition p can be but also can fail to be evidence for p. Let us start with two positive examples.

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 - .    ö A More Detailed Milk Example Assume the total evidence of Hanna, tevHanna, is irrelevant for the question of whether p₁: there is milk in the fridge. Then Hanna sees Carina searching the fridge for milk and returning empty handed. Thereby Hanna obtains (higher-order) evidence of unknown, unspecified, complementary evidence for p₁. That is, that there is some proposition e* that is included in the total evidence of Carina (i.e. tevCarina ⊨ e*) but not in the total evidence of Hanna (i.e. tevHanna ⊭ e*) such that PrCrCarina(p₁) = PrR(p|e*) > PrR(p₁|tevHanna) = PrCrsanna(p₁). Here it is safe to assume that whatever the content of e* is, the probability of tevHanna is independent of e* on the condition of p₁ and p₁, and thus that the relevant factor is neutral. In this case the (higher-order) evidence of unspecified, complementary evidence is evidence for p₁. Even stronger support for some proposition p can be obtained by unknown, unspecified complementary evidence e* for the proposition in question, if one has reason to assume that it coheres with the rest of one’s evidence in the light of p (at least more so than in the light of p). For such a case, let us look at a second example: Lazy Referees Example Suppose two lazy referees need to review the intermediary report of a research project and they decide to divide the whole report in two parts and each assess only one part. The first referee, Silvia, studies the quantity and quality of the published articles, the second referee, Wolfgang, assesses all other criteria for a successful research project. Then they meet again and find out that both their total evidence supports p₁, the proposition that the research project has been run successfully in the past. Silvia received evidence e and e supports p₁. Now Silvia learns that Wolfgang obtained some unspecified complementary evidence e* that also supports p₁. (Let us assume that all their other evidence is irrelevant to the question of whether or not p₁.) The fact that they received evidence from diverse areas and that they both came to the conclusion that p₁ leads referee Silvia to expect that her specified evidence e and the unspecified evidence e* of Wolfgang are coherent in light of the assumption that p₁, and very incoherent under the assumption that p₁. Thus, in this example it seems appropriate that Silvia considerably increases her credence in p₁. She has received evidence e for p₁ and higher-order evidence of unknown unspecified evidence e* for p₁, and she can assume that only the hypothesis p₁ renders these distinct pieces of evidence (the specified and the unspecified evidence) coherent. Thus, all three factors mentioned in the above Theorem are positive, and she should considerably increase her credence in p₁. Finally, let us consider a negative case that shows that the EEE Slogan is incorrect for some higher-order evidence of unknown, unspecified, complementary evidence; let us consider a variation of Feldman’s Criminal Case Example: Another Criminal Case Example Suppose Melina is a detective in a case similar to Feldman’s criminal case. Melina believes that she has considered all evidence e that is relevant to the question at hand, and that e points clearly in the direction of p₁, the proposition that Lefty is the culprit. Melina has refused, however, to consider evidence offered to her by the chief of police, because Melina knows

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that the chief is engaged in a secret love affair with Lefty. Then suppose the second detective Vincent states that he has collaborated with the chief of police and that they found evidence e* that incriminates Righty—as opposed to Lefty. Thus, Melina receives higher-order evidence of unspecified, complementary evidence in favour of p₁. For Melina it is reasonable to assume that all of this unspecified evidence e* is true but misleading (the witnesses might be lying, the coroner might be paid off, the police might have planted evidence). Part of Melina’s evidence indicates that her evidence e and the unspecified evidence e* are incoherent given the assumption p₁, but coherent given the assumption p₁. After all, p₁, that Lefty is the culprit, and the knowledge about the secret love affair predict not only her evidence but also why Vincent found all the (misleading) evidence pointing at Righty, but none of the evidence for p₁. So in this case higher-order evidence of unspecified evidence for p₁ is not evidence for p₁. Melina has reasons to believe that this unspecified evidence does not fit with her evidence, at least not under the assumption that p₁. Only p₁ makes the evidence fit together. Since Melina’s evidence already supports p₁ anyway, Melina does not revise her credence in p₁. This final example shows that evidence of evidence for some proposition is not always evidence for the proposition: the EEE Slogan is incorrect for some (higherorder) evidence of unspecified, complementary evidence. These results of course depend on Our Characterization of EE for p.

. Conclusion The EEE Slogan captures one of the most important—albeit vague—ideas in social epistemology. It is an idea we refer to when we defer to the judgements of others who we believe to be better informed than we are. Furthermore, to a large extent the division of labour in scientific inquiry rests on the idea that if we receive evidence that someone else has received evidence for p, this is also evidence for us that p is true. Yet despite its intuitive appeal, and its importance in everyday life and in science, most results so far indicated that the EEE Slogan is incorrect. In a first step, we provided a new framework for understanding and modelling evidence of evidence, which we called Dyadic Bayesianism. Within this framework we proposed Our Characterization of EE for p. We understand evidence of evidence for p as higher-order evidence for the existence of some (specified or unspecified) unknown evidence in the evidential state of another agent that supports p. In a second step, we investigated under which conditions evidence of evidence for p is evidence for p. Given two idealizing assumptions (i.e., that all evidence is true and that agents share the same reasoning commitments) we argued (i) that evidence of more comprehensive evidence for p is always evidence for p, and (ii) that evidence of complementary evidence for p is not always evidence for p. These results demonstrate the strength of our approach and our framework of Dyadic Bayesianism as regards the project to understand evidence of evidence and investigate the EEE Slogan.

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 - .    ö

Acknowledgements Special thanks go to Mattias Skipper and Asbjørn Steglich-Petersen for detailed commentaries on previous versions of the chapter, which helped us to improve it considerably. We are also grateful to Branden Fitelson for fruitful discussions on higher-order evidence that influenced this chapter. Anna-Maria A. Eder’s research was funded by the Austrian Science Fund (Erwin Schrödinger Program) through the research project Higher-Order Evidence (reference number J -G) at Northeastern University, Boston. Peter Brössel’s research on the chapter has been generously supported by the German Research Foundation (Emmy-Noether Program) through the research project From Perception to Belief and Back Again (reference number BR /-.) at the Ruhr-University Bochum.

References Adler, J. (). “Epistemological Problems of Testimony.” In E. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Winter  edition). Barnett, D. (forthcoming). “Perceptual Justification and the Cartesian Theater.” In: Oxford Studies in Epistemology , Oxford University Press. Brössel, P. (). “The Problem of Measure Sensitivity Redux.” In: Philosophy of Science (), pp. –. Brössel, P. (). “Keynes’s Coefficient of Dependence Revisited.” In: Erkenntnis (), pp. –. Brössel, P. (). “Rational Relations between Perception and Belief: The Case of Color.” In: Review of Philosophy and Psychology (), pp. –. Brössel, P. (forthcoming). Rethinking Bayesian Confirmation Theory. Springer. Brössel, P. and A.-M. A. Eder (). “How to Resolve Doxastic Disagreement.” In: Synthese , pp. –. Christensen, D. (). “Higher-Order Evidence.” In: Philosophy and Phenomenological Research , pp. –. Comesaña, J. and E. Tal (). “Evidence of Evidence is Evidence (Trivially).” In: Analysis , pp. –. Dorst, K. and B. Fitelson (ms). “Evidence of Evidence: A Higher-Order Approach.” Unpublished manuscript. Eder, A.-M. A. (forthcoming). “Evidential Probabilities and Credences.” In: The British Journal for the Philosophy of Science. Elga, A. (). “Reflection and Disagreement.” In: Noûs , pp. –. Feldman, R. (). “Reasonable Religious Disagreements.” In L. Antony (ed.), Philosophers Without God: Meditations on Atheism and the Secular Life, Oxford University Press, pp. –. Fitelson, B. (). “The Plurality of Bayesian Measures of Confirmation and the Problem of Measure Sensitivity.” In: Philosophy of Science , pp. –. Fitelson, B. (). Studies in Bayesian Confirmation Theory, Ph.D. thesis, University of Wisconsin. Fitelson, B. (). “Evidence of Evidence is not (Necessarily) Evidence.” In: Analysis , pp. –. Huttegger, S. (). “In Defense of Reflection.” In: Philosophy of Science , pp. –. Keynes, J. M. (). A Treatise on Probability. Macmillan. Lackey, J. and E. Sosa (). The Epistemology of Testimony, Oxford University Press. Lyons, J. (). “Epistemological Problems of Perception.” In E. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Spring  edition). Moretti, L. (). “Evidence of Expert’s Evidence is Evidence.” In: Episteme , pp. –.

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Moretti, L. (). “Tal and Comesaña on Evidence of Evidence.” In: The Reasoner , pp. –. Pryor, J. (). “The Skeptic and the Dogmatist.” In: Noûs , pp. –. Roche, W. (). “Evidence of Evidence is Evidence under Screening-off.” In: Episteme , pp. –. Shogenji, T. (). “A Condition for Transitivity in Probabilistic Support.” In: The British Journal for the Philosophy of Science , pp. –. Shogenji, T. (). “Is Coherence Truth Conducive?” In: Analysis , pp. –. Tal, E. and J. Comesaña (). “Is Evidence of Evidence Evidence?” In: Noûs , pp. –. Titelbaum, M. (forthcoming). Fundamentals of Bayesian Epistemology, Oxford University Press. Van Fraassen, B. (). “Belief and the Will.” In: The Journal of Philosophy , pp. –.

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4 Fragmentation and Higher-Order Evidence Daniel Greco

The concept of higher-order evidence—roughly, evidence about what our evidence supports—promises epistemological riches; it has struck many philosophers as necessary for explaining how to rationally respond to disagreement in particular, and to evidence of our own fallibility more generally. But it also threatens paradox. Once we allow higher-order evidence to do non-trivial work—in particular, once we allow that people can be rationally ignorant of what their evidence supports— we seem to be committed to a host of puzzling or even absurd consequences. My aim in this chapter will be to have my cake and eat it too; I’ll present an independently motivated framework that, I’ll argue, lets us mimic the particular case judgments of those who explain how to accommodate evidence of our fallibility by appeal to higher-order evidence, but without commitment to the absurd consequences.

. Road map My strategy will be as follows. I’ll start by introducing the idea of higher-order evidence (HOE), along with some of the examples which it is often thought to illuminate. In particular, my focus will be on the claim that, due to limited higherorder evidence, we are sometimes rationally uncertain about what our evidence supports. I’ll then review some puzzles this claim engenders. I’ll divide them in two categories. First, there are the synchronic puzzles. In particular, it seems to entail that epistemic akrasia—believing some claim, while simultaneously believing that you shouldn’t believe it—can be rational. But epistemic akrasia seems paradigmatically irrational (Greco b; Horowitz ; Littlejohn ). Second, there are a host of diachronic puzzles. Having presented an overview of reasons to be skeptical of the idea that we can be rationally uncertain about what our evidence supports, I’ll introduce an alternative strategy for making sense of the cases. In a nutshell, it involves the idea that evidence can be available for some purposes or tasks, but not others, and that we often do better to explicitly relativize talk about a subject’s

Daniel Greco, Fragmentation and Higher-Order Evidence In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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evidence to those purposes or tasks which the evidence is available to guide.¹ I’ll then argue that when this independently motivated framework is applied to the sorts of cases typically discussed in the higher-order evidence literature, it lets us offer plausible descriptions of cases that are usually characterized in terms of rational uncertainty of what our evidence supports, without the puzzles that characterization brings in its wake.

. Higher-order evidence As a first pass, we might say that HOE is evidence about evidence.² But this rough characterization obscures crucial distinctions—some species of HOE, so defined, raise distinctive epistemological problems that others do not. For instance, if I read in a newspaper that paleontologists have discovered powerful evidence that the cretaceous extinction was not, after all, caused by a meteor impact, I’ve plausibly gotten higher-order evidence in this minimal sense—I’ve gotten evidence (what I read in the newspaper) about evidence (what the paleontologists have discovered). But, for reasons that should become clear, this sort of case doesn’t raise the tricky issues that have typically been the focus of the recent literature on HOE. For that reason, I’ll work in this chapter with a narrower characterization of HOE. In the target sense, HOE for a subject S is evidence that bears on what S’s evidence supports. To receive higher-order evidence, in the sense I’m interested in, is for her to receive: . Evidence about which body of evidence S herself has, or . Evidence about evidential support relations—in particular, evidence about which propositions are supported to which degrees by a body of evidence that, for all S knows, is her own. In this narrower sense, when I read about paleontological discoveries, I don’t seem to get any higher-order evidence. I get new evidence to be sure, but that doesn’t seem helpfully characterized as either (a) evidence about what my evidence is, or (b) evidence about what that evidence supports.³ While this characterization of HOE may look oddly disjunctive, I don’t think it is. There are two sorts of reason I might be uncertain about what my evidence supports. First, while I might know what’s supported by each possible body of evidence, I might fail to know which body of

¹ This is a special case of the idea that our mental lives are “fragmented.” For some defenses of various versions of this view, see Lewis (), Stalnaker (; ; ), Egan (), Gendler (), Rayo (, ch. .), Greco (a; b), and Elga and Rayo (ms a; ms b). ² E.g., Richard Feldman () characterizes HOE as “evidence about the existence, merits, or significance of a body of evidence.” ³ See also Christensen (a), who offers a different characterization of HOE, but which would agree with mine in not counting the newspaper case. In particular, he claims that what’s distinctive about HOE is that it “rationalizes a change of belief precisely because it indicates that my previous beliefs were rationally sub-par” (p. ). And that’s not going on in the newspaper case—in coming to suspend judgment about whether a meteor strike caused the cretaceous extinction after reading the newspaper story, I needn’t believe that my previous confidence on that point was rationally sub-par.

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   evidence is mine. Second, while I might know which body of evidence is mine, I might fail to know what that given body of evidence supports.⁴ The two-clause characterization of HOE mirrors the two potential sources of uncertainty about what one’s evidence supports.⁵ Ultimately—counterintuitively, I think—I’ll suggest that epistemologists can and should avoid appeal to HOE, as characterized above. That is, so long as we’re careful not to equivocate on “evidence,” we can safely theorize in a framework in which a subject’s evidence always settles (a) what her evidence is, and (b) what that evidence supports, and so in which HOE has no non-trivial role to play. But for reasons that should become clear, it’s extremely tempting to equivocate on “evidence.” Appeals to HOE in the recent literature should typically be understood as involving a kind of subtle equivocation—a subject may lack evidence (in one sense) about what her evidence (in some other sense) is, or supports. But I’m getting ahead of myself—reasons to accept these surprising claims will only emerge later in the chapter. For now, having offered a general characterization of HOE, I’ll turn to some familiar examples in which it seems natural and fruitful to theorize in terms of it.

.. Examples ...  Probably the best-known examples that have been thought to illustrate the importance of HOE concern disagreement. Here’s a typical one:⁶ J: You’re a juror in a complex civil case in which a great deal of evidence was presented. On the basis of that evidence, you believe that the accused was at fault, so your tentative plan is to find for the plaintiff. And let’s stipulate that your reading of the evidence is, in fact, maximally rational; if an ideal intellect—an epistemic angel—were to look at your evidence, she’d come to the same conclusion you did. But when you sit down to deliberate, you find that many of the other jurors think otherwise—they believe that the accused was not at fault. Upon further discussion, you realize it’s not that anybody is bringing to bear unshared, background evidence—rather, you just disagree about what conclusion the shared body of evidence presented at court points to. Moreover, the other jurors seem like generally sensible people, and they’re not committing any obvious blunders or misunderstandings. At this point, what should you think about whether the accused was at fault? How might we analyze this case—in particular, capturing plausible verdicts about what it’s rational to believe both before and after hearing about the other jurors’ ⁴ Of course, I might also be ignorant in both ways. ⁵ So I agree with Dorst (this volume), who frames his discussion in terms of higher-order uncertainty, rather than higher-order evidence. ⁶ Perhaps the most famous example of disagreement is David Christensen’s (b) “Restaurant Case,” but I’m avoiding it on purpose. In short, that case raises the problem of logical omniscience, in addition to problems concerning disagreement. While I do think the diagnosis I’ll ultimately offer of this case could extend to that one, it would be complicated, and would go via the view defended in Elga and Rayo (ms a; ms b). Because I don’t have the space here to summarize their treatment of the problem of logical omniscience, nor how I would appeal to it in treating cases of disagreement, I’ll stick to cases of disagreement that don’t pose that problem.

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views—in terms of HOE? Let’s suppose that you are rational in believing that the accused is at fault prior to hearing about the other jurors’ views, and then rational in suspending judgment on that question afterwards—this pattern of judgments is typical of “conciliationism,” in the lingo.⁷ For what it’s worth, this will make my task harder—while there are extant views that downplay the significance of HOE, they tend to do so by treating other jurors’ views as irrelevant to what, in fact, your evidence supports concerning the liability of the accused.⁸ In this case it’s natural to say that you and the other jurors know what the evidence is. We can suppose you all have a very good memory, and that if you were asked about whether the body of evidence presented at court included this or that fact, you would always be able to respond accurately. And none of that changes when the other jurors express their views. So HOE in the first sense—evidence about which body of evidence is yours—doesn’t seem to be at issue. But, prima facie, it’s plausible that HOE in the second sense is at issue. While you may know what your evidence is, because you’re not an ideal evaluator of evidence—or at least, if you are, you don’t know that—it’s not transparent to you what the evidence supports concerning fault. As a matter of fact you judged correctly that it supported the belief that the accused was at fault, but you hold that belief about evidential support only defeasibly, and rightly so. When you learn that the other jurors came up with a different answer, your belief is defeated. You obtained misleading higher-order evidence to the effect that your evidence doesn’t support the claim that the accused was at fault after all. And if you’re rational, you’ll respond by suspending judgment, both about fault, and about what your evidence supports concerning fault. The fact that a reliable inquirer with the same evidence as you disagrees with you suggests you may have made a mistake in judging what your evidence supports. But disagreement isn’t the only way you can get evidence of such a mistake. The literature abounds with cases in which, due to learning that you’ve been drugged,⁹ or are suffering from oxygen deprivation,¹⁰ or sleep deprivation,¹¹ or some other malady, you get evidence that is naturally interpreted as suggesting that you made a mistake concerning evidential support relations—you took a body of evidence to support some proposition that, in fact, it does not—even when you have in fact made no such mistake.

...   While I characterized HOE as coming in two species, aimed at rectifying two sorts of ignorance—ignorance about what your evidence is, and ignorance about evidential support relations—the cases discussed in subsection ... only concerned the latter species. What about the former? They tend to turn on facts about our limited perceptual discriminatory capacities. Here’s an example from Salow (b), who is adapting a case from Williamson (). I use Salow’s version because I’ll later appeal to his explanation of what’s unattractive about the HOE-based interpretation of the case: ⁷ E.g., Christensen (b), Elga (), Feldman (). ⁸ I have in mind Titelbaum () and Smithies (), both of whom hold that any body of evidence maximally supports all a priori truths about evidential support relations. ⁹ Christensen (a). ¹⁰ Elga (ms). ¹¹ Christensen (a).

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   Imagine that you are faced with an unmarked clock, with a single hand that can point in any one of  slightly different directions. Your ability to discriminate where it is pointing is good, but not unlimited. If you are to be reliable in your judgements, you need to leave yourself a margin of error. For example, if the hand is in fact pointing at  minutes, you can reliably judge that it is pointing somewhere between  and  (inclusive), but are unreliable about claims stronger than that. The same is true of every other position the hand could be in. It is somewhat natural to identify your evidence with the strongest claim about the hand’s position which you can reliably get right . . . If the hand is in fact pointing at , my evidence will be that it is within [, ]; and if it is pointing at , my evidence will be that it is within [, ]. (Salow b, pp. –)

What’s crucial about the setup, for present purposes, is that you’re not in a position to know what your evidence is.¹² Suppose, for example, the clock is in fact pointing at , so your evidence is that it is between  and  (inclusive). We may assume that you know the general facts about the setup—you know about your impressive but not unlimited discriminatory capacities, know that the hand could be pointing anywhere from  to , etc. But in that case, if you were to know that your evidence is that the hand is between  and , then you would be in a position to know that the hand is actually pointing at , contradicting the stipulations of the case—by stipulation, that is your evidence only when the clock is pointing precisely at . So you must not know that your evidence is that the hand is between  and . Rather, if the strongest thing you know about where the hand is pointing is that it’s between  and , then instead of knowing exactly what your evidence is, there are three possibilities compatible with your knowledge concerning what your evidence is— either your evidence is that the hand is between  and , or your evidence is that it’s between  and , or your evidence is that it’s between  and .

. Puzzles Now that we’ve been introduced to the notion of HOE, and to the sorts of examples it’s typically used to illustrate, it’s time to turn to the diffculties it brings in its wake.

.. Epistemic akrasia Perhaps the most frequently discussed puzzle in this neighborhood concerns “epistemic akrasia”—believing some claim while also believing that one shouldn’t believe it. This is often generalized to involve other sorts of mismatch between one’s firstorder doxastic attitudes, and one’s attitudes concerning which attitudes one should have.¹³ Epistemic akrasia can seem paradigmatically irrational. But as many writers have pointed out, if we accept the evidentialist claim that an agent should believe whatever her evidence supports, and we also accept the existence of non-trivial higher-order evidence—that is, if we accept that agents can be rationally uncertain ¹² Like both Salow and Williamson, I’ll assume E = K—your evidence is what you know. So failing to know what your evidence is amounts to lacking evidence concerning what your evidence is. ¹³ Worsnip () and Rinard (forthcoming) offer similar generalizations of the anti-akrasia constraint, both in terms of having some doxastic attitude while failing to believe that the attitude is supported by one’s evidence.

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  - 



either about what their evidence is, or what that evidence supports—then it’s hard to avoid holding that agents should sometimes be epistemically akratic.¹⁴ The connection has been spelled out at greater length elsewhere, so I’ll just give the flavor of the idea here. First, suppose agents can be rationally uncertain about what their evidence is. Say you don’t know whether your total evidence is A, B, or C. You rationally think that each is equally likely to be your evidence. A supports the belief that P, but B and C do not. In fact, your evidence is A. If you have all the beliefs that are supported by your evidence, you’ll believe that P (after all, that’s supported by A, which is your evidence), while also believing that you probably shouldn’t believe that P (after all, you think it’s more likely than not that your evidence is either B or C, neither of which support believing that P).¹⁵ Things go much the same if agents can be rationally uncertain about what their evidence supports. Suppose you know what your total evidence is. But there are three hypotheses about what that evidence supports—A, B, and C—in which you rationally invest equal credence. On hypothesis A, your evidence supports the belief that P, but on hypotheses B and C, it does not. A is true—your evidence really does support the belief that P. Given those stipulations, if you have all the beliefs supported by your evidence, then you’ll find yourself believing that P (since A is true, believing that P is supported by your evidence) while believing that you probably shouldn’t believe that P (since you think it more likely than not that hypothesis B or C is true, and according to those hypotheses your evidence does not support belief in P). At this stage there are a wide variety of responses in the literature. Some authors simply bite the bullet, accepting that these sorts of epistemically akratic states are unavoidable—no adequate epistemological theory can avoid commitment to their possibility.¹⁶ Others hold that while there’s something to be said for the akratic states, there’s also something to be said against them—perhaps the sorts of cases just discussed present a kind of dilemma, in which not all epistemic demands can be met.¹⁷ And others try to contest the assumptions that led to the problem. Weaker versions of this third response will propose principles constraining just what combinations of first-order and higher-order evidence are possible, in the hopes that they can allow for some non-trivial HOE, but without licensing the sorts of seemingly irrational akratic states I’ve been discussing.¹⁸ A very strong version of this third response will involve claiming that agents can neither be rationally uncertain of what their evidence is, nor of what it supports.¹⁹ Before presenting a version of the third response, I’ll turn to a different set of puzzles.

¹⁴ See Salow (a, appendix B), Dorst (ms). ¹⁵ The limited discrimination example in section . could be such a case, with A, B, and C corresponding to [, ], [, ], and [, ], and P being “the hand is not pointing to .” ¹⁶ E.g., Lasonen-Aarnio (). ¹⁷ E.g., Christensen (a) holds that subjects with misleading higher-order evidence can’t help but violate a rational ideal. Worsnip () holds that such subjects cannot both be rational and believe what their evidence supports. ¹⁸ Versions of this response include Elga (), Horowitz (), and Dorst (ms). ¹⁹ Titelbaum () is explicit that subjects cannot be rationally uncertain about evidential support relations, but doesn’t discuss uncertainty about what one’s evidence is. Salow (a; b) comes close—he

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  

.. Diachronic puzzles In addition to supporting incoherent belief states at a given time, non-trivial HOE leads to odd consequences concerning changes of evidence and belief over time. In this section I’ll focus on odd diachronic consequences of ignorance concerning what your evidence is, rather than what a given body of evidence supports. But the latter phenomenon has been discussed in the literature too.²⁰

...   One of three doors, A, B, or C, is hiding a prize. The other two hide trash. Monty instructs you to pick a door.²¹ Before you can open it, Monty picks one of the remaining doors that he knows not to conceal a prize—there’s guaranteed to be at least one—and opens it, revealing trash. He now gives you the choice of sticking with your initial choice, or switching to one of the remaining doors (either the open door with trash, or the closed door that might contain a prize). For familiar reasons, you should switch to the closed door. There are many explanations, but the one I prefer runs as follows. Suppose you initially picked the wrong door—one that does not conceal the prize. Then there are two doors left, one of which contains the prize, and one of which does not. Given Monty’s strategy, he’ll have to open the door that does not contain the prize, leaving the door that does contain the prize as the only one left. If you switch to the closed door, you’ll switch to the door with the prize. Since you’ll initially pick the wrong door two-thirds of the time, switching will lead to the prize two-thirds of the time. Let’s recast that explanation in terms of confidence. For each door, you start out with a credence of 13 that the prize is behind that door. Suppose your initial pick is A. Your credences are unchanged—you have a credence of 13 that the prize is behind door A, and 23 that it’s behind one of the other doors. Now, Monty reveals that the prize is not behind door C. Given what you know about Monty’s strategy, you retain your credence of 23 that it’s behind door B or C, but now that you know it’s not behind door C, all of that credence goes to B—you end up with a credence of 23 that the prize is behind B, and 13 that it’s behind A. So far, so familiar. But what if you could learn that the prize isn’t behind door C, without learning that Monty revealed this to you? Would that undermine the argument for switching? Of course, there are some ways this could happen such that it obviously would. The door might be blown open by a gust of wind, not by Monty’s design. Or you might have planted a hidden camera in the room, or hired a

argues against the possibility of rational uncertainty about evidential support relations, and argues against the possibility of many (but not all!) forms of rational uncertainty about what one’s evidence is. ²⁰ See, e.g., Horowitz (, §.). ²¹ This section is inspired by Bronfman (), who notes the oddity of combining () the view that updating properly proceeds by conditionalization, () the view that KK failures are possible, and () the standard diagnosis of the Monty Hall problem. His solution is to hold onto  and , and propose a novel, non-conditionalization-based form of updating. My aim in the present section is to make the sorts of KK failures that would be involved in Monty Hall cases seem odd enough that we’d be better off sticking with () and (), and abandoning ().

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  - 

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spy to look behind door C and report back. In cases like these, when you learn via a non-Monty source that the prize isn’t behind door C, you have no reason to switch from A to B. But suppose you know for sure that the only way you could find out what’s behind door C is by Monty’s showing you, and you know how Monty operates (i.e., you know that he’ll only open door C if the prize isn’t there). In this case, is it still possible to learn that the prize isn’t behind door C, without learning that Monty revealed this to you? And if so, would that undermine the argument for switching? It’s very hard to imagine how this could be. And yet, if we think that cases of limited discrimination provide illustrations of how one can gain evidence, without its becoming part of one’s evidence that one has gained evidence, then it’s hard to rule this out. For example, suppose Monty operates as follows. Rather than throwing a door wide open to view, he renders a door ever so slightly transparent. Your perceptual discriminatory capacities are powerful enough, based on looking at the ever so slightly transparent door C, to give you the evidence that the prize isn’t behind it—you can just barely discriminate the look of trash through a slightly transparent door from the look of a prize through a slightly transparent door. So when Monty renders the door slightly transparent, it becomes part of your evidence that the prize isn’t behind door C. But your powers of discrimination are limited enough that it doesn’t become part of your evidence that it’s part of your evidence that the prize isn’t behind door C. If this is possible, how should you reason in such a case about whether to stick with door A or switch to B? You certainly aren’t in a position to justify switching via the argument sketched earlier. That argument went via the premise that Monty revealed that the prize isn’t behind door C, that is, that he allowed you to learn this. But in the imagined case, that’s not part of your evidence—you learned that the prize isn’t behind door C, but you didn’t learn that you learned this. The following reasoning sounds borderline incoherent, but it’s hard to see what’s wrong with it, given the assumption that limited discriminatory capacities can let you gain evidence without gaining evidence that you gained that evidence: Clearly I shouldn’t switch to C, because it doesn’t contain the prize, but should I stick with A, or switch to B? If I knew C didn’t contain the prize, then switching to B would be the sensible thing to do, because I’d only know that C didn’t have the prize if Monty had revealed it to me, and a familiar argument would establish that I’d have a 23 chance of gaining the prize by switching to B . . . but I can’t tell whether I know that C doesn’t contain the prize. For that matter, I can’t tell whether I know that B doesn’t contain the prize. Well, C doesn’t contain the prize, so I guess I’ll flip a coin to choose between A and B. The next example, due to Bernhard Salow, has a similar flavor—it concerns odd consequences of allowing that one can gain evidence, without gaining evidence that one has gained evidence.

... -  It’s a commonsense piece of epistemology that an investigation can only provide support for a hypothesis if, had the results of the investigation been different, it could

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   have undermined that hypothesis. Karl Popper () famously criticized research programs that seemed to be able to accommodate any possible body of data; to use two of his favorite examples, if Freudian theories of psychodynamics or Marxist theories of history cannot be undermined by evidence, they cannot be supported by evidence either. While Popper’s particular development of this insight was problematic,²² similar ideas about “no-lose” investigations have been defended in the recent epistemological literature.²³ But, as Salow (b) explains, once we take on the idea that you can gain evidence without gaining evidence that you gained evidence, we seem to be committed to the possibility of various sorts of no-lose investigation. Here’s his explanation of how this works in Williamson’s unmarked clock case, described earlier: My friend knows whether I’m popular; and I would like to have additional evidence that I am, regardless of whether it is true. So I construct an unmarked clock of the kind Williamson describes, and I ask my friend to set the hand in the following way: if people like me, he will set it to ; if they don’t, he will flip a coin to decide whether to set it to  or to . Having given the instructions, I know that the clock will be set somewhere between  and  . . . Next, I take a look. If people actually like me, the hand will be set to , and so my evidence will only tell me that it is somewhere between  and , which I knew already. So if people like me, I get no new evidence. But if people do not like me, it will be set either to  or to . Suppose it is set to ; then my evidence will allow me to rule out that it’s set to , since  is far enough away from the actual setting. But I knew that there was a fifty-fifty chance that it would be set to  if people didn’t like me. So seeing that it isn’t set to  gives me some evidence that I am popular. Moreover, my evidence cannot discriminate between the hand being set to  and its being set to , so that I get no evidence against my being popular. So, if the hand is set to , I will get evidence that I am popular; by similar reasoning, I will also get such evidence if the hand is set to . So if people don’t like me, I will get evidence that I am popular. Again, I have successfully set up a no-lose investigation into my popularity. (Salow b, pp. –)

If we ask how the subject of Salow’s example should reason, or might rationally act on his evidence—imagine that he’s deciding whether to book a large venue for his birthday party, which would only be necessary if he’s popular—we’ll run into the same sorts of perplexities we saw in the case of Monty Hall. I don’t pretend to have provided anything like a comprehensive overview of the terrain. I’ve just tried to point to various sorts of awkwardness that ensue when we allow that subjects can be rationally ignorant of either (a) which body of evidence is theirs, or (b) which beliefs are supported by which bodies of evidence. But the idea that subjects can’t be rationally ignorant of these things can sound hard to swallow. So in section . I’ll present a framework that lets us capture the truths that we express when we describe subjects as being rationally ignorant of (a) and (b), but without the awkward consequences.²⁴

²² See Salmon (). ²³ See, e.g., Titelbaum (), Salow (b). ²⁴ Does it have other awkward consequences? Certainly.

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  - 

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. Fragmented evidence What does it take to have some proposition as part of your evidence? First, assume the debatable but defensible view that E = K—your evidence is your knowledge.²⁵ Also assume that belief is a necessary condition on knowledge—you only know that P if you believe it. In this section I’ll try to show how broadly “fragmentationist” views about belief can lead to similarly fragmentationist views about evidence, and I’ll then apply those views to the putative cases of HOE discussed earlier in this chapter. What does it take to believe that P? A debatable, but attractive answer is that it involves behaving in ways that make sense if P. For example, someone who believes that there’s beer in the fridge will be disposed, all else equal, to walk to the fridge if she wants a beer, to respond “yes” to the question “is there any beer left?,” to offer beer to thirsty guests, to be surprised upon opening the fridge and failing to see beer, etc.— these are behaviors that make sense if there’s beer in the fridge.²⁶ But one can act as if P in some situations but not others, or when engaged in certain tasks but not others. For example, one might talk like a P-believer, but walk like a P-disbeliever.²⁷ We might capture this by saying that, for the purpose of talking, you believe that P, but for the purpose of walking you believe that ~P—your doxastic life is fragmented. And we can cut more finely. Even within the category of verbal behavior, or non-verbal behavior, some of one’s actions might make sense if P, but not others.²⁸ This kind of fragmentationist view about belief leads naturally to a fragmentationist view about knowledge and (if E = K) evidence. That is, suppose S has a dispositional profile that amounts to believing that P for some purposes, but not others. And assume that P is true, and that S’s (fragmentedly) believing that P has an appropriately non-accidental connection to the truth of P—the sort of connection necessary for knowledge. Then it’s natural to say that S knows that P—and thereby has P as part of her evidence—for some purposes, but not others. Jack Marley-Payne (ms) offers a nice example that illustrates the idea: Take the example of the inarticulate tennis player—let’s call her Serena. Over the course of a rally she can execute a complex plan which involves hitting repetitive shots to first ground her opponent in one position and then wrong foot him in order to win the point. Moreover, she can calibrate her play in response to the court conditions, the abilities of her opponent, whether she desires to humiliate him etc. However, she is unable to explain what she was doing—indeed she may even say things about her play that turn out to be false. Her non-verbal

²⁵ Famously defended by Williamson (). ²⁶ The example is from Schwitzgebel (), who defends a view about belief broadly similar, I think, to the one sketched in this paragraph. There he says believing that P involves “fitting the dispositional stereotype” of someone who believes that P. While I’m inclined to think that’s right, I’m more optimistic than he is about the prospects for giving a somewhat systematic account of which dispositions get stereotypically associated with which beliefs, along broadly radical interpretationist lines, where we can say something systematic about what sorts of dispositions “make sense” for a subject who believes that P. See Lewis (), Stalnaker (), and Dennett (). Also, unlike the previous authors, Schwitzgebel stresses that he thinks dispositions to have irreducibly phenomenal states are among the dispositions associated with having various beliefs. For my purposes in this chapter, I don’t think I need to take a stand on that question. ²⁷ See, e.g., Schwitzgebel (). ²⁸ See, e.g., Elga and Rayo (ms a; ms b).

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   behaviour suggests a belief that a cross-court shot will wrong foot her opponent while her verbal behaviour does not.

Suppose that a cross-court shot will wrongfoot her opponent, and it’s no accident that her non-verbal behavioral dispositions reflect sensitivity to this fact; rather, it’s the product of years of training, which included seeing how opponents reacted to similar situations in the past. In that case, it’s natural to say that Serena knows that a cross-court shot will wrongfoot her opponent. While she’s unable to articulate her knowledge, this doesn’t relegate it to the realm of mere reflex; after all, her choice of a cross-court shot is sensitively dependent on her background beliefs and desires in a way that mere reflexes aren’t. If we think the characteristic functional role of beliefs is to combine with other beliefs to lead you to choose actions that will satisfy your desires—while mere reflexes manifest themselves in rigid ways that aren’t so sensitive to the rest of an agent’s psychology—then what Serena has is belief, rather than mere reflex. And because that belief is non-accidentally true, it’s knowledge. It just happens to be knowledge that she can’t draw on for verbal reports.²⁹ We finally have enough conceptual machinery on the table to start applying fragmentation to the topic of higher-order evidence. With the fragmentationist view of belief and evidence in hand, we can distinguish two readings of the claim that non-trivial HOE is impossible. Let x and y be purposes relative to which a subject might have evidence. Fixed Purpose: It’s never the case that S’s evidencex supports the claim that P, while her evidencex supports the claim that her evidencex doesn’t support the claim that P. Variable Purpose: It’s never the case that S’s evidencex supports the claim that P, while her evidencey supports the claim that her evidencex doesn’t support the claim that P. In the remainder of the chapter, I’ll argue that Fixed Purpose is what we need to rule out the rationality of epistemic akrasia, and the bizarre verdicts about Monty Hall and the possibility of no-lose investigations. But we don’t need Variable Purpose, and familiar putative examples of non-trivial HOE can be interpreted as counterexamples to it. My hope is that this amounts to having our cake and eating it too. We can agree with a version of the intuitively plausible idea that you’re not always in a ²⁹ The idea that what you know depends in some sense on what task you’re engaged in is reminiscent of the version of “contextualism”—in contemporary terminology, probably best interpreted as a version of sensitive invariantism—defended by Michael Williams (; ). He defends a view on which our inability to answer skeptical challenges amounts to lacking knowledge, but only while we are engaged in the epistemological project of answering skeptical challenges. When we turn our attention to other inquiries— history, ornithology, or just what to have for lunch—our knowledge returns. While I’m sympathetic to much of Williams’s discussion, a crucial difference between his approach and the present one concerns higher-order knowledge and evidence. He understands his view as crucially connected to the possibility of failures of KK—cases in which a subject knows, but fails to know that she knows. E.g., “Thus stated, contextualism implies a kind of externalism, for though appropriate contextual constraints will have to be met, if a given claim is to express knowledge, they will not always have to be known, or even believed, to be met” (Williams , p. ). “Externalism thus drives a wedge between knowing something or other and knowing that one knows it” (p. ). By contrast, in the remainder of this chapter I’ll argue that fragmentationist views about knowledge and evidence open up room for reinterpreting cases that seemed to involve straightforward divergences between first-order and higher-order epistemic status.

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  - 

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position to tell what your evidence supports, and we can apply that idea in broadly the ways it’s been applied in the literature. But by distinguishing one version of that idea from a closely related one, we can explain why, properly interpreted, it doesn’t lead to absurdity. I’ll start with the case of the unmarked clock from section .... On the interpretation I’ll suggest, the subject of the example has different information available for different purposes. On my way of fleshing out the case, the subject is guided by one body of information in her assertions and judgments, but a different body of information in her visuomotor behavior. If we add the subscripts aj and vm to “evidence,” so that the subject’s evidenceaj is the knowledge available for guiding assertions and judgments, and her evidencevm is the knowledge available for guiding visuomotor tasks, then the diagnosis will be as follows: the subject’s evidenceaj fails to settle what her evidencevm is. But nothing in the case suggests that her evidenceaj fails to settle what her evidenceaj is, and, as I’ll explain, that’s enough to avoid awkward consequences concerning Monty Hall, or no-lose investigations. Next, I’ll turn to cases of putative ignorance about evidential support relations, such as the JUROR case from section .... I’ll argue that a fragmentationist can make sense of the idea that one ought to be agnostic about whether the accused is at fault, but without understanding that verdict as straightforwardly conflicting with the requirement of total evidence, and without the more general threat of rational akrasia posed by limited evidence concerning epistemic support relations. The basic idea will be that, in an important sense, the relevant evidence the subject has available for determining liability is a relatively sparse one, rather than the richer body that includes all of the evidence that was presented at court. The subject has that richer body of evidence in a sense, but only a sense—she has pieces of that body of evidence available for various different purposes, but she doesn’t have the whole body of evidence available for the purpose of determining liability.

.. Limited discrimination revisited In Salow’s description of the unmarked clock case, which mirrors Williamson’s, we’re told that when the clock’s hand is in fact indicating that it is  minutes past the hour, your powers of discrimination enable you to rule out that it indicates  minutes or fewer, and also to rule out that it indicates  or more, but they don’t allow you to discriminate any more finely than that—you cannot rule out , , or . What form does this discrimination take—how do your powers of perceptual discrimination manifest themselves? While this question isn’t given much attention by either Williamson or Salow, I’ll try to show that it is in fact of central importance. A natural answer is that they manifest themselves in occurrent judgments, and/or assertions—when the clock is in fact pointing at , then you will be disposed to inwardly judge, and/or outwardly assert, claims like the following: “it’s not at  or lower,” “it’s not at  or higher,” “it might be at , , or .”³⁰ And likewise, more generally, for other positions—when the clock is in fact pointing towards n, you’ll be ³⁰ While Williamson mainly describes the case in terms of “discrimination” and “knowledge” without indicating how they might be manifested, Salow explicitly talks about what the subject can “reliably judge” (Salow b, p. ).

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   disposed to judge and/or assert that it’s not at n –  or below, not at n +  or above, but nothing stronger than that. So far, so good. However, this way of thinking about how her powers of discrimination manifest themselves is inconsistent with another crucial feature of the case, as Williamson and others who’ve written about it understand it.³¹ Namely, you yourself—the subject of the case—could know the general epistemic features of the case; you can know that when the clock is pointing to n, your powers of discrimination enable you to know that it is pointing between n –  and n + , but nothing stronger. Intuitively, this is plausible—you could have reliable information about just how good your vision is, perhaps provided by an optometrist. Why is the idea that your powers of discrimination manifest themselves in judgment and assertion inconsistent with the idea that you can know the general epistemic features of the case? For the following reason. Suppose the clock is pointing at . Then your powers of perceptual discrimination will manifest themselves in your making the assertions and judgments mentioned in the previous paragraph—explicitly judging and/or asserting that it might be between  and , but nowhere else. But if you make those judgments or assertions, you can also notice that you’ve made them. And making those assertions and judgments is only compatible with knowing that it is between  and —on the assumption that your perceptual knowledge manifests itself in judgment and assertion, then if you had different perceptual knowledge you’d make different assertions and judgments. But then, contra the stipulation of the case, you’re in a position to know what your evidence is—namely, that the clock is pointing between  and . And because you know the general epistemic features of the case, we get the absurd result that you are in a position to know that the clock is pointing to —since you know that your evidence is only [, ] when it’s actually pointing to . What went wrong? One quick response is just to deny that it’s possible for there to be any version of the case in which both (a) your powers of perceptual discrimination are limited, and (b) your visual evidence is perfectly sensitive to the actual position of the clock. To motivate this response, we might say that the thought that your powers of discrimination are limited—that is, that you can’t tell exactly where the clock is pointing just by looking—is intimately bound up with the idea that, compatible with your having the visual evidence you have, there are various different positions the clock might be pointing. So if we want to hold on to the idea that you have limited powers of discrimination, we should reject the idea that there’s a – function from positions of the clock to bodies of evidence you get when you look at the clock. This is the route taken by Stalnaker () in the course of responding to a similar case in Hawthorne and Magidor (). Applying his response to the present case would lead to the result that, rather than being determined by the actual position of the clock, your evidence is determined by your best guess as to the position of the clock, which you are in a position to know. Moreover, there’s no strict mapping from where the clock is pointing to what your best guess will be. So the question “what is the

³¹ See especially Christensen (b, pp. –).

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  - 



subject’s evidence when the clock is in fact pointing to position n?,” will have a range of possible answers, rather than a unique one. While I’m sympathetic to this response, I think we can be a bit more concessive to the intended interpretation of the case. That is, we can allow that there’s a sense in which the subject’s evidence depends on the actual position of the clock, which the subject is nevertheless not in a position to know, while still resisting the central lesson about higher-order ignorance. We were led into absurdity when we assumed that your powers of perceptual discrimination would manifest themselves in a way that you were in a position to notice, as when you notice what you say or judge. So suppose they don’t. Suppose that your powers of perceptual discrimination manifest themselves in your visuomotor behavior—for example, they manifest themselves in the direction your hand will move if you decide to reach for the pointer, and other tasks that require integrating visual information with motor activity. For example, suppose your hand-eye coordination isn’t perfect, and if asked to very quickly reach out and touch the clock’s pointer, you won’t always hit it (imagine the hand is quite thin, and you’re asked to move very quickly). However, when the pointer is in fact pointing at n, you’ll never touch a position below n – , or above n + . And suppose this isn’t just a quirky fact about a particular task—for a wide range of visuomotor tasks, when the clock is pointing at n, your visuomotor behavior will be as if the clock is pointing somewhere in [n – ; n + ]. This lets us vindicate part of the description of the case—the part concerning how your evidence about where the clock is depends on where the clock is actually pointing. But what about the description of the case in terms of higher-order evidence? That is, can we make sense of the idea that, when the clock is pointing at , your evidence leaves open various possibilities concerning what your evidence is? If we stick to the idea that what evidence you have is manifested in your visuomotor behavior, it’s hard to see how to do so. That is, while it’s not so hard to see how, insofar as we’re interested in explaining your visuomotor behavior, we might fruitfully interpret you as having the evidence that the clock is pointing somewhere in [, ], it’s much harder to see how, insofar as we’re interested in explaining your visuomotor behavior, we might fruitfully interpret you as lacking evidence about what your evidence is. That is, it’s just not clear how we should expect information or lack thereof concerning what your evidence is to be manifested in visuomotor behavior. So let’s return to assertion and judgment. Plausibly, when you are in fact completing visuomotor tasks as if the clock is pointing in [, ], you won’t be in a position to reliably assert or judge that you are doing so. For example, suppose the clock is pointing to , and you’ve just been asked to reach out and touch the pointer ten times in a row. Each time you touch a position somewhere in [, ]. Now you’re asked: “did you touch a position below  any of those times?” Plausibly, you’ll respond: “I don’t know, maybe. My best guess is that the clock is pointing to , but that’s just a guess, so it might be pointing to . And given that my hand-eye coordination isn’t perfect, if it’s in fact pointing to , I probably hit  a few times.” And your coherently giving that speech is consistent with your being happy to assert that if the clock is in fact pointing at , then you did not touch a position below  on any of those occasions. Unlike when we were thinking of the

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   manifestation of one’s knowledge solely in terms of assertion or judgment, now we can see how one’s knowledge that the clock is pointing in [, ] could be reliably manifested, without your being in a position to learn from that reliable manifestation that you know that the clock is in [, ]. To keep all this straight, it will be helpful to introduce subscripts. Let your evidenceaj be your knowledge available for guiding assertions and judgments, and your evidencevm be the knowledge available for guiding visuomotor tasks. The diagnosis I’ve been offering can be stated as follows. When the clock is in fact pointing at n, your evidencevm is that it is pointing somewhere in [n – ; n + ]. Moreover, that general fact about how your evidencevm sensitively depends on the actual position of the clock is part of your evidenceaj. And when your evidencevm is that the clock is pointing somewhere in [k – ; k + ], your evidenceaj doesn’t settle that this is your evidencevm—rather, there will be various different possibilities compatible with your evidenceaj concerning what your evidencevm is. Which will those be? Here, we cannot give a unique answer, because while your evidencevm sensitively depends on the actual position of the clock, your evidenceaj does not—that way lies absurdity, as we saw. What does your evidenceaj depend on? Here I’m happy to stick with Stalnaker’s model—perhaps it depends on your “best guess” as to the position of the clock.³² The present diagnosis lets us avoid awkward consequences concerning Monty Hall. We can get a version of the Monty Hall case going where one “learns” that the prize isn’t behind door C—in the sense that one’s visuomotor dispositions now involve being disposed to complete visuomotor tasks as if it’s not there, in some way non-accidentally connected to the fact that it’s really not there—but where one doesn’t gain this information in a form that would make it available for assertion, judgment, and crucially, deliberation. And since the apparent absurdity in the case involved trying to imagine the sort of planning and deliberation that would be appropriate for an agent who had the evidence that the prize wasn’t behind door C, but lacked the evidence that she had this evidence, that absurdity vanishes when the only sense in which she “has” the evidence that the prize isn’t behind door C is a sense that wouldn’t be manifested in her deliberation. The diagnosis of Salow’s “no-lose inquiry” case would proceed along much the same lines. Taking a step back, in what sense does my diagnosis do without non-trivial HOE? It’s true that there’s a sense in which my explanation of the case makes appeal to non-trivial evidence concerning what your evidence is. But the two occurrences of “evidence” in the previous sentence require different readings—that is, I say that in the case of the unmarked clock, you have non-trivial evidenceaj concerning what your evidencevm is. But if we hold fixed the subscript, we don’t have non-trivial ³² As I hope is clear enough, it’s not central to my diagnosis that it’s visuomotor information that we contrast with information available for guiding assertion and judgment. It’s just that there be some such contrast—some type of task that one’s evidence (in a sense) can be used to guide, but whose manifestation can’t be reliably noticed and remarked upon by the subject who has that evidence. And since assertions and judgments can be noticed and learned from—when you assert or judge that P, you’re in a position to know that you’ve asserted or judged that P—for a diagnosis of my sort to work, we need some way that knowledge can be manifested other than in assertion and judgment. Visuomotor behavior is just one convenient, natural alternative.

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  - 



higher-order evidence: we saw that non-trivial evidenceaj concerning one’s evidenceaj led to absurdity, and it’s not clear how to interpret talk of evidencevm concerning one’s evidencevm. So while my diagnosis lets us say that in the case of the unmarked clock, your evidence doesn’t settle what your evidence is, that turns out to be a less than maximally perspicuous description of the situation. And my suspicion is that this will hold quite generally—cases that are tempting to describe as involving a subject’s evidence failing to settle what her evidence is will always turn out to be cases where we can distinguish different senses of “evidence,” and so distinguished, they will be cases in which one’s evidence—in one sense—fails to settle what one’s evidence—in some other sense—is. Given the distinction drawn earlier, they will be counterexamples to Variable Purpose, but not Fixed Purpose. While I can’t offer a full defense of that claim here, I hope I’ve at least shown the following: absent some such distinction, we get puzzles and paradoxes, as illustrated in section ... And in a general, schematic sort of case meant to illustrate the possibility of rational ignorance concerning one’s evidence—the case of the unmarked clock—the diagnosis I’ve offered is a natural and attractive one that avoids the paradoxes. So it’s natural to hope that similar diagnoses should be available in other cases.

.. Epistemic support relations revisited What about the cases meant to illustrate higher-order uncertainty not via ignorance about what one’s evidence is, but instead about what it supports? Can the strategy of the subsection .. be of any help? I believe it can, though I admit the fit here is a bit less natural. There is a constellation of closely related suggestions that a number of writers have defended for how to deal with HOE that prompts uncertainty about epistemic support relations—sometimes it’s put in the language of “bracketing” (Elga ; Christensen a), sometimes “calibration” (Sliwa and Horowitz ; Schoenfield ), sometimes in other terms.³³ The suggestion that we should use these methods is typically treated as a kind of sui generis epistemological principle, which may conflict with other more general principles—most notably, the requirement of total evidence (Kelly ). My ultimate goal in this section is as follows. I want to show how the kind of “bracketing” recommended by the authors just mentioned fits naturally with the idea, defended in section ., that evidence can be available for guiding some tasks, but not others. Since this idea is plausible and defensible independently of considerations having to do with higher-order evidence, my aim is to make “bracketing” seem more principled, and less ad hoc. I also hope that my discussion will render the prima facie conflict with the total evidence principle more palatable, in part by making it a bit less clear just how the total evidence principle should be interpreted, once we adopt the fragmentationist view of evidence. Return to the case of JUROR, described in section .... The recommendation of the bracketer is that, after hearing about the disagreement of her peers, the juror should form her opinion as follows. She should set aside the particular details of the ³³ Roger White () calls it “treating oneself and others as thermometers.” In this volume, Sophie Horowitz discusses a similar suggestion couched in terms of “perspective.”

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

 

evidence presented at trial and why she thinks those details point to the liability of the accused. Instead, she should reason from the sparse body of essentially psychological evidence, which includes only (a) the fact that she formed the judgment she did, (b) the fact that her peers formed the judgments they did, and (c) general facts that bear on her and her peers’ reliability in matters like these. And that sparse body of evidence will support, I assume, agnosticism about whether the accused is at fault. This recommendation looks like it straightforwardly conflicts with the requirement of total evidence.³⁴ There are facts that are part of the juror’s total evidence, and which bear on the question of whether the accused is at fault, that the bracketer says should nevertheless be set aside or ignored when reasoning about whether the accused is in fact at fault. In light of section ., however, I believe we should be a bit uneasy when we see phrases like “the juror’s total evidence.” After all, we saw there that it can be helpful to distinguish between the evidence a subject has available for some tasks from the evidence she has available for others. And in such cases, “the subject’s total evidence” will threaten to be ambiguous, or liable to refer to different bodies of evidence depending on which task we are contemplating the subject performing. While in that section we only distinguished between two broad sorts of tasks—discursive versus visuomotor—there’s no principled reason we can’t distinguish more finely. Elga and Rayo (ms a; ms b), following up on some suggestions of Stalnaker (; ), argue that solving the problem of logical omniscience requires distinguishing the information a subject has for verbally answering some questions from information a subject has for verbally answering others.³⁵ How does this possibility bear on the case of JUROR? Here’s the idea. While the individual items of evidence presented at trial are each available to the juror for various purposes—answering pointed questions about whether, for example, the accused drove a sedan, or whether the plaintiff was in Tallahassee on March —they may not be collectively available for answering the question: “is the accused at fault?” For that latter task, perhaps the only evidence available is the sparse body of broadly psychological evidence that, according to the bracketer, should determine the juror’s view. Phenomenologically at least, this strikes me as plausible. When dealing with a simple body of evidence, I feel like I can base my beliefs directly on that evidence—for example, if I see that the streets are wet, I’ll straightaway think that it recently rained, and it’s clear to me how my evidence supports my belief. But when dealing with a suffciently complex and multifaceted body of evidence, with different pieces pointing in different directions and in ways that can’t be straightforwardly weighed against one another, the process feels quite different. I’ll think about it for a while, and eventually I’ll find myself stably inclined to think one thing or another, but not in a way where it’s at all transparent to me how the particular pieces of evidence combine to generate my resultant doxastic inclination. Suppose I accede to that inclination— for example, I’m in the position of JUROR before having heard from my peers, I find myself inclined to believe that the accused is probably guilty, and I go on to believe that. ³⁴ See Williamson () on the dangers of “psychologizing the evidence.” ³⁵ While not put in quite these terms, Rayo () is in a very similar spirit, and nicely complements my strategy in the text for addressing the JUROR case.

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  - 



I’m tempted to say that in a case like this my ultimate belief is based not directly on my evidence—as it is in the simple cases—but is instead based on an inclination prompted by reection on that evidence. While the individual items of evidence are available to me in a sense—I can base some beliefs directly on them, taken individually or in small groups—they’re not collectively available to me for answering the target question. I didn’t mention disagreement in the previous paragraph. But it’s not hard to factor in. Just as I can base a belief that the accused was at fault on the fact that I was inclined to so believe that after reecting on the evidence, I can base agnosticism about whether the accused was at fault on the facts that (a) I was initially inclined to believe that the accused was at fault, (b) my peers were inclined to believe the opposite, and (c) none of us have any distinctive advantage when it comes to evaluating evidence of this sort. I admit that this view complicates the total evidence requirement, and threatens to reduce it to triviality—if a subject can always avoid the charge of violating the requirement by saying that the total evidence wasn’t really “available” for the task in question, then what does the requirement rule out? But rather than a reason to reject the strategy I’ve been outlining for how to think about disagreement, I think this is a genuine diffculty with how to interpret the total evidence requirement once we allow for the possibility of fragmented belief and evidence, which I think we have independent reason to do. We don’t want the total evidence requirement to rule that a subject who lacks the ability to integrate the information available to her for guiding visuomotor tasks with the information available for guiding explicit verbal reasoning thereby counts as irrational; there’s nothing irrational about being able to hit a bullseye without being able to explain what you’re doing. But once we admit that, it becomes diffcult to say just what sorts of failures to integrate bodies of information available for distinct tasks count as failures of rationality, of the sort that the total evidence requirement rules irrational. And absent an answer to that question, it’s not clear whether, properly interpreted, the total evidence requirement will rule out the kind of “bracketing” that’s been discussed in the literature on disagreement and HOE more generally. How broadly will this strategy work, if it works at all, for vindicating “bracketing” as a response to HOE prompting uncertainty about evidential support relations? My diagnosis of J depended on the idea that the subject in question is dealing with a complex body of evidence whose significance for the target question is diffcult to discern, but not all putative cases of rational uncertainty about epistemic support relations have that structure. For example, what about a case where my evidence is that the streets are wet, and I’m inclined to believe that it rained on that basis, but I then find out that I’ve taken a drug that makes me very bad at evaluating evidence?³⁶ This case, I think, is much harder to fit into the mold of the previous one, since it’s not clear in what sense I might fail to be in a position to base a belief that the streets are wet directly on the fact that it rained, even after taking the drug. I mention the case only to set it aside—I myself have a very hard time wrapping my head around such cases, and am inclined to think not much weight should be put on them. It’s not clear to me what we should say about what evidence is available to such a subject for what purposes, nor is it clear to me what we should say such a subject should believe

³⁶ This case is inspired by Christensen (a).

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

 

in such a case. While I feel the force of the bracketer’s position in cases like J, I think the fragmentationist can nicely handle those cases. I’m not sure what the fragmentationist can or should say about cases like this one, but that uncertainty matches my uncertainty about what the right verdict is. What about the description of disagreement cases in terms of HOE? It seemed natural to say that the juror’s evidence supported a particular belief, but failed to support the claim that it supported that belief. How should we revise this claim if we’re fragmentationists about evidence, along the lines I’ve been discussing? The closest thing we can say, I think, is the following. The evidence the juror would have for the purpose of determining whether the accused was at fault, if she had cognitive powers far vaster than ours—vast enough for the inference from that evidence to the accused’s fault to be as transparent and direct for her as the inference from wet streets to rain is for us—supports the claim that the accused is at fault. But the evidence the juror in fact has available for the purpose of determining whether the accused was at fault fails to support the claim in the previous sentence—it fails to support the claim that the evidence she would have if she had more impressive cognitive powers supports the claim that the accused is at fault. So there’s no single reading of “evidence” on which the juror’s evidence supports P, while also supporting agnosticism concerning whether it supports P—no counterexample to Fixed Purpose. Rather, the sense in which it supports P, and the sense in which it supports agnosticism, are quite different. And that lets us avoid the threat of epistemic akrasia. Once we offer the present diagnosis, we’re under no pressure to say that the juror’s evidence supports believing some claim, while also believing that she shouldn’t believe it. Rather, we can say the evidence she has available for determining whether the accused is at fault unequivocally supports agnosticism on that question, and supports believing that she should be agnostic on that question. Of course, if the subject had more impressive cognitive powers, then she’d have a different body of evidence available for determining whether the accused is at fault—one that would support an affirmative answer. But that’s different from her already having a body of evidence that would rationalize an affirmative answer (while failing to rationalize the claim that it would rationalize an affirmative answer), and so we don’t need to say that her presently available evidence—for any purpose—supports an akratic state.

. Conclusions When we say without qualification that a subject’s evidence can fail to settle what her evidence supports—either because it fails to settle what her evidence is, or because it fails to settle questions about which bodies of evidence support which propositions— we run into troubles. But to flatly deny that this is possible beggars belief.³⁷ My strategy in this chapter has involved following the old dictum: “whenever you meet a contradiction, draw a distinction.”³⁸ By allowing counterexamples to Variable Purpose—allowing one version of the idea that there can be non-trivial HOE—we ³⁷ See Dorst (this volume).

³⁸ William James attributes it to the Scholastics.

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  - 

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can have a framework for describing a host of cases of epistemological interest. But by ruling out counterexamples to Fixed Purpose—rejecting a different version of the idea that there can be non-trivial HOE—we can avoid the troubles that the more familiar descriptions engender.

References Bronfman, A. (). “Conditionalization and Not Knowing That One Knows.” In: Erkenntnis , pp. –. Christensen, D. (a). “Does Murphy’s Law Apply in Epistemology? Self-Doubt and Rational Ideals.” In T. Gendler and J. Hawthorne (eds), Oxford Studies in Epistemology I, Oxford University Press, pp. –. Christensen, D. (b). “Epistemology of Disagreement: The Good News.” In: The Philosophical Review , pp. –. Christensen, D. (a). “Higher-Order Evidence.” In: Philosophy and Phenomenological Research , pp. –. Christensen, D. (b). “Rational Reflection.” In: Philosophical Perspectives , pp. –. Dennett, D. (). The Intentional Stance, The MIT Press. Dorst, K. (this volume). “Higher-Order Uncertainty.” In M. Skipper and A. Steglich-Petersen (eds), Higher-Order Evidence: New Essays, Oxford University Press. Dorst, K. (ms). “Evidence: A Guide for the Uncertain.” Unpublished manuscript. Egan, A. (). “Seeing and Believing: Perception, Belief Formation and the Divided Mind.” In: Philosophical Studies , pp. –. Elga, A. (). “Reflection and Disagreement.” In: Noûs , pp. –. Elga, A. (). “The Puzzle of the Unmarked Clock and the New Rational Reflection Principle.” In: Philosophical Studies , pp. –. Elga, A. (ms). “Lucky to be Rational.” Unpublished manuscript. Elga, A. and A. Rayo (ms a). “Fragmentation and Information Access.” Unpublished manuscript. Elga, A. and A. Rayo (ms b). “Fragmentation and Logical Omniscience.” Unpublished manuscript. Feldman, R. (). “Reasonable Religious Disagreements.” In A. Louise (ed.), Philosophers Without Gods, Oxford University Press. Feldman, R. (). “Evidentialism, Higher-Order Evidence, and Disagreement.” In: Episteme (), pp. –. Gendler, T. (). “Alief and Belief.” In: The Journal of Philosophy , pp. –. Greco, D. (a). “Iteration and Fragmentation.” In: Philosophy and Phenomenological Research (), pp. –. Greco, D. (b). “A Puzzle about Epistemic Akrasia.” In: Philosophical Studies , pp. –. Hawthorne, J. and O. Magidor (). “Assertion, Context, and Epistemic Accessibility.” In: Mind , pp. –. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs , pp. –. Horowitz, S. (this volume). “Predictably Misleading Evidence.” In: M. Skipper and A. SteglichPetersen, Higher-Order Evidence: New Essays, Oxford University Press. Kelly, T. (). “Peer Disagreement and Higher Order Evidence.” In: R. Feldman and T. Warfield (eds), Disagreement, Oxford University Press. Lasonen-Aarnio, M. (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research , pp. –.

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Lewis, D. (). “Radical Interpretation.” In: Synthese , pp. –. Lewis, D. (). “Logic for Equivocators.” In: Noûs , pp. –. Littlejohn, C. (). “Stop Making Sense? A Puzzle about Evidence and Epistemic Rationality.” In: Philosophy and Phenomenological Research (), pp. –. Marley-Payne, J. (ms). “Task-Indexed Belief.” Unpublished manuscript. Popper, K. (). Conjectures and Refutations, Routledge. Rayo, A. (). “A Puzzle about Ineffable Propositions.” In: Australasian Journal of Philosophy , pp. –. Rayo, A. (). The Construction of Logical Space, Oxford University Press. Rinard, S. (forthcoming). “Reasoning One’s Way out of Skepticism.” In: Brill Studies in Skepticism. Salmon, W. (). “Rational Prediction.” In: British Journal for the Philosophy of Science (), pp. –. Salow, B. (a). “Elusive Externalism.” In: Mind. Online first. Salow, B. (b). “The Externalist’s Guide to Fishing for Compliments.” In: Mind (), pp. –. Schoenfield, M. (). “A Dilemma for Calibrationism.” In: Philosophy and Phenomenological Research , pp. –. Schwitzgebel, E. (). “In-Between Believing.” In: Philosophical Quarterly , pp. –. Schwitzgebel, E. (). “A Phenomenal, Dispositional Account of Belief.” In: Noûs , pp. –. Sliwa, P. and S. Horowitz (). “Respecting All the Evidence.” In: Philosophical Studies , pp. –. Smithies, D. (). “Ideal Rationality and Logical Omniscience.” In: Synthese , pp. –. Stalnaker, R. (). Inquiry, The MIT Press. Stalnaker, R. (). “The Problem of Logical Omniscience, I.” In: Synthese , pp. –. Stalnaker, R. (). “The Problem of Logical Omniscience II.” In: Stalnaker, Context and Content, Oxford University Press. Stalnaker, R. (). “On Hawthorne and Magidor on Assertion, Context, and Epistemic Accessibility.” In: Mind , pp. –. Titelbaum, M. (). “Tell Me You Love Me: Bootstrapping, Externalism, and No-Lose Epistemology.” In: Philosophical Studies , pp. –. Titelbaum, M. (). “Rationality’s Fixed Point (Or: In Defense of Right Reason).” In T. Gendler and J. Hawthorne (eds), Oxford Studies in Epistemology V, Oxford University Press. White, R. (). “On Treating Oneself and Others as Thermometers.” In: Episteme , pp. –. Williams, M. (). Unnatural Doubts: Epistemological Realism and the Basis of Scepticism, Blackwell. Williams, M. ([]). Groundless Belief, Princeton University Press. Williamson, T. (). Knowledge and its Limits, Oxford University Press. Williamson, T. (). The Philosophy of Philosophy, Blackwell. Williamson, T. (). “Improbable Knowing.” In: T. Dougherty (ed.), Evidentialism and its Discontents, Oxford University Press. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research , pp. –.

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5 Predictably Misleading Evidence Sophie Horowitz

Evidence can be misleading: it can rationalize raising one’s confidence in false propositions, and lowering one’s confidence in the truth. But can evidence be predictably misleading? Can a rational agent with some total body of evidence know that this evidence makes it rational for her to believe a (particular) falsehood? It seems not: plausibly, rational agents believe what their evidence supports. Suppose for reductio that a rational agent can see ahead of time that her evidence is likely to point towards a false belief. Since she is rational, if she can anticipate that her evidence is misleading, then it seems she should avoid being misled. But then she won’t believe what her evidence supports after all. That is to say, if evidence were predictably misleading, it wouldn’t be misleading in the first place. So, it seems, evidence cannot be predictably misleading. The argument sketched above has a lot of intuitive appeal. But it poses a problem for another compelling epistemological view: the view that so-called “higher-order” evidence can require us to revise our beliefs. As I will argue, higher-order evidence is predictably misleading. Insofar as higher-order evidence rationalizes changing one’s beliefs, this change tends to result in less accurate beliefs—and we can know this on the basis of an a priori argument. This gives us a new and powerful reason to worry about the significance of higher-order evidence. In this chapter I will develop and examine this objection, and explore some possibilities for addressing it.

. Higher-order evidence Higher-order evidence, as I’ll understand it, is evidence that bears on the functioning of one’s rational faculties, or on the significance of other evidence that one has. The following is an example of what I have in mind: Cilantro: Sam’s trustworthy roommate leaves a Tupperware container of chicken curry in the fridge. A sticky note on top reads: “If the following logical proof is valid, the green specks in this curry are cilantro. If not, they are mint.” A valid proof follows. Sam (sadly) hates cilantro—it tastes soapy to him—so he works through the proof before risking a bite. He correctly ascertains that the proof is valid and concludes that C: the curry contains cilantro. Then Sam notices a headline in the newspaper: “Gene Causing Soapy Cilantro Taste Linked to Poor Logical Reasoning Abilities.” The story goes on to detail the results of a study Sophie Horowitz, Predictably Misleading Evidence In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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showing that people with Sam’s aversion to cilantro perform abnormally poorly on exactly the types of logical reasoning required to assess the proof on the sticky note. Let’s call Sam’s first-order evidence (the sticky note, etc.) “E,” and his higher-order evidence (the newspaper story) “HOE.” Many people have the following intuitive reaction to cases like Cilantro: After examining E, Sam should be highly confident in C. After examining both E + HOE, Sam should reduce confidence in C. Much recent literature has defended this intuitive reaction, examining epistemic principles that could explain it and emphasizing the odd consequences of rejecting it.¹ But there has also been much debate over how this intuitive reaction could possibly be correct. Some epistemologists suggest that if Sam does reduce confidence in C, he is throwing away or ignoring his first-order evidence;² he may be violating plausible epistemic norms like consistency or probabilistic coherence;³ and it is hard to see what kind of “epistemic rule” could be guiding him.⁴ A common thread in the higher-order evidence literature seems to be that there is something wrong with Sam if he does not reduce confidence in C, but that it is hard to accommodate this thought in a single, consistent picture of epistemic rationality. A number of authors—including those who take higher-order evidence to have rational import—have remarked that higher-order evidence seems different from first-order evidence, in important ways. I take the discussion that follows to be in line with these thoughts. The present challenge, however, develops the problem of higherorder evidence in an especially conspicuous and troubling way.

. The problem for HOE: it is systematically, predictably misleading. I’ll take as a working assumption that HOE does make it rational to reduce confidence in C, as the intuitive reaction suggests. Given that assumption, it’s easy to see how HOE is misleading in our paradigm case, Cilantro: specifically, it is misleading regarding C, the proposition for which it has distinctly higher-order import. Sam’s first-order evidence indicated that the curry contained cilantro, which, given the trustworthiness of his roommate, is highly likely to be correct. But after reading the newspaper, Sam became much less confident that the curry contained cilantro; his overall belief state moved farther from the truth. We can even suppose that Sam’s new, lower level of confidence in C rationalized eating some of the curry; in that case it looks like respecting HOE was a real tragedy for Sam!

¹ See, e.g., Christensen (; ; ) and elsewhere, Elga (), and Feldman (). ² Kelly (, p. ); Christensen (). ³ Christensen () and elsewhere. ⁴ Lasonen-Aarnio ().

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Sam’s situation isn’t just an unlucky feature of this particular case, Cilantro. Here is a more general argument: Higher-order evidence is misleading Premise  First-order evidence tends to point towards the truth. Premise  Higher-order evidence tends to point away from what the first-order evidence supports. Conclusion Higher-order evidence tends to point away from the truth. As I will argue here, the conclusion of this argument is a true generalization about higher-order evidence. Higher-order evidence tends to be misleading. And since the argument above is made on a priori grounds—P and P simply follow from reflections on the nature of evidence and how it works—higher-order evidence is predictably misleading. Indeed, Sam himself could go through this argument and figure out that his own higher-order evidence is probably misleading. Before defending the premises, I would like to pause to clarify a few things: two about the argument’s terminology, and another about the form of the argument itself. First, the phrase “tends to” in this argument is deliberately loose and informal. There are a number of ways in which one could make it more precise. For example, for the first premise, one might say: “the evidence supports P if it is a reliable indicator of P.” Or one might say: “when the first-order evidence supports P highly, the expected accuracy of P is high.”⁵ And so on. For present purposes, I won’t endorse any particular one of these. A second terminological note concerns what it means to say that higher-order evidence is misleading. As I understand it here, a piece of evidence is not just misleading, full stop, but misleading with respect to a proposition, in an evidential context. As I mentioned above, I am interested in whether higher-order evidence is misleading with respect to those propositions for which it has distinctly higher-order import. In Cilantro, it is natural to talk about the newspaper story, or some proposition concerning the newspaper story, as a piece of evidence. That piece of evidence rationally affects Sam’s attitudes about a number of different topics, in a number of different ways. For example, it might rationally raise his confidence that some scientists are studying the cilantro-soap-taste gene, or that the soap-taste gene is connected to logical ability. A piece of evidence might be misleading regarding some of these propositions and not others.⁶ Here, I am most interested in the way higher-order evidence affects the proposition(s) that it targets qua higher-order evidence. The second qualifier, “in an evidential context,” is necessary for the familiar reason that the import of a piece of evidence can change based on what other evidence one has. A piece of evidence can be misleading against the backdrop of one body of evidence, but not another. Here, I am interested in a particular backdrop as well: the accompanying first-order evidence. So when I say that higher-order evidence is misleading, what I mean is that against the backdrop of its accompanying ⁵ See Schoenfield (a) for a similar discussion, put in terms of expected accuracy. Schoenfield argues that respecting higher-order evidence does not maximize expected accuracy. ⁶ Because evidence has many effects all at once, it would be more accurate to talk about “higher-order rational effects.” To stay in line with the literature, I will stick with “higher-order evidence.”

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first-order evidence, higher-order evidence tends to make one’s rational opinion about the proposition(s) that it targets qua higher-order evidence less accurate. (Again: all of this is just to articulate the claim I will defend. I haven’t yet defended it!) Finally, note that this argument is not deductively valid; I formalize it here not to show a logical implication, but to isolate the assumptions that support the claim I am interested in. For it could be that we only, or almost always, receive higher-order evidence in those rare situations where the first-order evidence is misleading. This is the case with undercutting or rebutting defeaters: we tend to get evidence that the lighting is tricky when it is tricky, for instance, and the jellybean isn’t red after all. In fact, one might think that ordinary defeat works precisely by alerting us to the fact that we are in an odd situation where our first-order evidence is misleading.⁷ But this is not how things are for higher-order evidence. There is no reason to think that when we encounter higher-order evidence should be at all correlated, positively or negatively, with whether our first-order evidence is misleading. Misleading evidence is hard to spot and comes along more or less randomly (unless someone is trying to trick us—but that is not correlated with receiving higher-order evidence). Therefore, since higher-order evidence tends to point away from what the first-order evidence supports, we should expect that higher-order evidence tends to point away from the truth.⁸ This looks bad. Plausibly, epistemic rationality involves believing what one takes to be true, from one’s own perspective. Epistemic rationality also, plausibly, involves believing what one’s evidence supports. Cases like Cilantro, which appear to be cases of predictably misleading evidence, bring out a tension between these plausible thoughts. How could it be rational to believe what our evidence supports, if we know that doing so is likely to lead us away from the truth? Before jumping into that challenge, though, let me say more to motivate it. I will defend P and P in subsections .. and ...

.. Defending P First, here is P again: P

First-order evidence tends to point towards the truth.

⁷ Allan Coates () makes a similar observation: he points out that ordinary defeat indicates that one’s earlier belief was justified, but that one would no longer be justified in holding it. Higher-order defeat suggests that one’s earlier belief was not justified to begin with. ⁸ At this point I have sometimes encountered the following objection: couldn’t we make a parallel argument about any old new piece of evidence? After all, new evidence often makes it rational to change our beliefs. By this argument, isn’t “the new piece of evidence” predictably misleading? This argument sounds fallacious, for much the same reason that Kripke’s dogmatism argument sounds fallacious. However, the argument here is different from this fallacious argument. In particular, the analogue of P in the “new evidence” argument is false. New evidence does not tend to point away from what the rest of one’s evidence supported; in fact, if P is true, new evidence should tend to point in the same direction as the original evidence. The force of my main argument here comes from the fact that higher-order evidence tends to go against what the first-order evidence supports; so the “new evidence” argument is not analogous after all. Thanks to Paolo Santorio and James Garson for helpful discussion on this point.

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I do not have much to say in defense of P. Some skeptics might deny it. But anyone who thinks that evidence is relevant to what we should believe, and often rationalizes changing our opinions about the world—anyone who herself strives to believe what her evidence supports, and revises her beliefs on its basis—should accept P. After all, what we believe is what we take to be true. So it seems we can’t coherently set out to believe what our evidence supports unless we think that our evidence is generally a guide to truth.

.. Defending P I take P to be the more controversial premise in the argument above. Here it is again: P Higher-order evidence tends to point away from what the first-order evidence supports. I will explain what P means first, and then why we should hold it. To rephrase P: if one’s first-order evidence rationalizes a certain change in one’s beliefs, then higherorder evidence will tend to rationalize making a change in the opposite direction. For example, if one’s total first-order evidence makes it rational to increase confidence in P (relative to one’s prior confidence in P, before receiving that evidence), then higherorder evidence bearing on that first-order evidence will tend to make it rational to decrease confidence in P (relative to one’s prior confidence in P given the first-order evidence alone). Again, it is important to note that in P, I am just focusing on the propositions that higher-order evidence targets qua higher-order evidence. In this case, that means Sam’s belief about C: whether the curry contains cilantro. My claim in P is that, insofar as higher-order evidence has a distinctive sort of impact, it works by defeating, neutralizing, or weakening the effect of the first-order evidence that it targets. (But can’t higher-order evidence confirm? I’ll get to this in Objection (A), below.) In Cilantro, Sam’s first-order and higher-order evidence point in opposite directions. Sam’s first-order reasoning supports P, and after going through the proof it is rational for Sam to become highly confident in P. But after going through the proof and reading the newspaper, it is rational for Sam to become less confident in P. So Sam’s higher-order evidence counteracts the effect of his first-order evidence, concerning C. (But doesn’t it just counteract the effect that Sam thinks his first-order evidence has? I’ll get to this in Objection (B).) I take it that if someone denies P, it is because they have an alternative understanding of higher-order evidence, on which P is false. So I will argue for P by arguing against two objections of this form. Both of these objections say, in different ways, that the phenomenon I am discussing is a special feature of Cilantro, rather than a more general truth. In denying these views I don’t take myself to have definitively proven P, but to cast doubt on the two most plausible ways to deny it.

...  (): - ? P says that higher-order evidence tends to undo or counteract the effect of firstorder evidence. Let’s call this phenomenon “higher-order defeat.” (Cilantro is a case of higher-order defeat, as are most of the cases of higher-order evidence discussed in

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the literature.) But in some cases, higher-order evidence instead confirms what the first-order evidence supports. Call that phenomenon “higher-order confirmation.” So one might object as follows: There’s no reason to think that higher-order defeat is more common than higher-order confirmation. Therefore, we have no reason to believe P. In fact, we can’t say anything general about what rational effect higher-order evidence “tends” to have, relative to what the first-order evidence supports. Sometimes it defeats, and sometimes it confirms. I agree that higher-order confirmation is possible. Here is an example: Peer agreement: We all take a test, and I’m not sure that I got the right answer to question . In fact, I did get it right, and I am rationally pretty confident in the right answer—though I do have some doubt, so I’m not completely confident. Then I talk to the rest of the class, afterwards, and find out that everyone else got the same answer that I did. My classmates’ agreement is higher-order evidence: it bears on the significance of my first-order evidence, and suggests that I did in fact accommodate it rationally. And intuitively, it should make me more confident in the right answer than I was before. It therefore points in the same direction as the first-order evidence.⁹ However, I don’t think that the possibility of higher-order confirmation removes, or even mitigates, the central challenge. First, even if we agree with the objector that higher-order confirmation is just as common as higher-order defeat, we could reformulate the challenge by restricting our attention to cases of higher-order defeat. Higher-order defeating evidence still tends to be misleading. And we can tell when our higher-order evidence is defeating, not confirming: so, we can predict when it is misleading in our own case. I think that there is a stronger response available, as well. That is: even though higher-order evidence can provide confirmation, it is still true that higher-order evidence is predictably misleading in a very important sense. That is: insofar as one’s total higher-order evidence rationalizes changing one’s opinion—that is, insofar as it rationalizes holding a different opinion than what would be rational without it—it tends to be misleading. To illustrate the point, let’s look more carefully at Peer agreement. In that case, I started off with some doubt about my rational abilities, which is why I was not entirely confident in my answer. The higher-order evidence I received from my classmates counteracted the higher-order doubt that I had previously. So there is one piece of higher-order evidence that points to the truth in this case—but it is only able to do so against the backdrop of my prior higher-order doubt.¹⁰ In Peer agreement, my belief about the answer to question  becomes more accurate only ⁹ My case is based on one presented in Christensen (). ¹⁰ If you’re not convinced, compare this case to another one, in which I go into the math test with no reason for higher-order doubt. If I had no higher-order doubt at all, what should I believe about the answer to question ? Plausibly, I should believe exactly what my evidence supports. So I should be highly confident of the answer. Now suppose that, just as in the original Peer agreement story, I talk to everyone else after class and find that they put down the same answer. Should I become more confident? It is hard to see why my credence should change at all in this case. This suggests that higher-order confirmation (in the sense of raising one’s confidence) in general can only happen against a backdrop of higher-order defeat.

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because the net effect of my higher-order reasoning gets smaller. This is very strange: with first-order evidence, we tend to become more accurate as the net effect of that evidence gets larger. This means that insofar as higher-order evidence makes a net difference to what’s rational for you to believe, it’s for the worse.

...  (): -    ? Another way to deny P focuses not on different kinds of cases, like peer agreement, but on different kinds of people. This objection says: Higher-order evidence is misleading for people like Sam, in cases like Cilantro. However, there are other people for whom the very same higher-order evidence would not be misleading in cases like Cilantro. To illustrate this view, suppose Sam has another roommate, Sally, who also hates cilantro. Suppose Sally is now in just the same circumstances as Sam is in: she sees the curry in the fridge, reads the note, and completes the proof. But where Sam correctly concluded that C (the curry contains cilantro), Sally makes a logical mistake, judges the proof to be invalid, and concludes ~C. (Perhaps this is because Sally, unlike Sam, is compromised in her logical reasoning abilities due to the cilantro-soapy-taste gene.) Then Sally reads the story in the newspaper, and becomes worried about the reasoning she has just completed. For her, the effect of the newspaper story is to reduce her confidence in ~C, thereby increasing confidence in C. Sally’s situation is therefore one in which higher-order defeat leads to a more accurate first-order belief, and a first-order belief that is more in line with what the first-order evidence supports. So, according to this second objector: in Sally’s case, the higher-order evidence points to C—that is, it points in the same direction as the first-order evidence. Whether higher-order evidence agrees or disagrees with first-order evidence depends on whether you are like Sam, or like Sally. Therefore, P is false.¹¹ There is a lot to say about Sally’s case, and how it should be explained within our larger theory of epistemic rationality.¹² Here, I will just point out two reasons to resist the present objection. First: as some authors have noted, there seems to be an asymmetry between agents like Sally and agents like Sam. At the end of the story, Sam seems more rational than Sally, even though they both did something procedurally sensible in response to their higher-order evidence.¹³ The present objection, which treats the two cases as equals, will have a hard time explaining the sense in which Sam’s final belief state is more rational. Second, and more importantly, the objection relies on a certain interpretation of “where the evidence points,” which I will argue is mistaken.¹⁴ This objector seems to ¹¹ Furthermore, this argument seems to have the resources to say that, not only is it false that higherorder evidence tends to be misleading, but in fact it is true that higher-order evidence tends to point to the truth. Actual, fallible agents who receive HOE are in Sally’s position more often than we’re in Sam’s. That means that for us, higher-order evidence will tend to point towards the truth. Thanks to Charity Anderson for very helpful discussion here. ¹² For further discussion, see Horowitz & Sliwa () and Schoenfield (b) and Christensen (a). Also see Kelly () and Christensen () for an exchange on this point re: peer disagreement. ¹³ See Christensen (), Kelly (), Horowitz & Sliwa (). ¹⁴ Thanks to Justin Fisher for helpful discussion here.

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want to say that the evidence points one way for Sally and another way for Sam, despite Sally and Sam sharing their evidence. So in this case, the very same evidence points to both truth and falsity. On the interpretation that I favor, Sam and Sally’s total evidence supports confidence in C, despite the fact that Sally ends up with high confidence in C after responding to that evidence. If my interpretation is right, it explains the asymmetry between Sam and Sally (Sam is more rational because his beliefs align with where the evidence points, or what it supports, while Sally’s beliefs do not) and also explains why the present objection is on the wrong track. Let me explain what I mean. In order to know where a certain sign or signal points to, we often need to know where it is pointing from. Different kinds of indications or signs, occurring in different contexts, lend themselves to different interpretations. Example : on a hike, you encounter a blue arrow painted on a tree, pointing up the mountain. This arrow is pointing from your present location, exactly where you are standing when you can see the arrow. The arrow’s meaning is something like, “from here, go up the mountain to follow the blue trail.” Example : you are an explorer on Treasure Island, following directions to a pirate’s hidden chest of jewels. Step  of the pirate’s instructions says, “Turn left and walk  paces. The treasure is buried here.” Obviously, this step of the direction is not pointing from one’s present location unless one has also correctly followed Steps –. Instead, Step  points to some location relative to the preceding steps. An important difference between these two examples comes out in cases where we fail to reach the destination. In the first example, suppose you followed the blue marker up the mountain and found yourself on the green trail rather than the blue trail. This is the trail marker’s fault; it pointed in the wrong direction. Now take the second example: suppose you turn left, walk thirty paces, and start digging. You find no treasure. Does this mean that the pirate’s instructions were wrong? Well, that depends: did you follow Steps –? If you did, then we can blame the pirate’s instructions. But if instead, you wandered randomly around the island, checked the directions, and jumped straight to Step , then it’s your own fault that you did not find the treasure. We might understand evidential pointing in either of these two ways. If a piece of evidence is like a trail marker, directing an agent from wherever she happens to be, it would be apt to describe Sally’s evidence as “pointing to the truth.” Sally responded to her higher-order evidence given the (not fully rational) beliefs she had at the time, and ended up closer to the truth about C. However, if a piece of evidence is more like a step in the pirate’s directions, this description would not be apt. On this second understanding, a piece of evidence points to one belief or another relative to the rest of one’s total evidence—not relative to one’s current “epistemic location.” While Sally may have done something right in response to her higher-order evidence, it is not accurate to say that higher-order evidence pointed to the truth in her case.¹⁵ Merely drawing this distinction, I think, puts pressure on the present objection: it is not obvious that Sally’s evidence is non-misleading, simply because it’s not obvious ¹⁵ Notice that if we take my view here, we can still make use of other forms of positive epistemic appraisal to describe Sally’s reasoning. James Pryor’s notion of “rational commitment” might be useful in this context. (See Pryor (), pp. –.)

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that we should interpret “evidential pointing” in the first way rather than the second. Moreover, I am inclined to think that the second interpretation of “evidential pointing” is the more fundamental one: we should see evidence as more like the pirate’s directions than like the trail marker. When we talk about whether a body of evidence is misleading, we care about which of our epistemic failures or successes can be credited to the evidence itself, rather than to the person assessing that evidence. So we want to know whether a bit of evidence points to the truth relative to the rest of an agent’s evidence—not relative to her (possibly irrational) current state. Perhaps both interpretations of “evidential pointing” are legitimate. If you are inclined to think so, we can think of the present challenge as focusing on that sense of “pointing” that the evidence does all on its own. If what I have argued is correct, we should think of our evidence as a map to the truth, and higher-order evidence is something that makes this map systematically less reliable.

.. Summing up I have claimed that we should accept both P and P. First-order evidence tends to point to the truth, and higher-order evidence tends to point in the opposite direction from the first-order evidence that it targets. We should also expect higher-order evidence to crop up at any time—not only when the first-order evidence is misleading. So we should accept the claim that higher-order evidence tends to be misleading.

. Two-norm solutions How can higher-order evidence have a rational effect if it is predictably misleading? The first sort of view I’ll discuss embraces both of these apparently contradictory claims. According to this type of view, there are two different senses of “rationality” at work in cases of higher-order evidence. In one sense of “rational”—the one that corresponds to evidential support—it is rational to ignore higher-order evidence. It is this first sense of “rational” that makes the following sentence true: “Rationality is a guide to the truth.” But in another, derivative sense of “rational,” it is rational to respect higher-order evidence. I’ll call views of this style “two-norm views.” The idea that our apparently inconsistent normative judgments track two (or more) norms is familiar from ethics, where philosophers often acknowledge a difference between “subjective” and “objective” moral norms or “oughts.” In the current literature on higher-order evidence, epistemologists have drawn two-norm distinctions in a number of ways: (a) objective vs subjective; (b) reasons vs rationality; (c) an evidential norm vs the dispositions that generally lead one to follow it; (d) the norm one should follow vs the norm that one should try to follow; (e) the “best method” vs the best method to adopt; (f ) the plan you should conform to vs the plan you should make.¹⁶ In many of these cases, one norm is understood as “ideal” and the ¹⁶ See Sepielli (ms) for (a); Worsnip () for (b); Lasonen-Aarnio () for something close to (c) and for (e); Williamson () for something close to (d); and Schoenfield ( a) for (f ). LasonenAarnio (ms) speaks more generally of the evidential norm vs a “derivative norm,” with the details of how derivative norms are derived left for later. Some of these views are motivated by the kind of puzzle I raise here, and some by other puzzles raised by higher-order evidence.

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other as “non-ideal.” The upshot of all of these proposals, of course, is that the first norm in the pair tells Sam to maintain high confidence in C. The second norm tells him to reduce confidence in C in response to his higher-order evidence.

.. A challenge for two-normers Two-norm solutions are somewhat unsatisfying for one obvious reason: they leave open the question of what we should believe. There is therefore a simple argument against these views: since epistemology’s primary aim is to answer this question, twonorm views have failed.¹⁷ I am sympathetic to this simple argument, but it is a bit too quick; after all, perhaps these views are right to say that we have been up to now confused about the aim of epistemology, and that the question of what we should believe is ambiguous or equivocal. Indeed, the fact that we seem to have run into a contradiction in answering the question is good evidence that it is equivocal. So I would like to focus instead on a related challenge. What is it about these two norms that makes them both distinctly epistemic, and both worth caring about? To answer this challenge, I take it that defenders of two-norm views need to explain how each norm is related to the pursuit of truth. If some mode of rational evaluation is not connected to truth, it is either not distinctively epistemic or else it is not worth caring about. Some two-normers might readily admit that this challenge can’t be met. For such people, perhaps only the first norm is meant to be interesting or worth caring about. One gets this sense from, for example, Timothy Williamson, who often writes about various epistemic rules being the right ones to “try to follow” (as opposed to being the right rules to follow), but does not seem to think that this more subjective category forms anything like a theoretically useful or coherent whole.¹⁸ Or perhaps the first norm is epistemic, and the second is practical.¹⁹ However, I am currently interested in the prospects for solving (rather than giving up on) the problem at hand. So I will focus here on the possibility that both norms are important and distinctly epistemic. It seems to me that the most promising two-norm approach should say that both norms are connected to the truth, but in different ways. This is how I interpret some of the suggestions above, particularly those that draw the distinction along ideal/nonideal lines. What is good advice for ideal agents might be terrible advice for non-ideal agents, and vice versa. So, if we think of epistemic norms as giving guidance or advice, it is plausible that we would end up with different norms for different kinds of agents. Miriam Schoenfield develops this approach ((f), in the list above), specifically addressing the connection between higher-order evidence and truth or accuracy. I will focus on her view here. Schoenfield argues that when we are making epistemic plans, or asking for epistemic guidance, we should consider the effect of making one plan or another. We should consider the effect because, in Schoenfield’s view, epistemic rationality aims to achieve certain goals (namely, accuracy). So what’s relevant to assessing a plan is how well it achieves those goals. If we are programming an infallible robot, we can just think about the effects of following a certain plan. But for fallible agents, who ¹⁷ Kvanvig (, ch. ) develops this view in detail. ¹⁸ Williamson (). See also Weatherson () for a similar view about morality. ¹⁹ Smithies (ms) develops this view.

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don’t always follow through on our plans, we should consider the best plan to make. Just like a chronically late person who sets her clock ahead to compensate, we should sometimes make epistemic plans that compensate for our expected shortcomings.²⁰ Schoenfield argues that in cases of higher-order evidence, the best plan to conform to is believing what one’s first-order evidence supports. But since we can’t expect to follow that plan perfectly—after all, we know that our reasoning might be impaired in various ways—we should make the plan to respect higher-order evidence as well. Why should we agree that the best plan to conform to has us ignore higher-order evidence? Schoenfield’s argument is in line with the argument I have given above: higher-order evidence is predictably misleading. Schoenfield imagines a “perfectly rational ignorant agent” who is programming an infallible robot to form opinions in response to evidence. If this agent only cares about her robot’s accuracy, she will program the robot to ignore higher-order evidence. That’s because, in cases like Cilantro, respecting higher-order evidence would result in the robot having less accurate beliefs. I agree with Schoenfield’s assessment here. What about the second part of Schoenfield’s view: the idea that the best plan to make involves revising our beliefs in light of higher-order evidence? This is where problems arise. First of all, it is not obvious that respecting higher-order evidence is, in fact, the plan that is best-to-make in Schoenfield’s sense. The question of which plan we should make is an empirical question, which depends on the particulars of our own psychology and our tendencies to live up to, and fall short of, our plans. These factors are complex and hard to predict, and vary from agent to agent. A plan to respect higher-order evidence might have the desired effects for some level-headed agents, for example, but might send insecure agents into a tailspin of self-doubt, and might leave some arrogant agents’ irrational beliefs intact. It would be remarkable if there was any one plan that was best-to-make for more than one person, and that plan ended up being anything like the general rules for higher-order evidence that have been defended in the literature. Given these worries, it is questionable to what extent Schoenfield’s suggestion can really vindicate our intuitions in cases like Cilantro.²¹ However, even setting this issue aside, Schoenfield’s view faces other challenges due to its general structure. In particular, the view’s focus on consequences leads it to overgeneralize in intuitively problematic ways. First: although this type of two-norm view focuses on epistemic consequences—in this case, consequences for true belief or accurate credences—it is not clear why the norms that it supports need to be purely epistemic. For example, I tend to reason badly when I’m hungry. So maybe I should make this plan: Sandwich Plan:

Have a sandwich before engaging in difficult reasoning.

²⁰ Schoenfield (a, p. ). See her sections  and  for discussion of how this distinction bears on higher-order evidence in general. ²¹ Schoenfield acknowledges this general type of worry (a, pp. –), and points out that even though we cannot be sure of the actual consequences of making any given plan, we can always assess a plan’s expected consequences using whatever evidence we have. However, it is still not clear that the expected consequences of responding to higher-order evidence come out ahead.

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My Sandwich Plan is a good one: I know that I reason better when I’m not hungry, and end up with more accurate beliefs if I have a sandwich first. I’m also quite good at following through on my plans to eat, so the Sandwich Plan is not only a good plan to conform to, but also a good plan to make. With due respect to its obvious merits, however, the Sandwich Plan is not the sort of plan we are interested in in the present context. Epistemology should not tell me to have a sandwich. If we really are focused on the consequences for true belief, several kinds of inquiry would serve us better: psychology, optometry, etc. More generally, then, it is hard to see why this rationale for a two-norm view yields two distinctly epistemic norms, rather than practical ones.²² Schoenfield could rule out the Sandwich Plan by restricting her view to purely epistemic plans, whose inputs and outputs are purely epistemic (for instance: evidence, beliefs and credences, and inductive standards). This move would eliminate the Sandwich problem. But other counterintuitive consequences would remain. For instance, consider Ivan, who has severe arachnophobia and reasons badly whenever he believes that a scary spider is nearby. Given his particular shortcomings, perhaps this is the best plan for him to make: Spider Plan:

Never, ever believe that there is a spider nearby.

Suppose Ivan could get himself to conform to this plan, training himself to ignore all evidence of nearby spiders. (This might be easier than training himself to get over his arachnophobia!) He would thereby end up with many more true beliefs than he would have if he took the spider-evidence into account. Ivan’s plan looks rational by Schoenfield’s lights. While the Spider Plan is a bad plan for infallible rational agents, it’s a good plan for Ivan. It helps him compensate for his own fallibility, and leads to more accurate beliefs in the long run. But this is the wrong result. It is epistemically irrational for Ivan never to believe that there are spiders nearby, even when he can clearly see a spider right in front of him, receives reliable testimony that there is a spider in the vicinity, or hears the distinctive pitterpatter of little hairy legs.²³ The upshot is that two-norm views face a challenge defining the second norm: making it distinctly epistemic, clearly tied to the truth, and plausible.

.. Another “two-norm” view: epistemic dilemmas? Similar problems arise for a different sort of view, which says that in cases like Cilantro, agents face an “epistemic dilemma.” According to this view, which has been defended by David Christensen, there are two rational requirements that can’t be

²² See Lasonen-Aarnio (forthcoming) for a similar suggestion. ²³ The irrationality of Ivan’s Spider Plan raises a broader potential worry: it suggests that there is a problem with accuracy-based views in epistemology more generally. I think that this worry can be addressed, as long as it is possible to think about the importance of accuracy in a non-consequentialist way. (For further discussion of how this might be accomplished, see Carr () and Konek and Levinstein (forthcoming).) But while all accuracy-based accounts might have a spider problem, explicitly consequentialist accounts definitely do.

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satisfied at the same time in cases like Cilantro.²⁴ So no matter what Sam believes, he is guaranteed to violate at least one rational obligation. The dilemma view is not exactly a two-norm view. It does not say that there are two modes of rational evaluation, and it does not attribute puzzles of higher-order evidence to any kind of equivocation. But it has similar trouble explaining why our epistemic obligations in these cases are genuine obligations, and why we should care about meeting them. Christensen writes that in cases like Cilantro, although we can’t perfectly satisfy all of our epistemic obligations at once, there is nevertheless a maximally rational response.²⁵ Christensen thinks that the maximally rational response here is to decrease confidence in C. But why is it maximally rational to decrease confidence in C? And if it is, why should we care about being maximally rational? As I’ve argued, if it is rational to revise confidence in response to higher-order evidence, rationality is predictably and systematically misleading. On Christensen’s view, maximal rationality is predictably and systematically misleading. The suggestion that Sam faces a rational dilemma does justice to the thought that, if he responds to higher-order evidence, Sam is doing something wrong. But the dilemma view still can’t explain why Sam is also doing something right in decreasing confidence in C—and it certainly can’t explain why Sam’s response is maximally rational. The original puzzle remains.

. Calibration views In this final section I will explore the possibility that existing, mainstream accounts of higher-order evidence have the resources to address this challenge. To do so, I will argue that these accounts will need to draw a distinction between a rational agent’s perspective and her body of total evidence. They will also need to deny that it is always rational to believe what our evidence supports. It might help to make a few things more explicit. Throughout sections .–., I have been working with the following assumptions: () () () ()

Evidence tends to point to the truth. Rationality requires believing what’s likely from one’s own perspective. Rationality requires believing what’s likely given one’s evidence. Rationality requires revising one’s beliefs in light of higher-order evidence. I also argued for the following:

() Higher-order evidence is predictably misleading: it rationalizes changing one’s beliefs in a way that, predictably, tends to make those beliefs less accurate. If we accept (), this seems to create a problem—most likely, it might appear, a problem for claim (). In sections . and ., I focused on developing this problem. In section . I looked at a way we might resolve the problem, by positing an equivocation in what () and () mean by “rationality requires.” ²⁴ For example, see Christensen (), (), and (b). ²⁵ If there were no maximally rational response, the dilemma view would be very similar to the twonorm views.

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This section will be focused on a different response, aimed at claims () and (). So far I have not separated these claims. () is almost tautologous: to believe something that one takes to be unlikely (i.e., to believe something that is unlikely from one’s own perspective) seems paradigmatically irrational. And () looks, at first glance, like just one way to spell out () in more detail. If one’s “perspective” just is one’s total evidence, then rationality requires believing what is likely given one’s total evidence. My suggestion here will be that () is not just a more precise statement of (); and that, moreover, we can reject (). In so doing, we can keep (), (), (), and (). The resulting view says that higher-order evidence does change what’s likely from your perspective, and hence (because we are keeping ()) what you should believe. But, on this view, higher-order evidence does not change what’s likely on your total evidence. So higher-order evidence should not be thought of as providing ordinary evidential support.²⁶

.. Total evidence and rational perspectives Let’s return to Cilantro, and attend to an important feature of the case that I have so far ignored. That is: although Sam can know that his evidence is misleading, he cannot know that he is being misled by his misleading evidence. A person is only misled by misleading evidence if she believes what that evidence supports. And Sam does not regard himself as believing what his total evidence supports. He thinks he made a mistake: that’s what the newspaper article suggests. This means that when Sam decreases confidence in C, he doesn’t recognize this as a departure from the truth. From Sam’s perspective at the end of the story, C is not highly likely to be true. To return to the analogy from section ., the position Sam thinks he is in is like that of someone who has deviated from the pirate’s instructions, but found the treasure anyway. More specifically, what actually happened to Sam was this: he accommodated his first-order evidence rationally, moving towards the truth. Then he accommodated his higher-order evidence rationally, moving away from the truth. But what Sam rationally thinks happened is something else: he accommodated his first-order evidence irrationally, moving away from the truth. Then he accommodated his higher-order evidence rationally, moving back towards the truth. Given what Sam rationally thinks has happened to him, he should regard his low confidence in C, at ²⁶ In this section I will take the phrases “what the evidence supports” and “what the evidence makes likely” to be synonymous. So, I will take () to be equivalent to: (0 ) Rationality requires believing what your evidence supports. We could choose instead to use talk of “evidential support” in another way, according to which one could coherently hold one of () and (0 ) and deny the other. For instance, we could instead say that “what the evidence supports” is equivalent to “what one should rationally believe given one’s evidence”. (Christensen () writes: “HOE really is best thought of as evidence. It is information that affects what beliefs an agent (even an ideal agent) is epistemically rational in forming” (p. ).) On that view, (0 ) is analytic, but not obviously equivalent to (). For example, if a body of evidence entails some proposition, then it makes that proposition likely. But one might deny that it is rational to believe everything entailed by one’s evidence. I find it natural to read () and (0 ) as equivalent, but I have no particular objections to interpreting them differently. Someone who feels strongly that () and (0 ) are not equivalent should substitute talk of “evidential support” in this section with talk of “what is likely, given one’s evidence.”

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

the end of the story, as highly accurate. But since (so he thinks) he arrived at this accurate belief by departing from what the evidence supported, he should not regard his final doxastic state as fully supported by the evidence. Building on the pirate analogy: the position Sam thinks he is in is like someone who deviated from a mistaken set of instructions, and found the treasure anyway. This is not an obviously irrational attitude to have. Compare: there would be something odd about searching for treasure by precisely following a set of instructions that one takes to contain (particular) mistakes. But there would not automatically be anything odd about searching for treasure using a mistaken map that one was not following precisely. Similarly, it is not incoherent for Sam to believe P while taking himself to have evidence that misleadingly points away from P.²⁷ How can we incorporate this observation into our account of higher-order evidence? We want to say that it is rational for Sam to take C to be unlikely: from his perspective, C is probably false. But given Sam’s total evidence, C is still highly likely: his evidence makes C probable. This second conclusion, that C is likely given Sam’s evidence, seems hard to deny. How could one dispute it? One might try the following Very Easy Argument to argue that C is not highly likely given Sam’s evidence: Very Easy Argument: To determine what’s likely given some body of evidence, we should just think about what an agent with that total evidence should believe. Sam should have low confidence in C after accommodating his total evidence. Therefore, C is not highly likely, given Sam’s total evidence. Unfortunately, the Very Easy Argument is nonsense. Depending on how we set up the case (and how reliable Sam takes his roommate to be), E + HOE may in fact entail C. Take the strongest case, where E entails C; adding HOE to E does not destroy that entailment.²⁸ So we should accept that C is highly likely, given Sam’s evidence. To hold both of these views about Sam’s situation, we have to say that a rational agent’s perspective is distinct from her total evidence; so, what she should believe is distinct from what her total evidence supports. What Sam should believe is what’s likely from his perspective; this might or might not align with what’s supported by his total evidence. If we make this distinction, we can see why it might not be so bad after all to say that higher-order evidence is predictably misleading. While one can predict in the abstract that higher-order evidence makes it rational to move one’s ²⁷ In coming to have this attitude, Sam is “epistemically akratic”: he has a belief that he believes is unsupported by his evidence. Many have taken for granted that such beliefs are irrational. I have argued (in Horowitz ()) that it is irrational to have attitudes of this form—but only true in cases where our evidence tends to be “truth-guiding.” In cases where evidence is “falsity-guiding,” i.e. predictably misleading, these attitudes are not so strange. If my argument here is right, higher-order evidence in general is falsity-guiding. See also Christensen (a), Horowitz and Sliwa (), Worsnip (), Lasonen-Aarnio (ms), and Smithies (ms) for further discussion of epistemic akrasia in cases of higher-order evidence. ²⁸ Of course, some have argued that in some cases it is not rational to believe what is entailed by one’s evidence. See, for instance, Schechter (). My position here is fully compatible with this view. According to my present suggestion, we should agree with Schechter and others that it is not always rational to believe what is entailed by one’s evidence. But it is still true that one’s evidence makes its entailments highly likely. That is, given Sam’s proof, note from his roommate, and the newspaper report, the curry probably does contain cilantro. See Christensen () for further discussion of this point.

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 

beliefs farther from the truth, one cannot recognize in one’s own case when this is happening. The suggestion to distinguish between evidential support and an agent’s rational perspective is not completely novel. In fact, I think this is a natural interpretation of a popular class of views about higher-order evidence, which says that rational agents should calibrate the strength of their credences to the expected reliability of their own reasoning (meaning, roughly: the extent to which that reasoning tends to point to the truth). Following recent literature, let’s call the view “Calibrationism.”²⁹ According to Calibrationism, the assessment of one’s own reliability must be independent from the evidence and reasoning in question. If Sam could rationally rely on his first-order evidence while assessing his reliability, he would be able to reason like this: “The newspaper says I’m likely to reach a false conclusion about C. But this proof is valid, and my roommate said that if it was valid, C would be true. Therefore, the newspaper is wrong: I’m immune from the effects of the cilantro/soapy-taste gene!” That reasoning looks irrational—just like it would be irrational for someone to conclude that he is not colorblind just by looking at objects and noting their apparent colors. To assess the reliability of one’s reasoning in some domain, Calibrationists hold that one needs to “bracket” or “set aside” the evidence and reasoning in that domain.³⁰ I think it is natural to interpret this feature of Calibrationism as forcing a separation between one’s evidence and one’s rational perspective. Once evidence is “set aside,” it is no longer part of one’s perspective; it can’t be appealed to in forming beliefs. But advocates of Calibrationist views are often careful to insist that the evidence doesn’t disappear when it is “bracketed.” So bracketing is a change in perspective without a change in evidence. This means that, for the Calibrationist, the way in which higher-order evidence rationalizes belief change can’t be explained by the normal notion of evidential support (again, in the sense of what the evidence makes likely). Nevertheless, higher-order evidence has a rational effect on first-order beliefs. Its effect is to change what is likely from a rational agent’s perspective, by changing the expected reliability of the rational agent’s process of reasoning.³¹

.. Upshots for Calibrationism In this final section, I will look at some potential upshots and unsolved problems for Calibrationism, given this new interpretation of the view.

²⁹ See White (), Schoenfield ( b), Horowitz & Sliwa (), and Lasonen-Aarnio (ms) for further discussion; “calibrationism” is a generalization of the “conciliationist” or “equal weight” views defended by Christensen and Elga. There are different versions of the view, which I will set aside for the moment. ³⁰ The notion of “bracketing” is due to Elga, and has a large role in Christensen’s work. For further discussion of independence, see Christensen (), Christensen (), and Elga (). See Vavova () for helpful discussion of how exactly these independence principles should be formulated to allow them to avoid skepticism. ³¹ Notice that on this view, what’s rational to believe still supervenes on one’s total evidence. So Calibrationism is compatible with evidentialism, if the latter is understood as a supervenience claim.

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An interesting feature of Calibrationism is that it seems to treat reasoning as just another source of beliefs, alongside perception, memory, and so on. This raises questions about just how far the analogy extends. For example: in cases of higherorder defeat, a rational perspective is one that disregards some evidence. Should we think of perceptual (undermining) defeat in the same way? It seems to me that there is reason to answer “yes.” To avoid bootstrapping, we need to say that an agent’s estimate of the reliability of her perception must be independent of the perceptual evidence in question. If that is true, it seems that perceptual defeat requires us to set aside evidence—and, therefore, the phenomenon of perceptual defeat already motivates the thought that we should believe what is likely given our perspective, rather than what is likely given our evidence.³² But even if there is a strong analogy between reasoning and other sources of belief, there are important disanalogies as well. One disanalogy is that while we are epistemically responsible for reasoning well on the basis of our evidence, we are arguably not epistemically responsible for seeing well, hearing well, remembering well, and the like. How can we maintain the thought that reasoning is epistemically assessable, if these other belief-forming mechanisms are not? One plausible option for the Calibrationist is to say that we are rationally required to reason well, and we are also rationally required to calibrate our beliefs to the expected reliability of their sources (including reasoning). This begins to look like a version of a two-norm view, in that there are two separate kinds of epistemic evaluation at work in cases like Cilantro. But identifying the two modes of evaluation in this way may have better potential for answering the central challenge for two-normers, which was to explain why both norms are distinctly epistemic, and both worth caring about. Reasoning is clearly tied to the truth, in that it is the process by which we discover what follows from, or what is made probable by, our evidence. It makes sense that we care about reasoning well, from a purely epistemic point of view. But reasoning alone doesn’t necessarily give an agent the best shot at truth, from her own perspective. Calibration does that. And it is rational to calibrate. Finally: one might wonder, why is it rational to calibrate, given that calibrating is predictably misleading?³³ We can know a priori that a rational agent who calibrates on the basis of higher-order evidence, and revises her belief through that calibration, will have less accurate beliefs than an agent who does not. So why is calibrating a good idea? Here I think the Calibrationist’s best defense is to say that, contra Carnap, rationality is not about programming the best robot. (Evidential support, on the other hand, might be.) Instead, it is about the search for truth from one’s own perspective— a perspective that is, unfortunately for us, sometimes limited. Calibrating in light of higher-order evidence is rational because, from the point of view of the agent, she is correcting her errors and moving towards the truth. But when everything is going perfectly, that is precisely when there are no errors to correct; so in fact the agent’s beliefs are becoming less accurate.

³² See Lasonen-Aarnio () for related discussion. ³³ Thanks to George Sher and referees for this volume for raising this worry.

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 

A calibrationist might therefore be able to predict that if she is calibrating perfectly, she will be misled. But she can also predict that those will be precisely the cases in which she will not be able to tell that she’s calibrating perfectly. This sits well with an intuitive way to think about misleading rational requirements: they should only mislead us when we are unaware.

. Conclusion I have argued that higher-order evidence is predictably misleading. Insofar as we are rational, accommodating this evidence correctly will tend to lead us away from the truth. So why should we believe what higher-order evidence supports? I have argued that one standard view of higher-order evidence may be able to answer this challenge. However, doing so comes at a cost: to maintain the standard view, we must say that we should not always believe what our evidence makes likely. This cost must be weighed against the benefits of the view: maintaining our intuitions about higherorder defeat in cases like Cilantro, and maintaining that such defeat is a distinctly epistemic phenomenon, with straightforward ties to the truth.

Acknowledgements For helpful feedback and discussion, thanks to Charity Anderson, David Christensen, Sinan Dogramaci, Kenny Easwaran, Justin Fisher, Miriam Schoenfield, Paulina Sliwa, and Katia Vavova, as well as audiences at Brandeis University, Rice University, the University of Houston, the  Texas Epistemology eXtravaganza, and the  Central Division Meeting of the APA.

References Carr, J. (). “Epistemic Utility Theory and the Aim of Belief.” In: Philosophy and Phenomenological Research (), pp. –. Christensen, D. (). “Does Murphy’s Law Apply in Epistemology? Self-Doubt and Rational Ideals.” In: Oxford Studies in Epistemology , pp. –. Christensen, D. (). “Disagreement as Evidence: The Epistemology of Controversy.” In: Philosophy Compass , pp. –. Christensen, D. (). “Higher-Order Evidence.” In: Philosophy and Phenomenological Research , pp. –. Christensen, D. (). “Disagreement, Question-Begging and Epistemic Self-Criticism.” In: Philosophers’ Imprint (). Christensen, D. (). “Epistemic Modesty Defended.” In: D. Christensen and J. Lackey (eds), The Epistemology of Disagreement: New Essays, Oxford University Press. Christensen, D. (a). “Disagreement, Drugs, etc.: From Accuracy to Akrasia.” In: Episteme (), pp. –. Christensen, D. (b). “Conciliation, Uniqueness, and Rational Toxicity.” In: Noûs (), pp. –. Elga, A. (). “Reflection and Disagreement.” In: Noûs (), pp. –. Feldman, R. (). “Respecting the Evidence.” In: Philosophical Perspectives , Blackwell, pp. –. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs (), pp. –.

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Horowitz, S. and P. Sliwa (). “Respecting All the Evidence.” In: Philosophical Studies (), pp. –. Kelly, T. (). “Peer Disagreement and Higher Order Evidence.” In R. Feldman and T. Warfield (eds), Disagreement, Oxford University Press, pp. –. Konek, J. and B. Levinstein (forthcoming). “The Foundations of Epistemic Utility Theory”. In: Mind. Kvanvig, J. (). Rationality and Reflection, Oxford University Press. Lasonen-Aarnio, M. (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research (), pp. –. Lasonen-Aarnio, M. (forthcoming). “Enkrasia or Evidentialism.” In: Philosophical Studies. Pryor, J. (). “What’s Wrong with Moore’s Argument?” In: Philosophical Issues , pp. –. Schechter, J. (). “Rational Self-Doubt and the Failure of Closure.” In: Philosophical Studies (), pp. –. Schoenfield, M. (a). “Bridging Rationality and Accuracy.” In: Journal of Philosophy (), pp. –. Schoenfield, M. (b). “A Dilemma for Calibrationism.” In: Philosophy and Phenomenological Research (), pp. –. Sepielli, A. (ms). “Evidence, Reasonableness, and Disagreement.” Unpublished manuscript. Smithies, D. (ms). “The Irrationality of Epistemic Akrasia.” Unpublished manuscript. Vavova, K. (). “Irrelevant Influences.” In: Philosophy and Phenomenological Research (), pp. –. Weatherson, B. (). “Running Risks Morally.” In: Philosophical Studies , pp. –. White, R. (). “On Treating Oneself and Others as Thermometers.” In: Episteme (), pp. –. Williamson, T. (). Knowledge and its Limits, Oxford University Press. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research (), pp. –.

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6 Escaping the Akratic Trilemma Klemens Kappel

. The Akratic Trilemma Consider the following familiar cases: The Pill: Sarah finds the right solution to a mathematical problem, but is then told that she might have ingested a reason-distorting pill, which, imperceptible to her, makes her unable to reason cogently about this type of mathematical problem (cf. Christensen ). Sleepy Detective: Sam is a detective who finds the right answer after working all night: Jones is the culprit. However, a colleague then tells Sam that he is unable to think clearly about the case because he is too exhausted from lack of sleep (cf. Horowitz ; Weatherson ). These cases highlight two pervasive features of our cognition. First, we are fallible cognitive agents in that we sometimes get the evidence wrong. Second, we often have higher-order awareness of our fallibility based on evidence that concerns the functioning of our cognition. So, we are epistemic creatures in the sense that we form beliefs on the basis of appreciating evidence that the world presents us with, but also in the sense that we are self-reflective epistemic individuals that regularly consider evidence concerning the proper functioning of our first-order cognitive capacities. These cases raise the suspicion of epistemic akrasia, cases in which a subject S believes some proposition p on some evidence e, while S at the same time on good evidence e0 believes that there is some serious defect in the grounds for her belief in p. A common pre-theoretical intuition holds that epistemic akrasia can never be rational, and many epistemologists agree: The Anti-Akratic Constraint: “it can never be rational to have high confidence in something like P, but my evidence doesn’t support P.” (Horowitz ) The Akratic Principle: “No situation rationally permits any overall state containing both an attitude A and the belief that A is rationally forbidden in one’s current situation.” (Titelbaum ) Finally, Weatherson writes: . . . the anti-akratic thinks that it is “wrong to both be confident in p and in the proposition that the evidence for p is not strong, no matter which proposition p is, and no matter what the agent’s background.” (Weatherson , p. , without endorsement) Klemens Kappel, Escaping the Akratic Trilemma In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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Until recently, the purported irrationality of epistemic akrasia was widely agreed upon, or was not widely questioned, at any rate. This has changed, however. We can begin to see why by way of what I will call the Akratic Trilemma (modifying Sliwa & Horowitz ): (E) S’s credence in p should rationally reflect e and only e, where e is S’s evidence bearing on the truth of p. (E) S’s credence in higher order propositions p’ should rationally reflect S’s evidence e0 and only e0 , where e0 is S’s higher-order evidence bearing on the truth of p0 . In (E) a higher-order proposition is a proposition concerning epistemic matters of a first-order proposition. So, higher-order propositions are individuated by their content. For now, let us say that the content of higher-order propositions might be the relation between a first-order proposition p and the evidence e supporting that proposition, S’s capacity to process e, or epistemic norms governing e and p, or similar matters. Higher-order propositions differ from first-order propositions simply in having a different subject matter. Evidence for a proposition is evidence bearing on the truth of that proposition. Note that (E) is a simple implication of (E): when a proposition p in (E) is of higher order, we get (E). Thus, principles (E) and (E) both reflect the fundamental idea that rational belief in a proposition should reflect whatever evidence speaks for the truth or falsity of the proposition, and nothing else. Note also that evidence is divided into first order and higher order by content of the propositions that they bear upon. By implication the same piece of evidence may both speak to the truth of a first-order proposition and the truth of a higher-order proposition, making the evidence both first order and higher order. For example, my evidence for my belief that I have been drinking nothing but water all evening is evidence for this belief, but also for my higher-order beliefs about the state of my cognitive capacities. This may seem confusing, but it is a natural implication of the suggested way of individuating first-order and higher-order evidence, and it will not affect the argument below. With these comments in mind, (E) and (E) seem innocuous. However, when (E) and (E) are combined with The Enkratic Requirement: It is epistemically irrational for S to have high credence in p on the basis of some body of evidence e, and at the same time have high credence that S’s processing of e is not functioning properly. we get an inconsistent triad. When S possesses a body of first-order evidence e indicating p, while S at the same time has higher-order evidence e0 indicating that S has misjudged or misapprehended the evidence e, respecting (E) and (E) will lead to a violation of the Enkratic Requirement, and respecting the Enkratic Requirement will involve violating either (E) or (E). So, on the face of it we cannot accept both (E), (E) and the Enkratic Requirement. Yet, they all seem plausible. The conjunction of (E), (E) and the Enkratic Requirement is what I call the Akratic Trilemma. Plan of the chapter. The existence of the Akratic Trilemma is the reason it is not so obvious after all that we should accept the Enkratic Requirement. However, I will argue that we should avoid the Akratic Trilemma by retaining the Enkratic Requirement and modifying (E) in certain ways (and ultimately also (E)). This is the stance central to the position known as Calibration, or at least a version of this position.

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

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I argue for this version of Calibration mainly by examining an argument against Calibration pressed recently by Maria Lasonen-Aarnio (Lasonen-Aarnio ), but also by developing certain aspects of Calibration. I present Lasonen-Aarnio’s main argument in section ., and develop a response to it along with my favoured version of Calibration in section .. To situate the discussion, it will be helpful first to review five ways of responding to the Akratic Trilemma and I do so in section .. I end with a brief summary in section ..

. Five responses to the Akratic Trilemma There are, I suggest, five different general responses to the Akratic Trilemma to be discerned in the literature, and it will be helpful to briefly review those responses first, before turning to the main discussion. Calibration. The overall idea in Calibration is that levels should be adjusted to fit one another in such a way that one does not violate the Enkratic Requirement. Consider Sam, the Sleepy Detective. The version of calibration I wish to defend holds that Sam should reduce his high confidence that he identified the culprit in response to his evidence that he might not be properly cognitively functioning due to being exhausted from lack of sleep. This implies Sam’s first-order credences are not determined solely by his first-order evidence pointing towards various hypotheses about culprits, but are also determined by Sam’s higher-order evidence that he is mentally exhausted from lack of sleep, which is clearly evidence that does not pertain to the truth or falsity of the proposition that Jones or any other suspect is the true culprit. Yet, it in part determines the rational credence that Sam should have in those propositions. The higher-order evidence and the first-order evidence meshes and determines what first-order credence Sam should have regarding Jones. Thus, Calibration responds to the Akratic Trilemma by retaining the Enkratic Requirement, and consequently by revising (E) in ways that I shall say more about below.¹ Calibration is the general view that first-order credences and higher-order credences should be adjusted to avoid the sort of incoherence generated when levels conflict. Note though, that there are two very different ways of achieving this. One has a top-down direction: when Sam has higher-order evidence that he is mentally exhausted from lack of sleep, this should make him reduce his confidence that Jones is the culprit. However, another way would be a bottom-up adjustment: when Sam appreciates first-order evidence correctly, this first-order evidence—or the very fact that he has identified it correctly—tends to make him justified in his higher-order belief that he has indeed correctly identified the evidential force of the first-order evidence, and therefore also that his lack of sleep has not impaired his cognition. Elsewhere I have argued that this bottom-up direction of influence is implausible. To get an indication of why, consider again Sarah reflecting over a mathematical ¹ For the same reason calibration should abandon (E) as well. This is not because rational credence of higher-order beliefs is affected by first-order evidence, but because we might have third-order evidence: evidence pertaining to our capacity to appreciate and process higher-order evidence correctly, which should be accommodated when we settle on higher-order credences.

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problem, while considering the possibility that she has ingested a reason-distorting pill. It seems very hard to believe that Sarah can become rationally confident that she hasn’t ingested a reason-distorting pill after all, or that her reasoning was not affected by a pill, merely by correctly identifying the solution to the mathematical problem she is grappling with, even if her beliefs about that solution are in fact based on impeccable appreciation of entailing evidence. So, correctly appreciating first-order evidence does not in general make us justified in any particular higher-order beliefs (cf. Kappel . For an opposing view, see Kelly , ). Throughout this discussion I will assume that Calibration endorses a top-down adjustment, and not a bottom-up adjustment, of levels in cases like Sarah working on a mathematical problem in the Pill, and Sam trying to figure out who the culprit is. To simplify, and make the discussion more manageable, I will simply assume that Calibration endorses what for lack of a better name I will call: Defeat by higher-order evidence: When S has high credence in p on the basis of evidence e, but higher-order evidence e0 tells S that her processing of e is not functioning properly, then S should have less credence in p than otherwise warranted by e. Note that Defeat by higher-order evidence is a special case of the Enkratic Requirement. Defeat by higher-order evidence says that if you have a high credence in p on the basis of some evidence e, and you have higher-order evidence e0 that tells you that your processing might be unreliable, then you should have less credence in p than otherwise warranted by e. Thus, failing to adjust your first-order credence in this situation is irrational. But this is just an instance of the Enkratic Requirement. I don’t mean Defeat by higher-order evidence to be a complete and precise specification of how higher-order evidence interacts with first-order credences. The point is merely to state a plausible principle that allows for a general discussion of a version of Calibration. In recent work, Horowitz and others have defended Calibration by showing that denying it is highly counterintuitive (Horowitz ; Sliwa & Horowitz ). I find these arguments compelling, though like so many other arguments in philosophy, they are not conclusive (see the lucid discussions in Weatherson ). Below I will defend Calibration against certain other arguments. Level-splitting. Maybe the most prominent alternative to Calibration is the idea that we should split the levels. According to this view, there is nothing inherently irrational in believing a proposition as a result of a correct appreciation of the firstorder evidence for that proposition, while also believing (falsely, upon misleading evidence) that one’s first-order cognition is flawed in some way. One can be rationally justified in both beliefs, and there is no rational tension between them, no demand for adjustment. When the two levels are in tension this is merely an unfortunate fact about how the evidence is lined up, and is no occasion for revising either the first-order belief nor the higher-order belief. So, epistemic akrasia may be fully rational. In regard to the Akratic Trilemma, this amounts to retaining (E) and (E), while rejecting the Enkratic Requirement. Versions of such views have recently been defended by, among others, Maria Lasonen-Aarnio, Timothy Williamson, Ralph Wedgwood, and Brian Weatherson. As we shall see in section ., level-splitting is

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indirectly supported by arguments against the coherency of higher-order defeaters proposed in (Lasonen-Aarnio ), and my discussion will focus on those arguments. I hope to show in section . that while one sense of level-splitting is plausible, it is a sense which is consistent with Calibration. No conflicting higher-order justification. In a recent paper Michael Titelbaum has argued, roughly, that if our first-order belief rationally reflects the evidence, then we cannot be rationally justified in believing that it does not (Titelbaum ). With respect to the Akratic Trilemma, this can be seen as retaining (E) and the Enkratic Requirement (ER), but then either denying (E) by denying that higher-order propositions are rationally responsive to evidence pertaining to their truth, or to claim that while (E) is true there can nonetheless never be cases where first-order beliefs and higher-order beliefs conflict in a way that violate the Enkratic Requirement. Titelbaum’s arguments for his view are complex, and I cannot discuss them in the detail they deserve here. However, I do think that his argument essentially boils down to asserting (E) and the Enkratic Requirement (ER) as premises, and then showing that (E) must go to avoid inconsistency. (E) and the Enkratic Requirement together imply that one cannot have a rational but false belief about what one’s first-order evidential situation requires. Pluralism. While he doesn’t put it in these terms, Alex Worsnip in effect argues that we should avoid the Akratic Trilemma by accepting a form of pluralism (Worsnip ). (E) and (E) require that first-order belief and higher-order belief are responsive to evidence, whereas the Enkratic Requirement is a form of coherence requirement applying to beliefs. Evidence responsiveness and coherence might be said to concern different normative properties. In Worsnip’s terms evidence responsiveness concerns epistemic reasons, whereas the coherence requirement concerns epistemic rationality, but all that matters here is that we assert that they are different normative properties. If it is true that the lemmas relate to different normative properties, then the derivation of inconsistencies underlying the paradox involves an equivocation. Once we see this, we can retain all three lemmas and hold that the conflict is only apparent. I will not discuss pluralism in detail, but if my defense of Calibration succeeds, the case for Pluralism will be undermined. Imperfection. Christensen has argued that there are cases where one violates an epistemic rule no matter what one does. Such cases can arise when we have misleading higher-order evidence against our epistemic rules. No matter how the correct epistemic rules are stated there seem to be possible cases where S is subject to an epistemic rule, but has misleading higher-order evidence that she should follow some other rule. When this happens, says Christensen, it might both be true that we are bound by a particular epistemic rule and also that we should not comply with it. Rule-violation may be unavoidable—even ideally rational agents will be imperfect (Christensen , ). Christensen doesn’t discuss the Akratic Trilemma as I have stated it, but I think that the underlying problem is the same. Imperfection suggests that we respond to the Akratic Trilemma by accepting all parts of it, while acquiescing to the fact that they may conflict such that one cannot avoid violating a lemma. This is the epistemological equivalent to tragic moral dilemmas. Again, if my defense for Calibration works, we don’t have to assume imperfection.

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There is at least some reason to think that the five options above exhaust the possible general avenues one might opt for in response to the Akratic Trilemma, at least the more obvious ones. The three first options (fully or partially) deny one of the lemmas, and accept the two others. The two remaining views accept all lemmas and the ensuing conflict, but offer two different interpretations of it, a pluralist and a nonpluralist interpretation. If we agree that the root problem has the structure outlined in the Akratic Trilemma, then the five options outlined seem to exhaust the space of logically possible types of responses, though each type permits many different variations.

. An objection to Calibration: can we make sense of defeaters? In the remainder of the chapter I defend a version of Calibration by first considering an objection to the view. As I said, we can think of Calibration as endorsing Defeat by higher-order evidence. In her paper, Maria Lasonen-Aarnio () has argued we cannot make sense of the idea that higher-order evidence impacts the rationality of first-order beliefs, and this obviously poses a problem for Calibration. Lasonen-Aarnio’s overall argumentative strategy is as follows: Epistemic rationality is a matter of following correct epistemic rules. To accommodate situations where first-order beliefs are defeated by higher-order evidence, we need a two-tiered theory. A two-tiered theory asserts rules about what is rationally required in various situations, and then adds a second layer of provisions taking defeating higher-order evidence into account. However, a two-tiered theory is impossible to state in a satisfactory manner within the Rule-governance framework. Therefore, we should abandon the idea that there can be defeating higher-order evidence. Turn now to Lasonen-Aarnio’s argument (in my discussion I omit many interesting details of Lasonen-Aarnio’s elaboration and concentrate on what I consider to be the core argument). Assume first: Rule-governance: A doxastic state S is epistemically rational only if it is the result of following correct epistemic rules (Lasonen-Aarnio , Condition , p. ). To account for cases where one follows the correct rules, but comes across misleading higher-order evidence that one does not, we need the following idea: Within the context of a rule-driven picture, the rough idea will be that a doxastic state is justified just in case (i) it is the product of following a correct epistemic rule, and (ii) one doesn’t have evidence that it is flawed. (Lasonen-Aarnio , p. )

Lasonen-Aarnio argues that this idea turns out to be impossible to make sense of once we accept Rule-governance. To see this, consider what Lasonen-Aarnio calls the Puzzle: Assume that a correct epistemic rule R recommends (requires) believing p in circumstances C, and that Suzy is in fact in circumstances C. Suzy comes to believe p as a result of applying R. But she then acquires evidence that her doxastic state was not the result of applying a correct rule and is, therefore, flawed (this may or may not be evidence that R itself is flawed). Assume that despite possessing this higher-order evidence, Suzy continues to be in circumstances in which R recommends believing p. (Lasonen-Aarnio , pp. ff.)

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Assume for the sake of argument that Calibration is true and implies that Suzy should suspend judgment about p once she receives the higher-order information. By the assumptions of the case, R correctly tells Suzy to believe p in C. Yet, we assume that because of her misleading higher-order evidence, Suzy should not believe p, despite her still being in C. According to Rule-governance, there must be a rule governing what Suzy should do, a rule which is friendly to Calibration, and which accommodates defeating higher-order evidence. But how is this additional step to be captured in epistemic rules? Lasonen-Aarnio discusses and rejects three options: The Über-rule View: R is not the correct rule after all. Rather, there is an Überrule R telling Suzy what to believe in C when the additional higher-order information is added. The Epistemic Dilemma View: There are conflicting rules in the Puzzle: one rule telling Suzy to believe p, and another rule telling Suzy to suspend belief about p. The Hierarchy View: There is a hierarchy of rules, such that a higher-order rule R0 overrides R in this case. Let’s briefly consider why Lasonen-Aarnio discards each of these options. The Über-rule view holds that there is an Über-rule R stating how Suzy should deal with evidence in C, but also saying how Suzy should react on any possible higherorder evidence, including misleading higher-order evidence and various types of defeating evidence, as well as iterated layers of higher-order evidence. LasonenAarnio advances two objections to the Über-rule View. The first is that the Überrule is bound to be highly complex and therefore hardly cognitively manageable. The second is this. Suppose that Suzy finds herself in a situation where she receives misleading higher-order evidence regarding the Über-rule itself. What should Suzy do? Assume that the Über-rule initially tells Suzy to believe p. Upon the addition of higher-order evidence against the Über-rule, presumably the Über-rule tells her not to follow it. Follow the Über-rule, and you will violate it. It seems that no matter what Suzy does, she will violate the Über-rule. This is the Über-rule problem, and I will return to it later, but for now simply note that Lasonen-Aarnio takes this problem to be a reason to reject the Über-rule View. Turn then to the Epistemic Dilemma View, which says that in cases like The Puzzle, the “correct epistemic rules are incompatible” (Lasonen-Aarnio , p. ). The Epistemic Dilemma View holds that we should accept both of the following: ) If a subject has evidence that her doxastic state is flawed, then the state fails to count as epistemically rational, and she ought to revise it. ) If an epistemic rule is correct, and the rule tells one to believe p in circumstances C, and one is in fact in circumstances C, then one ought to believe p. (Lasonen-Aarnio , p. )

Clearly, () and () can conflict, and this is what they do in the Puzzle. The Epistemic Dilemma View simply grants that such conflicts may arise. In this particular context, the main problem with the Epistemic Dilemma View is that it is not really helpful to Calibration; it fails to support the idea that first-order credences should sometimes be adjusted in response to higher-order evidence. There being an irresolvable conflict between the demands at the various levels is no help for

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Calibration. Of course, the Epistemic Dilemma View is a version of what I called Imperfection, asserting that there may be unresolvable conflicts between object levels and higher-order levels. Calibration and Imperfection are two very different responses to the underlying Akratic Trilemma, so it is not surprising that the Epistemic Dilemma View or Imperfection is not helpful for Calibration. Consider finally the Hierarchy View. This view holds that for a given epistemic situation C, there is a set of correct epistemic rules {R₁, . . . , Rn}, and an ordering relation of these rules, that is, a kind of meta-rule M telling one which rule to follow in cases of conflict between the rules. So, in Suzy’s case there are really three different rules: R₁ telling Suzy to believe p in C, a distinct rule R₂ telling her to suspend judgment about p as a result of the misleading higher-order evidence she has received in C, and finally a meta-rule M telling her when to apply the first and when to apply the second rule in cases where they conflict. Lasonen-Aarnio advances objections to the Hierarchy View similar to those we have already seen. Suppose that Suzy has misleading higher-order evidence telling her that meta-rule M is false or misapplied. What should Suzy do? Lasonen-Aarnio considers two options. One would be to say that meta-rules are immune to misleading higher-order evidence. The second would be to suggest that there are meta-metarules, which are rules giving a hierarchy of possible meta-rules, and saying which one should use in cases where one has higher-order evidence against a meta-rule. Lasonen-Aarnio rejects both of these options. It is difficult to see what could motivate the first option, and the second option, she notes, looks “like the beginning of an infinite regress” (Lasonen-Aarnio , p. ), and it is doubtful whether beings like us could be guided by such a complex hierarchy of rules (Lasonen-Aarnio , p. ). So, the Hierarchy View must go. In summary: Lasonen-Aarnio considers three ways in which Calibration could account for the Puzzle within the rule-driven framework, but she discards them all. Hence, if we adhere to Rule-governance, Calibration must go. This is the core of Lasonen-Aarnio’s argument against Calibration, and indirectly in favor of some version of level-splitting.

. The Fine-grained View I suggest that there is a simple and plausible way of dealing with the Puzzle within the Rule-governance framework. The basic idea is that we individuate epistemic circumstances and rules applying to them more finely. Consider again the Puzzle. Assume that Suzy finds herself in circumstances C, and that epistemic rule R correctly tells her to believe p in C. Let C0 be the new set of circumstances that arises when Suzy was in C and then receives misleading higher-order evidence concerning that R is false, or that she is likely to misapply R in C. Assume that Rule-governance is correct. Then there is a rule R0 telling Suzy what to believe in C0 . Assume, as before, that R0 tells Suzy to suspend belief about p. What we get, of course, is that there is no conflict between R and R0 . Surely, R requires believing p, while R0 requires suspending belief about p. Yet, there is still no conflict: R requires believing p in C, whereas R0 requires suspending belief about p in C0 , but since C and C0 are different situations there are no conflicting requirements. After receiving the misleading higher-order evidence,

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Suzy is no longer in C, but in C0 , and it is thus irrelevant for her what R says about C. Suzy should abide by R0 , and not by R, so, there is no tension. Call this the Fine-grained View. Note that while the Fine-grained View is fully consistent with Rule-governance, it is not a two-tiered system. So, by adopting the Fine-grained View we can reject Lasonen-Aarnio’s initial assumption that to account for defeaters or defeat by higher-order evidence within Rule-governance we need a two-tiered epistemology. Also, the Fine-grained View does not utilize a hierarchy of rules and a meta-rule to adjudicate between rules. So, there is no problem about an infinite regress of rules. The simple idea underlying the Fine-grained View is that if what we epistemically ought to do in a situation is determined by evidence, and if it is also a matter of following epistemic rules, then it is quite natural to think that new evidence presents you with new circumstances, and new circumstances may require new rules. This also gives an indication of how the Fine-grained View, in a fully specified version, would individuate epistemic circumstances and evidence. When new relevant evidence is added to some situation we get a new situation, and this changes what we epistemically ought to do. This is true both when we add new relevant first-order evidence to a situation, and when we add relevant higher-order evidence. Thus, we can assume that evidence and evidential situations are individuated such that whenever a change in the evidential situation makes a difference to what a subject should believe, this constitutes a new evidential situation, even if one cannot individuate evidential situations independently of how we think that changes in evidential situations should affect credences. Clearly, however, there might be concerns about the Fine-grained View, and I now want to consider three potential worries. One is that the Fine-grained View might seem inconsistent with a compelling form of evidentialism. The second objection is that the Fine-grained view may imply that epistemic rules are too complex to be psychologically manageable. Finally, one might be concerned about the Über-Rule Problem; indeed, the Fine-grained View may be interpreted as a version of the ÜberRule View, which makes it all the more pertinent that it deals with the Über-rule problem in a plausible way.

.. Respecting the evidence Assume that C and C0 are epistemic situations that are identical with respect to the first-order evidence in those situations, where first-order evidence is the evidence pertaining to the truth of some first-order proposition p that is under scrutiny in C. So, C and C0 differ only in that C0 contains additional higher-order evidence regarding S’s capacity to reliably process the first-order evidence available to her. Assume as before that some rule R applies to C and tells S to believe p. On the Finegrained View, this means that some other epistemic rule R0 applies to C0 . Assume that this rule R0 tells S to suspend belief about p. Clearly, this combination means that we have to deny a particular form of evidentialism, call it Narrow Evidentialism: S’s belief about p should reflect all and only evidence bearing on the truth of p. Rejecting Narrow Evidentialism is of course just another way of rejecting (E) in the Akratic Trilemma. But rejecting Narrow Evidentialism is compatible with insisting

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that our beliefs should heed the evidence, and only the evidence. This is because we can accept Broad Evidentialism: S’s beliefs about p should reflect all S’s evidence bearing on the rationality of believing p, including S’s higher-order evidence about S’s capacity to process evidence about p. Calibration says that cases like The Pill and Sleepy Detective are cases where our doxastic attitudes to a proposition should reflect evidence that does not bear on the truth of that proposition, but nonetheless bears on the rationality of the doxastic attitude in question. In these cases, we can still insist that agents respect the evidence in the broad sense if they respond to higher-order evidence by reducing credence in their first-order belief. Narrow Evidentialism is, one might suggest, motivated by the fundamental assumption that what we should rationally believe about some proposition depends only on our evidence for or against the truth of that proposition (see Weatherson  for an elaborate defense). While there is surely something intuitively compelling about this, one might suggest that Broad Evidentialism is motivated by the equally fundamental assumption that our rational beliefs should reflect all the evidence bearing on the rationality of holding the belief. Cases like The Pill and Sleepy Detective provide strong intuitive support to the idea that all evidence includes higher-order evidence concerning the reliability of one’s processing for first-order evidence. Without pretending that this alleviates all concerns Narrow Evidentialists may have, we can at least say that Broad Evidentialism is prima facie plausible for cashing out what respecting the evidence means.²

.. Complexity A second worry is that the Fine-grained View will be committed to the existence of epistemic rules that are not cognitively manageable and therefore not suitable for guiding our reasoning in the right way. After all, it might be said, epistemic rules are there to guide our reasoning, and this imposes constraints on how complicated rules can be. Are rules cognitively manageable and reason-guiding on the Fine-grained View? On the Fine-grained View, epistemic rules are highly specific, but have a very simple structure. Call this an SHS-structure of epistemic rules (for Simple and Highly Specific). We can represent this structure of epistemic rules as follows: SHS-structure: [in C, adopt D(p)], where D(p) is some doxastic attitude to p, and C is a finely individuated epistemic situation, The alternative to viewing epistemic rules as highly specific but simple is, it seems, to view them as more general but complex.³ Call this a GC-structure of epistemic rules (for General and Complex). We can represent this structure as: GC-structure: [in C, adopt D(p), unless . . . ], were D(p) is a doxastic attitude to p, and C is individuated only by first-order evidence, and ‘ . . . ’ is a placeholder for ² One potential issue for Broad Evidentialism is that it might have difficulties making sense of conditionalization, see Christensen (). Thanks to a reviewer for noting this. ³ This borrows from Hare’s discussion of universal and general moral principles, see Hare (, p. ).

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modifications of the general rule that applies to C when C contains various forms of additional higher-order evidence. So, according to the GC-structure, all situations involving the same set of first-order evidence is governed by the same epistemic rule, hence the generality of the rule. But the rule itself is complex in that it allows for modifications, exceptions, and special cases depending on what additional higher-order evidence the situation contains. In discussions of the Über-rule problem, it is often assumed that the Über-rule has GCstructure, but as we shall see below, one can also imagine an Über-rule with SHSstructure. The Fine-grained View proposes that epistemic rules have SHS-structure. One might wonder if the complexity of the SHS-structure of rules that comes with the Fine-grained View is a problem. In response, note first that no matter what structure of epistemic rules one prefers, the full package is going to be very complicated. Even if we imagine a system of epistemic rules taking into account only first-order evidence and nothing else, these rules are also going to be highly elaborate. If we insist that high complexity is a problem, then it is more likely the assumption of Rulegovernance is under pressure rather than any particular view of the structure of rules. But second and more importantly, I will suggest that it is not obvious that high complexity of epistemic rules is a problem at all. The underlying assumption is Rulegovernance, the idea that epistemic rationality is a matter of following correct epistemic rules. It is important to note that there are (at least) two possible readings of Rule-governance. Rule-following may be thought of as consciously entertaining the content of a rule and executing a particular activity in accordance with instructions embodied in the rule. This is what we might call the comprehend-and-execute reading of Rule-governance, according to which epistemic rationality requires that a subject S in some substantive sense understands the relevant epistemic rules and use the rules as guidance in her reasoning. But rule-following may also be given a much less intellectualist reading, where epistemic rationality merely requires that one’s activities comply with a rule, and do so in non-deviant ways. Call this the constitutive reading of Rule-governance, which says that for each evidential situation C, what S ought to believe is expressible in a rule requiring a doxastic attitude D(p), and for S to be epistemically rational in C is to respond with the proper attitude D(p), and do so in the right way. Both readings of Rule-governance are compatible with the two views of the structure of epistemic rules that I discussed previously. High degree of complexity of epistemic rules (whatever their structure) is clearly an issue for the comprehendand-execute reading of Rule-governance. On the constitutive reading, however, it is not clear that either form of complexity is a problem at all. In other domains of human activity plausibly governed by rules, the constitutive reading seems vastly more attractive that the comprehend-and-execute reading. Consider our competence to speak grammatically, or fairly correct, at any rate. Clearly, Rule-governance is plausible for this competency. But surely, this requires the constitutive reading; it is highly implausible to hold that normal speakers of a language possess grammatical competence only if they consciously hold the rules of grammaticality before their minds and use these rules as guidance when they form sentences. Competence with

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respect to grammar is a capacity to conform to grammatical rules, and to do so in the right kind of way, but normally this does not involve comprehending and executing the rules as guidance to one’s effort to put together grammatical sentences. It seems that if we want to understand epistemic rationality in terms of Rule-governance, we need to say something similar about epistemic rules. If we accept the constitutive reading of the Rule-governance of epistemic rationality, complexity is no more a problem in epistemology than in grammar. Even if complexity renders it impossible for epistemic agents to comprehend the totality of the system of rules this might not be viewed as a theoretical problem, as long as epistemic agents can still comply with specific rules. If we can follow the rules (in the relevant sense), why would it be a problem that we cannot comprehend the rules in some intellectualistic sense?⁴ Hence, the Fine-grained View and the complexity of rules that comes with the SHS-picture of rules will not be a problem on the constitutive reading of Rule-governance (and neither would the complexity of those rules on the GC-picture be a problem).

.. The Über-rule problem Consider finally the Über-rule problem. How does the Fine-grained View handle the Über-rule problem? Assume that the Über-rule is a fully comprehensive rule that says what one should rationally believe in all cases. Assume also that the Über-rule generally requires agents to be sensitive to higher-order evidence, including misleading higher-order evidence. Suppose now that S finds herself in a situation in which the Über-rule tells her to believe p. Suppose that S also has misleading higher-order evidence against the Über-rule itself. What should S do? On the one hand, it seems that she should do as the Über-rule says—believe p. On the other hand, we are assuming that the Über-rule is sensitive higher-order evidence of just this kind, which suggests that S should do something else than believing p, perhaps suspending belief in p. If so, it seems that the Über-rule issues conflicting verdicts, telling S both to believe p and to suspend judgment about p, which seems paradoxical. This is the Über-rule problem. One might perhaps hope to avoid this sort of paradox by arguing that there cannot be misleading higher-order evidence against the Über-rule, or that the rule is somehow immune to defeat by higher-order evidence, but both these options seem to lack independent merit. Alternatively, one might respond to the Über-rule problem by endorsing level-splitting, pluralism, or imperfection, and each of these options has its proponents, as we have seen. I will not discuss the merits of these options, but instead suggest that the Fine-grained View supports a simple and plausible response to the Über-rule problem. On the Fine-grained view we should think of the Über-rule as nothing but an assignment of specific epistemic rules to finely individuated circumstances. So, the Über-rule R assigns an epistemic rule Ri to each circumstance Ci, where circumstances are finely individuated. That is individuated both by their first-order ⁴ As remarked by a reviewer, a potential problem with the Fine-grained View is that it might have difficulties explicating the structural or explanatory relations between various particular rules. I will set aside this problem as it cannot be discussed without an elaboration of what sort of theoretical unification of epistemic rules is required or desirable.

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

 

evidence and their higher-order evidence. Thus, the Über-rule is really a set of ordered pairs of finely individuated epistemic situations and epistemic rules. Call this the disjunctive interpretation of the Über-rule: The Über-rule R: {(C₁, R₁); (C₂, R₂); . . . (Cn, Rn)}, where Ci and Ri are finely individuated circumstances and rules. Suppose now that we have misleading evidence against the Über-rule. What, on the disjunctive interpretation, does this amount to? In one trivial type of case, misleading evidence against the Über-rule amounts to S being in C₁ and falsely but rationally believing about some other condition C₂ that R₂ does not apply to C₂. This is just a case in which S has a false belief about the content of the Über-rule, but where this does not affect what S should believe in C₁, and therefore it does not lead to the kind of paradoxes involved in the Über-rule problem. A more complicated case is this. Suppose S is in some situation C to which a rule R applies according to the Über-rule R. S then receives misleading evidence against R, where this evidence amounts to evidence that R does not apply to C after all, or that S misapplies R in C, say because S cannot properly process the first-order evidence in C. This is similar to the cases we have discussed above, and if what I argued above is correct, then it doesn’t present a genuine problem. When we add new evidence to C we get a new evidential situation C0 to which a new specific rule R0 applies which tells S what to do in C0 . Both (C, R) and (C0 , R0 ) are disjunctive parts of the Über-rule R, but R and R0 are clearly not in conflict, as they apply to different situations. Hence, there is nothing paradoxical about the situation. Yet, there is a further type of case.⁵ It seems that one can be wrong about which rule applies to one’s situation, and one can be wrong about which situation one is in. Both types of cases raise the (now familiar) question: when some epistemic rule applies, but you are also rationally committed to believing that it does not, what should you do? This question arises for any understanding of epistemic rules and epistemic situations allowing for us to have mistaken beliefs about them. While this problem is not specific to the Über-rule, it surely affects the disjunctive interpretation of the Über-rule. I now want to discuss this problem in more detail, and suggest a way in which Calibration should respond to it. Consider first the possibility of being wrong about one’s epistemic situation. Consider the following case: (C) (C) (C) (C) (C)

S is in C₁ R₁ applies to C₁ R₁ in C₁ requires S to believe p S (in C₁) mistakenly but rationally believes herself to be in C₂ R₂ applies to C₂ and requires S to suspend belief about p.

On the picture of epistemic rules and circumstances that go into the disjunctive reading of the Über-rule, it seems that (C–C) are co-possible. As the case is described, S is rationally warranted in believing p (from (C)), and yet S is also

⁵ This and the following sections were developed in response to remarks made in discussion by David Christensen, Clayton Littlejohn, and Julien Dutant. Thanks for a helpful and clarifying discussion.

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

rationally required to believe that she is rationally required to suspend belief about p (from (C) and (C) (if necessary we can assume that S knows or rationally believes the truth of (C)). So, the question arises: what should S believe about p? Just as one can be wrong about which epistemic situation one is in, so one can be wrong about which rules apply to one’s situation. Consider: (R) (R) (R) (R) (R)

S is in C₁ R₁ applies to C₁ R₁ in C₁ requires S to believe p S correctly takes herself to be in C, but mistakenly but rationally believes R₂ to apply to C₁ R₂ requires S to suspend belief about p

Again, (R–R) seem to be co-possible, and the question arises: what should S believe about p? Both (C–C) and (R–R) are versions of the sort of issues generated by the Über-rule problem: what should one do when following epistemic rules correctly leads one to believe that one should not follow an epistemic rule, which happens to be a correct rule? It might perhaps be replied that (C–C) and (R–R) are not really co-possible, but I find it difficult to see what would support this contention. There seems to be nothing in the nature of epistemic rules indicating that we could not have false but warranted beliefs about what they require, and it is hard to see why one cannot have false but justified beliefs about what evidence is available in one’s epistemic situation. It seems that the co-possibility of (C–C) and (R–R) raises a challenge for Calibration. These are cases in which object level and higher-order level clearly dissociate, and yet S is, by stipulation, fully rational in complying with the first-order level. If we admit that (C–C) and (R–R) are co-possible, isn’t this the end of Calibration, and an admission that level-splitting is right? I will argue that (C–C) and (R–R) are indeed co-possible, but that this is entirely unproblematic for Calibration, though it does show a sense in which level-splitting is exactly right. For brevity consider just (C–C)—the remarks I have to offer concerning (R–R) are analogous. One way in which (C–C) might appear quite problematic is simply that (C) S mistakenly but rationally believes herself to be in C₂ (C) R₂ applies to C₂ and requires S to suspend belief about p might seem to entail something like (C)

S is rationally required to suspend belief about p.

To be sure, if (C) were entailed by (C) and (C), we would have an outright inconsistency, as (C) contradicts (C–C), and this would render situations in which we are wrong about our epistemic situation paradoxical. We would then have to say that (C–C) are not co-possible after all, since (C–C) would entail the negation of (C–C). Any view implying the co-possibility of (C–C) would face a stern objection, and this would include Calibration.

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 

But clearly (C) doesn’t follow from (C) and (C), even if we add the assumption that S knows the truth of (C). It is instructive, however, to discuss an assumption that would make (C) follow from (C) and (C): Belief-dependence: If S rationally believes that normative rule R applies to C and that S is in C, then R applies to S in C. If S rationally believes that R does not apply to C or that S is not in C, then R does not apply to S in C. Belief-dependence together with (C) and (C) entails (C), and cases of being rationally wrong about one’s epistemic situation would be paradoxical. Using Beliefdependence as an additional premise we can derive contradictions from (C–C). Is Belief-dependence plausible? Surely it is not, and there are a couple of familiar reasons why not. First, Belief-dependence is highly doubtful in moral cases: morally wicked actions do not become morally right just because you believe on misleading evidence that they are. Second, Belief-dependence, as least as I have stated it here, is meta-incoherent. The view seems to apply to itself, in which case it leads to paradoxes. Suppose you don’t believe in Belief-dependence? Is it then the case that it doesn’t apply to you? Or does it apply to you all the same? But third, it is interesting to consider what in this context may make Belief-dependence seem intuitively plausible. One thought is the following. Consider what is involved in doing one’s best to follow the proper rules. It is very natural to think that for you to make a sincere effort to follow proper rules, you need as a minimum to consider the particular rule at hand a correct rule. Otherwise, how can you even conceive of letting yourself be guided by the proper rules if the rules in question are ones that you don’t know of or do not consider applicable to your situation? Conversely, if you want to do your best to follow the proper rules, and believe that a particular rule is the correct rule, there is a sense in which you cannot but do your best to follow that rule. This might give some motivation for Belief-dependence: when doing your best, you have to go by the rules that you consider correct. Note that this rationale for Belief-dependence seems to presuppose the comprehendand-execute reading of Rule-governance. If we rationally disbelieve a rule it cannot govern our effort to reason correctly. An epistemic rule that we disbelieve cannot function properly as an epistemic rule in the sense that we cannot aim to reason correctly by consciously entertaining the content of the rule and try to comply with it. Conversely, if there are rules that we rationally (though perhaps mistakenly) believe to be correct rules, it is difficult to see how we could be governed by any other rules in our effort to reason properly. So, it might seem that the comprehend-and-execute reading of Rule-governance actually underwrites a distinct functional constraint of epistemic rules: rules must be rationally believed, otherwise they cannot govern our effort to be reasoning correctly. If this is right, then at least one rationale for Belief-dependence arises from the comprehend-andexecute reading of Rule-governance. However, if what I have argued previously is correct, there are independent grounds for rejecting the comprehend-and-execute reading of Rule-governance in favor of the constitutive reading. Hence this rationale for Belief-dependence is undermined.

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

This strongly suggests, I think, that we should reject Belief-dependence. There are cases in which all of (C–C) are true, but Belief-dependence should be asserted as a reason to think that (C–C) paradoxically entails (C). Of course, this does not show that (C–C) are non-paradoxical, only that one natural reason one might have to think so is undermined. This provides some evidence for assuming that there are cases where one is bound by some epistemic rule, rationally but falsely believes that one is not bound by that rule, and yet there is nothing irrational or enigmatic about this. What we should believe in such cases is determined by the rules that apply to us, not by what we believe to be bound by. So, these cases instantiate what we might call: Rule-akrasia: S in C is bound by epistemic rule R, and S in C rationally believes that S is not bound by R. Assume now that Rule-akrasia may be fully rational. Saying this is another way of asserting that the levels are detached—what rules apply to one as a thinker is unaffected by one’s rational beliefs about what rules apply. By insisting that Ruleakrasia may be fully rational, it seems that we have simply abandoned Calibration in favor of a version of level-splitting. What has become of the idea, central to Calibration, that first-order level and higher-order level should integrate? What about the Enkratic Requirement, adherence to which is central to Calibration? I now discuss these questions, and show that Calibration can both embrace the idea that Ruleakrasia can be fully rational, while still insisting on the Enkratic Requirement. Recall that we should reject Belief-dependence. The truth of epistemic rules does not depend on whether we rationally believe them or not. But epistemic rules may still be sensitive to our beliefs or evidence about our own epistemic situation, including our beliefs or evidence about our capacity to process first-order evidence correctly. Epistemic rules that take as arguments not only first-order evidence but also higher-order evidence about one’s epistemic capacities are, as I will say, broadly evidence sensitive. We can reject Belief-dependence and still claim some of our epistemic rules are broadly evidence sensitive. One generic way in which epistemic rules might be broadly evidence sensitive is by conforming to what I called Defeat by higher-order evidence: when S firmly believes p on the basis of evidence e, but higherorder evidence e0 tells S that her processing of e might be defective, then S should have less credence in p than otherwise warranted by e. So, Calibration should assert that epistemic rules applying to cases like The Pill and Sleepy Detective are broadly evidence sensitive in that the rules applying to these cases accommodate the fact that agents rationally but falsely believe that something might be amiss with their firstorder cognition. Broad evidence sensitivity is very different from Belief-dependence. Belief-dependence says that the applicability or truth of an epistemic rule depends on whether we believe it. Broad evidence sensitivity is the different feature that epistemic rules are sensitive to higher-order evidence. Calibration should reject Beliefdependence, but endorse that epistemic rules can be broadly evidence sensitive. To illustrate the difference between Belief-dependence and broad evidence sensitivity, consider a slightly modified version of one of Weatherson’s cases (Weatherson , p. ):

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

 

Aki has been provided with excellent though misleading philosophical evidence in favor of a false skeptical thesis—testimonial skepticism—and is therefore confident that one cannot get reasons to believe propositions merely on the basis of testimony. Assume that testimonial skepticism is false. Suppose also that Aki cannot help but believe p because her friend Ben tells her that p a particular occasion. So, Aki believes p on the basis of Ben’s testimony, but also believes that testimony never grounds justified belief. So, on her own reasoning she is not rationally entitled to believe p on Ben’s saying so.

Weatherson argues that Aki is fully rationally in her belief in p, assuming that other relevant conditions for justified testimonial beliefs are met. But since, as we assume, Aki is also rational in adopting testimonial skepticism, Aki is what I called ruleakratic. I suggest that Calibration should agree with Weatherson that Aki is not irrational. Aki is guilty of a justified but mistaken rejection of an epistemic rule governing testimonial belief, but Aki’s thinking that some epistemic rule is false doesn’t make it so. Compare to a different case: Amy has been provided with excellent though misleading evidence that Ben is an unreliable testifier in the particular domain that Ben is talking about. Ben tells Amy that p. In fact, however, Ben is a fully reliable testifier in that domain. Normally, Amy would not, under those circumstances, be able to firmly believe p on Ben’s testimony. If Amy nonetheless were to place high credence in p on the ground of Ben’s testimony, Amy would clearly be irrational. Calibration can fully embrace this as well, as we have seen, and explain it as follows: the reason that Amy would not be rational in placing high confidence in p is that the epistemic rule governing testimonial belief is sensitive to Amy’s evidence concerning the reliability of Ben’s testimony that p. So, Calibration can accept that Aki is rational in believing p, while Amy is not. The crucial difference between Aki’s and Amy’s cases is that Aki has a rational but false belief about an epistemic rule, whereas Amy has misleading higherorder evidence about the reliability of the testimonial evidence she receives, and this affects the rationality of her first-order belief. False beliefs about epistemic rules do not affect what epistemic rules govern us, but some rules are sensitive to higher-order evidence. The sort of adjustment between levels that is the hallmark of Calibration is not due to epistemic rules being Belief-dependent, but to some rules being broadly evidence sensitive. It is easy to generate cases similar to Aki’s and Amy’s. Consider: René believes, on the basis of apparently good philosophical arguments, involving scenarios with evil demons, that knowledge requires infallible foundations, and he believes that this rules out genuine perceptual knowledge. Hence, René believes that none of his ordinary perceptual beliefs qualify as knowledge. Suppose that perceptual knowledge does not require infallible foundations, and that many of René’s own perceptual beliefs actually meet the standards required for perceptual knowledge. Should we say that René doesn’t have ordinary perceptual knowledge after all? I suggest that we should not. René believes a false very general epistemic rule, but this peculiar epistemological theory of his does not undermine his perceptual beliefs, or their status as knowledge. René might rationally believe that his ordinary beliefs do

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not count as knowledge, yet there is nothing irrational in forming those beliefs. Contrast this with: Rob rationally (though falsely) believes that certain of his perceptual beliefs are formed in completely unreliable ways due to the interference of an evil demon. In such a case, Rob should not have high credence in those beliefs, even if they are in fact reliably formed, and if even true they do not count as known. The difference between is that René’s case merely involves false beliefs about a particular kind of epistemic rules, whereas Rob’s case involve sensitivity to defeating higher-order evidence, albeit misleading evidence. What I suggest is that Calibration should treat these cases very differently. Return finally to the cases featuring Sarah trying to figure out the solution to a mathematical problem and Sam, the sleep-deprived detective. These cases are persuasive because they involve higher-order evidence concerning the reliability of the relevant cognitive abilities. The cases don’t involve false beliefs about epistemic norms. We can easily imagine variants of these cases involving false beliefs about epistemic norms, rather than beliefs about the malfunctioning of cognitive faculties: Jack is a detective, and the evidence available him overwhelmingly implicates Jones. So Jack firmly believes that Jones is guilty. However, Jack is in the grip of skeptical epistemology. He believes, weirdly, that for him to firmly believe that Jones is the perpetrator, he must be able to conclusively rule out any other possibility. Jack realizes that, though far-fetched, there are plenty of logical possibilities in which Jones is not involved in the crime, which is not excluded by Jack’s evidence. Again, we can say that Jack’s first-order belief is entirely rational in that it respects the evidence at hand. Jack’s first-order belief should respect his evidence, not his false beliefs about epistemic norms. This is entirely different from Sam, who suspects that his relevant cognitive capacities are not functioning well. This is why Sam and Sarah, Amy and Rob need to revise their first-order credences to be in compliance with the Enkratic Requirement, and why Aki, René, and Jack do not violate the Enkratic Requirement when they fail to adjust their first-order credences in response to their false beliefs about epistemic norms. This explains why Calibration should accept the Enkratic Requirement, and at the same time acknowledge the possibility of rational Rule-akrasia. It is one thing to have a false belief about an epistemic norm, but a very different thing to believe that one’s processing of evidence is not functioning well. The driver of Calibration is the idea first-order credences are sensitive to both first-order evidence and to higher-order evidence concerning the reliability of one’s first-order faculties, not the mistaken idea that we should adjust first-order credences in response to false beliefs about epistemic norms.

. Summary It may be helpful to summarize the most important junctions of the entire argument. Cases like The Pill and Sleepy Detective are puzzling because they give rise to the

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

 

Akratic Trilemma: (E) says that first-order beliefs should respond to first-order evidence, (E) says that higher-order beliefs should respect higher-order evidence, yet the Enkratic Requirement (ER) is inconsistent with the conjunction of (E) and (E). There are five different responses to the Akratic Trilemma defended in the literature: () Calibration (retain ER, but reject (E)); () Level-splitting (retain (E) and (E), but reject ER); () No conflicting higher-order level justification (retain (E) and (ER), but reject (E)); () Pluralism: retain all in a pluralist interpretation; () Imperfection: retain all in a non-pluralist interpretation, and live with the pain. Calibration asserts that higher-order evidence can defeat the justification of firstorder belief. Maria Lasonen-Aarnio has questioned whether this is compatible with Rule-governance. To account for defeat in a rule-governed framework, we need a two-tiered epistemology, where one rule tells us how to react on first-order evidence, but a second layer of rules then modifies that rule in response to higher-order evidence. We can try to capture defeat by higher-order evidence in terms of an Über-rule, conflicting rules, or a hierarchy of rules. But neither option works, and therefore Calibration must go. However, the Fine-grained View accounts for Defeat by higher-order evidence in terms of finely individuated epistemic situations to which highly specific rules apply. When significant higher-order evidence is added to a situation, this generates a new situation to which a new rule applies. There is no need for two-tiered epistemology, no hierarchy of rules, no conflict between rules, and no looming regress of rules. Each rule has a simple structure, yet the full system of epistemic rules is obviously complex. This not an objection once we go by the constitutive reading of Rule-governance which does not require that rules are reason guiding in an overly-intellectualistic sense. Since higher-order evidence can affect the rationality of first-order beliefs they are not responsive only to first-order evidence. But beliefs might still be responsive only to evidence. The Über-rule is a conjunction of assignments of specific epistemic rules to specific situations. What happens in cases where a rule R applies to a case, and yet we have misleading evidence which by another rule R0 tells S that R doesn’t apply? In such cases, Calibration should hold, we are bound by R. This is because Calibration should reject Belief-dependence—rules do not become true or binding just because we rationally believe that they apply. The implication is that Rule-akrasia may be fully rational, and Calibration should embrace this. While Calibration should reject Belief-dependence, it should embrace the different idea that epistemic rules are sometimes broadly evidence sensitive—they take as input both first-order and higher-order evidence. This is what accounts for the core idea in Calibration—that we should reduce first-order credence in cases like The Pill and Sleepy Detective. Epistemic rules that conform to Defeat by higher-order evidence are broadly evidence sensitive in this way. This is why Calibration can accept that Rule-akrasia may be rational, while at the same time embracing The Enkratic Requirement, which is entailed by Defeat by higher-order evidence.

Acknowledgements Earlier versions of the material in this chapter have been presented at workshops and conferences in Bled, Cologne, Copenhagen, and Edinburgh. Thanks to all participants for valuable input, in

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particular Bjrn Hallsson, Clayton Littlejohn, David Christensen, Fernando Broncano-Berrocal, Giacomo Melis, Josefine Pallivicini, Julien Dutant, Matthew Chrisman, Ralph Wedgwood, Robert Audi, Thomas Grundmann, and Thor Grünbaum. Thanks in particular to editors and reviewers for this volume. Thanks to Alex France for help with language revision.

References Christensen, D. (). “Higher-Order Evidence.” In: Philosophy and Phenomenological Research (), pp. –. Christensen, D. (). “Disagreement, Question-Begging and Epistemic Self-Criticism.” In: Philosophers’ Imprint (). Christensen, D. (). “Epistemic Modesty Defended.” In D. Christensen and J. Lackey (eds), The Epistemology of Disagreement, Oxford University Press, pp. –. Hare, R. M. (). Moral Thinking, Oxford University Press. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs (), pp. –. Kappel, K. (). “Bottom-up Justification, Asymmetric Epistemic Push, and the Fragility of Higher-Order Justification.” In: Episteme (), pp. –. Kelly, T. (). “Peer Disagreement and Higher-Order Evidence.” In R. Feldman and T. Warfield (eds), Disagreement, Oxford University Press. Kelly, T. (). “Disagreement and the Burdens of Judgment.” In: The Epistemology of Disagreement: New Essays, pp. –. Lasonen-Aarnio, M (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research (), pp. –. Sliwa, P. and S. Horowitz (). “Respecting All the Evidence.” In: Philosophical Studies (), pp. –. Titelbaum, M. (). “Rationality’s Fixed Point (or: In Defense of Right Reason).” In T. Gendler and J. Hawthorne (eds), Oxford Studies in Epistemology , Oxford University Press. Weatherson, B. (). Normative Externalism. Unpublished manuscript. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research , pp. –.

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7 Higher-Order Defeat and Evincibility Maria Lasonen-Aarnio

. The plan Here is a package of views that feature notably in debates about higher-order evidence. First, it is possible to acquire higher-order evidence that I have made some sort of rational error in assessing my evidence and, hence, that my doxastic states fall short of a normative standard such as rationality. Second, such evidence has defeating force: sufficiently strong evidence that a belief is irrational defeats the rationality of that belief. As a result, the evidence calls for revising my opinions. For instance, if I become reasonably confident that my belief in p is irrational, I ought to suspend judgement about whether p.¹ Such views are often motivated by appeal to an ideal of epistemic modesty. . . . we may make mistakes in assessing evidence . . . reason to suspect that we’ve made a mistake in assessing the evidence is often also reason to be less confident in the conclusion we initially came to. The rationale for revision, then, expresses a certain kind of epistemic modesty. (Christensen , p. )

In what follows, my first aim will be to establish a claim that I think is almost selfevident once spelled out: that commitment to a systematic kind of defeat entails commitment to an epistemic access condition I call normative evincibility. My second aim will be to argue that there is a deep tension inherent in the view that while we can acquire misleading evidence regarding the normative status of our beliefs, their normative status is always evincible—that is, there is a deep tension inherent in views committed to defeat by higher-order evidence. I hope, then, to argue against the above package of views by providing a kind of reductio. Several authors (including myself ) have pointed out that those who endorse higher-order defeat, and the kinds of ‘level-connecting principles’ that go with it, appear to be committed to ruling out the possibility of certain kinds of epistemic

¹ See, e.g., Bergmann (), Christensen (a, a), Elga (ms), Feldman (), Horowitz (), and Schechter (). Maria Lasonen-Aarnio, Higher-Order Defeat and Evincibility In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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situations that arguably can arise.² But the discussions so far leave open the possibility of treating these seeming counterexamples as isolated, peculiar cases in which exceptions to generalizations are called for. My hope is to shift the dialectic: the kinds of cases that have been discussed are surface symptoms of a deeper, more endemic problem. I will begin by saying how I think of higher-order evidence, and by spelling out in more detail two distinct commitments of views endorsing higher-order defeat (section .). I then formulate a kind of epistemic access condition I call evincibility, arguing that endorsing higher-order defeat commits one to endorsing the evincibility of rationality, a kind of epistemic access to the rational status of one’s (present) doxastic states (section .). Finally, I discuss the internal coherence of views committed to defeat by higher-order evidence (section .).

. Higher-order evidence: a general characterization ‘Higher-order evidence’ (or ‘HOE’) is mostly epistemologist talk, understood in terms of an array of cases now somewhat canonical in the literature. Here is a case I take to be canonical enough: Resident Rezi is a medical resident in charge of diagnosing a patient and prescribing the appropriate treatment. After carefully reflecting on the patient’s symptoms, labs, and other relevant information, she becomes confident, and comes to believe, that the appropriate treatment is a  mg dose of Wellstrol. Rezi knows that due to her constantly sleep-deprived state, rarely an isolated cognitive blip will occur: an error in her reasoning that results in her arriving at a random conclusion by a cogent-seeming process. Rezi typically cannot detect such blips herself. She also know that just when such blips occur is itself random. As it happens, Rezi’s performance is being monitored by a team of neuroscientists who can see the fine cognitive workings of her mind. As she is about to prescribe Wellstrol for her patient, the neuroscientists inform her that a blip occurred: her diagnosis is the output of a process no better than a random guess at tracking the evidentially supported opinion. Though the neuroscientists are all but infallible, this time they are mistaken: Rezi’s original reasoning was impeccable, and a  mg dose of Wellstrol was appropriate given her evidence. The testimonial evidence that a blip occurred is higher-order evidence. Let me, however, flag at the outset that I do not want to imply that any body of evidence can be partitioned into two parts, the first-order evidence and the higher-order. It is better to think about higher-order evidence in terms of higher-order import. A single piece of evidence can have import regarding both first-order questions and higherorder questions. (And a piece of evidence can have different kind of import in ² See Christensen (b), Williamson (), Horowitz (), Worsnip (), and Lasonen-Aarnio (forthcoming).

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different situations, depending on other evidence one has.) What, then, is it for a piece of evidence to have higher-order import in a given situation? According to one popular slogan, higher-order evidence is evidence about evidence, or evidence bearing on evidential relations.³ Taking this view as a starting point, we could say that to have higher-order import is to bear on propositions regarding what one’s evidence supports. Applied to the above case, the suggestion would be that Rezi’s evidence regarding her cognitive blip now makes it less likely that her original medical evidence supported (made likely) that a  mg dose of Wellstrol is the appropriate treatment. But at the very least, I think this is far from clear. Evidence that her opinion was the result of a blip is evidence that her opinion is the output of a random process. But the fact that a random process produced the belief that a  mg dose of Wellstrol is the appropriate treatment has no bearing either on whether Wellstrol is the appropriate treatment, nor on whether it was likely on Rezi’s original evidence that it is appropriate. It does, however, have bearing on whether Rezi’s belief that Wellstrol is the appropriate treatment has, or ever had, epistemic statuses such as being rational, being justified, or constituting knowledge. Thinking about HOE exclusively as evidence about evidential relations blurs a more general phenomenon from view, one that is not specific to the epistemic realm. On a broader characterization that I favour, higher-order import is a matter of bearing on a relevant normative status of one’s mental states, or even of one’s actions. In the epistemology literature focus has been on evidence that bears on normative statuses such as rationality, justification, and perhaps knowledge. More generally, I suggest we think of HOE as any evidence bearing on whether one’s beliefs (intentions, actions) are permitted in some relevant sense, and hence, on whether they have some relevant normative status. There are two ways in which evidence might bear on whether, say, a belief has the status of being rational: the evidence might make it more likely that the belief has the status, or it might make it less likely. This talk of making likelier or less likely should be understood against the background of the total evidence one already has: if one thinks that updating happens by conditionalization, then what it is for a piece of evidence e to bear on a proposition p is for it to be the case that Pr(pje) 6¼ Pr(p) (here I am assuming that ‘Pr’ denotes something like evidential probabilities). I will leave it largely open what restrictions (if any) should be placed on the normative statuses that HOE bears on. I will assume that there is at least one sense of permissibility (and correspondingly, of being required and forbidden) pertaining to doxastic states, that there is a range of correct norms that concern permissibility in this sense, that evidence bearing on such permissibility counts as HOE, and that, according to views committed to defeat by HOE, sufficiently strong misleading evidence bearing on the permissibility of at least a range of doxastic states (such as beliefs) defeats their permissibility. Epistemologists typically talk of the normative statuses of rationality or justification in ways that fit this bill: a doxastic state is permitted in the relevant sense just in case it is rational or justified. Though the discussion below could be conducted using any term picking out a normative

³ One of the earliest proponents this idea was Feldman ().

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status with the desired role, I will use ‘rational’. I will use ‘irrational’ for a status had by a state that is not rational. (This might be a simplification, insofar as one thinks it is possible for a doxastic state to not be rational without being irrational, but I doubt anything of substance rests on it.) While my use of ‘rational’ is in line with its common use in epistemology, there is also another way of reading most of the rationality-talk below, namely, as a placeholder for any genuinely normative status of a doxastic state, such that evidence that a doxastic state lacks that status counts as HOE. By using the word ‘rational’, I do not intend to commit myself to any form of internalism, or even to the indispensability of the ideology of rationality. My characterization of HOE is somewhat schematic. But this broader characterization allows us to see parallels between various questions that have been discussed in different areas of philosophy. Before looking at these parallels, I will outline two commitments of a view on which there is a systematic phenomenon of higher-order defeat: the first is an existence claim I have implicitly assumed, a commitment to the possibility of acquiring even deeply misleading HOE; the second is a commitment to a systematic kind of defeat by such evidence. (I will set aside the further commitment to a positive recommendation regarding how one ought to adjust one’s doxastic states when faced with HOE.) () Acquiring evidence is a diachronic process: I start out with an initial body of evidence, and then acquire new evidence calling me to update my doxastic states. For instance, I may start out confident that a given belief is rational, but as a result of acquiring new evidence, it may then become unlikely on my new total evidence that my belief was (or, assuming that I still hold it, is) rational.⁴ What I take to be the central existence question regarding HOE concerns the possibility of acquiring deeply misleading evidence bearing on the normative status of one’s beliefs, actions, etc. Applied to the normative status of rationality, we can formulate this existence claim as follows: Acquisition It is possible to acquire evidence making it rational to believe, or at least be reasonably rationally confident, in falsehoods regarding the rational status of one’s doxastic states.⁵ Most of the literature on HOE in epistemology assumes Acquisition: I might hold a perfectly rational belief, but then acquire misleading evidence—whether in the form

⁴ There are delicate issues of time-indexing that arise here. Distinguish between the condition that my belief is rational, which can be true at one time and false at another, and the proposition my belief is rational at t. Upon acquiring HOE bearing negatively on the rationality of one’s belief, presumably the condition that my belief in p is rational becomes less likely. However, in such cases HOE is also supposed to have a kind of retrospective bearing on whether it was ever rational for one to believe p in the first place (cf. Christensen a). Hence, the proposition my belief is rational at t also becomes less likely. Note that this sort of thing often happens when we get evidence bearing on states that extend through time. For instance, I might think that I am angry at someone, but get evidence that I am not in fact angry, just disappointed. This is evidence bearing on the condition that I am angry, but it also bears on whether I was ever angry to begin with. ⁵ Note that by a ‘doxastic state’ I mean a state such as believing a proposition p, disbelieving p, or suspending judgement in p. I do not mean an entire state consisting of propositional attitudes to many different propositions.

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of peer disagreement, evidence that I am susceptible to cognitive biases, that I have been given a reason-distorting drug, etc.—that my belief is irrational, evidence that makes it rational for me to now be at least reasonably confident in the relevant falsehood.⁶ It is worth noting that some authors have denied the existence of at least a certain kind of misleading HOE, arguing that a deep kind of rational uncertainty about normative (as opposed to descriptive) matters is impossible. For instance, Elizabeth Harman (: –) claims that false moral beliefs that arise not from ignorance of non-moral facts, but from ignorance of moral facts, are not epistemically justified. Mike Titelbaum () argues that what he describes as a priori truths about what rationality requires can never be too unlikely on one’s overall evidence, for there is always a priori evidence in place for those truths.⁷ These views only deny that it is possible to acquire HOE that is misleading in virtue of pointing to false normative claims. But numerous discussions of defeat assume, for instance, that one can acquire even radically misleading evidence regarding support-facts.⁸ And the whole discussion of normative uncertainty (which I mention below), is propelled by the assumption that we often find ourselves in situations of even a deep kind of rational uncertainty about what normative (e.g., ethical) theory is correct.⁹ Let me now look at commitment to the defeating force of the relevant kind of HOE. () By views committed to defeat by higher-order evidence I mean views on which sufficiently strong HOE that one falls short of a relevant normative status more or less always, and necessarily, defeats the relevant normative status. I myself am sympathetic to a view on which normative statuses like knowledge and rationality are sometimes lost as one acquires HOE. However, the reasons for this have nothing to do specifically with HOE. For instance, my confidence in a proposition p might be rational to begin with, in virtue of the fact that p is likely on my

⁶ Christensen (: ) dubs a similar claim Respecting evidence of our epistemic errors: ‘This sort of ideal requires, for example, that in typical cases where one is initially confident that p, and one encounters good evidence that one’s initial level of confidence in p is higher than that supported by one’s first-order evidence . . . , one will give significant credence to the claim that one’s initial level of credence is too high. This sort of requirement applies even when one hasn’t actually made an error in one’s initial assessment of the evidence.’ ⁷ Littlejohn () holds a similar view, though on different grounds. If, for instance, R is a reason to believe p, and I have R, but I fail to believe p, then I manifest de re unresponsiveness to my reasons. But if I falsely believe that R is not a reason to believe p, Littlejohn thinks I manifest the same kind of error. (This is so even if I have arbitrarily strong evidence for the false view—Littlejohn denies that rationality is a matter of evidential support.) Such a view reflects remarks made by Harman (: ): ‘Believing a certain kind of behaviour is wrong on the basis of a certain consideration is a way of caring about that consideration.’ ⁸ For instance, in Feldman’s () example, a student acquires misleading evidence that her perceptual experience that p does not provide good support for believing p. See also Christensen (a) for uncertainty regarding logical entailment, Horowitz () for a case involving uncertainty about what a body of evidence supports. ⁹ See also, e.g., Littlejohn () for considerations in support of the idea that one’s evidence could provide arbitrarily strong support for a false normative view.

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evidence. I might then acquire evidence that bears on both the rationality of my confidence in p, and on p itself. If the result of such evidence is that p is no longer likely, then it is no longer rational for me to be confident in p. Or, my belief in p might constitute knowledge to start out with, but the result of acquiring HOE is that I no longer know p, because some condition on knowledge such as safety is no longer satisfied. Perhaps, for instance, I re-base my belief in such way that it is no longer safe from error. However, I very much doubt that such effects are systematic: sometimes HOE has no evidential bearing on the relevant first-order propositions, and sometimes one can continue to know despite the HOE (e.g., Lasonen-Aarnio , ). In fact, I think it is plausible that the Resident case described above is like this, for evidence that a blip occurred simply has no bearing on whether Wellstrol is the appropriate treatment. And I am inclined to think that Rezi can retain her knowledge, which is not to say that she is immune from criticism if she retains her belief. But here I won’t rely on these ideas: I bring them up to distinguish between views that endorse a (systematic) phenomenon of higher-order defeat, and views that do not. Further, not any view on which there is always something wrong with retaining a doxastic state despite having evidence that it is irrational is committed to what I am referring to as ‘higher-order defeat’. For instance, one might think that there is something blameworthy about Rezi if she retains her belief, despite having strong evidence that it is the result of a more or less random cognitive process. One might think that there is something incompetent or unvirtuous about it; or that it fails to reflect a good strategy for having beliefs that constitute knowledge, that are proportioned to the evidence, or that are rational. Closely related is the thought that it manifests some bad dispositions, dispositions that tend to lead one astray. Indeed, I think that Rezi manifests some bad dispositions, for given natural assumptions about her psychology, she cannot be disposed to retain her belief only when the higher-order evidence is misleading (see Lasonen-Aarnio forthcoming B). But such an answer does not commit one to the idea that the relevant epistemic statuses are in fact defeated—evidence that her belief is irrational need not zap its status as rational, or its status as knowledge. For a normative status N, we can ask whether HOE bearing in a specific way on whether one’s mental state or action has N has a systematic kind of defeating force with respect to that status. For instance, does evidence bearing in this way on whether an action is morally right defeat the moral rightness of the action? Does evidence bearing in this way on whether a belief is rational defeat its rationality? As we will see, there are disputes about exactly what this bearing should be. I will work with a simple view that explains common verdicts across standard candidate cases of higher-order defeat. Here is a pretty standard form that such cases take.¹⁰ Initially, one rationally believes a proposition p, or is rationally confident in p, based on evaluating some body of evidence, or of performing some reasoning. One then acquires new evidence the result of which is that it becomes reasonably likely that the doxastic process producing one’s belief was flawed and, hence, that one’s belief or high confidence in p ¹⁰ E.g. Christensen (b, a), Elga (, p. ), Feldman (), Kelly (), Lackey (), White (, p. ) Horowitz (, p. ). Not all of these authors defend defeat by HOE; the point is that examples they discuss have this structure.

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was irrational.¹¹ Typically, this is now likely to degree . (as in cases of peer disagreement) or above. On the view under consideration, then, there is a threshold r such that even if a state was in fact rational, its becoming likely to degree r or above that the state is irrational defeats the rationality of the state, rendering it irrational: Higher-Order Defeat Evidence making it likely to (at least) degree r or above ( < r < ) that a relevant doxastic state is irrational defeats the rationality of that state.¹² I will remain largely noncommittal about what the relevant threshold r is. One suggestion is that r is any value above .: evidence that make it likely that a state is irrational defeats the rationality of that state. But note that in some examples of higher-order defeat it is not even . likely that one has committed a rational error.¹³ Further, on such a view it would be difficult to make sense of conciliatory verdicts regarding cases of peer disagreement: in typical cases, a subject thinks it is equally likely that she committed a rational error as that her peer did. And there are numerous other examples in the literature besides in which defeat is assumed to occur in a situation in which the higher-order evidence makes it merely . likely that one’s initial belief is irrational.¹⁴ Paradigmatically, higher-order defeat is assumed to apply to belief: a belief cannot be rational if it is too likely on one’s evidence that it irrational. One might think that is true, but deny a corresponding thesis for suspension, or for credences (assuming such states are psychologically real): perhaps it can, for instance, be rational to assign a middling credence to p, even if it is likely that a middling credence is irrational.¹⁵

¹¹ A somewhat natural interpretation of the evidential dynamics of standard higher-order defeat scenarios is that it becomes likely to some sufficiently high degree that a given doxastic state one is in is irrational. This involves confirmation in both a comparative and absolute sense. However, I will assume that the (putative) defeating force of the new evidence derives from the fact that it is now sufficiently likely in the absolute sense that one’s belief is irrational, or that it is now rational for one to believe, or to be confident, that the belief is irrational. That it is now more likely than before that a given doxastic state is irrational follows from this assumption, together with the assumption that the state was previously rational (note that many descriptions of scenarios involving defeat by HOE explicitly assume this; as an example, see the Sleepy Detective case in Horowitz ). ¹² Parallel questions can be asked in a more propositional mode. For instance: ‘Is (strong) evidence that it is not (propositionally) rational for one to believe p compatible with its being (propositionally) rational for one to believe p?’ I will focus on doxastic normative statuses. On some views, defeat of a doxastic status like doxastic justification does not entail defeat of a propositional status such as propositional justification (see, e.g., van Wietmarschen ). ¹³ E.g., in Sliwa and Horowitz’s () example an anaesthesiologist should still think it is % likely that his initial judgement was correct. ¹⁴ E.g., examples motivating calibrationism, like Schoenfield’s () pilot who learns that she is at risk of hypoxia. ¹⁵ An alternative way of thinking works in terms of expectations. Some metanormativists have defended the view that one ought (it is rational, it is morally right, etc.) to perform the act with the highest expected value (e.g., Ross ); others have argued that one ought to go for the act with the highest expected choiceworthiness (e.g., MacAskill ). Similarly, in the epistemology literature some have defended Rational Reflection-type principles: one ought to adopt the credence that is one’s (rational) expectation of the rational credence, or some refinement of this (Christensen b, Elga , Dorst forthcoming). This chapter does not take on such principles (see Lasonen-Aarnio  for arguments against them). Even those who defend some form of Rational Reflection for credences could adopt

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It won’t matter much whether Higher-Order Defeat is assumed to apply to all doxastic states, as long as it applies to some. In what follows, I will assume that it applies at least to the state of belief. Note also that on a possible view of defeat, evidence as such that a state is normatively flawed is not sufficient for defeat; what constitutes a defeater is a belief, or sufficiently high degree of confidence, based on such evidence. Whether it is evidence as such, or doxastic states based on such evidence, that act as defeaters, won’t matter, for the main dialectic of this chapter can be played out on either view. Though the term ‘higher-order evidence’ is rarely used outside epistemology, the idea of defeat by higher-order evidence has clear repercussions in discussions of practical reason and ethics. Consider, for instance, Joseph Raz’s () discussion of what he calls exclusionary reasons.¹⁶ Raz argues that reasons to think one is incapacitated (drunk, fatigued, emotionally upset, etc.) can exclude certain first-order reasons for action from consideration. Consider Ann, who has excellent reason for signing a document that would commit her to an investment. She then acquires evidence that she is very tired and emotionally upset due to events that took place during the day—and that as a result, she cannot properly appreciate the force of her reasons. Hence, Ann has evidence that by signing the document she would be doing something she ought not to do, something she lacks reason to do. Raz seems to think that such evidence can make it the case that Ann ought not to sign. Or consider discussions of moral uncertainty. Assume for the sake of argument that a Kantian theory is correct. I start out confident in the theory, and confident that in my current situation I ought to use the money on my savings account to help out a friend who has fallen ill. I then discover literature on utilitarianism, and become confident that the Kantian theory is wrong: it is morally wrong for me to waste my resources helping my friend, when the same money could save the lives of numerous people who lack access to proper nutrition and basic medicine. Is it still morally right for me to use my resources to help my friend; is that still what I overall ought to do? If the action of helping my friend would no longer have the status of being morally right as a result of this evidence—or as a result of my rational response to it—then it looks like we have a case of defeat of moral rightness by higher-order evidence that an action is not morally right.¹⁷ Higher-Order Defeat (and a principle entailed by it, which I dub Level-connection below) as a constraint on rational belief. It is worth noting though that versions of Rational Reflection, together with a non-maximal creedal threshold view of belief, clash with so-called enkratic principles very widely defended, both by proponents of higher-order defeat, and philosophers who reject versions of acquisition (see LasonenAarnio () for examples of such a clash, and Lasonen-Aarnio (forthcoming A) for a critical discussion of enkratic principles). See also Dorst’s chapter in this volume. ¹⁶ The discussion is in chapter , ‘On Reasons for Action’. My case of Ann is based on an example discussed by Raz. It is striking how much recent discussions in epistemology about how HOE forces one to ‘bracket’ some of one’s first-order reasons resemble Raz’s idea of excluding certain first-order reasons. Thanks to Arto Laitinen for drawing my attention to Raz’s discussion. ¹⁷ Numerous authors have defended norms that recommend different actions depending on what credences one assigns to various first-order normative theories. Such a norm might initially recommend acting as the Kantian theory recommends, but then switching as a result of becoming confident in utilitarianism. However, there is no agreement in the literature regarding just what status it is that these ‘metanormative’ theories are supposed to govern. Some views may be committed to saying that what it is

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Hence, putative cases of defeat by higher-order evidence are not restricted to epistemology. I will now describe an epistemic access condition I call evincibility—in particular, the evincibility of normative statuses such as rationality. I will then argue that HigherOrder Defeat entails commitment to the evincibility of rationality.

. Normative evincibility Evincibility is a kind of epistemic access. The access in question is a matter of certain truths always being sufficiently likely—likely at least to some positive degree r—on one’s evidence. On the strongest construal, r = , and evincibility requires that these truths are always certain. Evincibility in the strongest sense entails that there can never be evidence bearing on the relevant class of truths. If we let r be some value very close to , then evincibility becomes very weak. Certain cases involving infinities aside, many find appealing a view on which evidence is factive, and the rational priors are never certain of falsehoods. While not equivalent to the weakest version of evincibility, such a view already guarantees that, setting infinite cases aside, all true propositions are probable to some degree above . A rather natural, substantive form of evincibility assumes that r > .: certain truths are always likely on one’s evidence. When it comes to these truths, our evidence always points us in the right direction. Another, still rather substantive, thesis assumes that r  .: certain truths can never be unlikely—our evidence can never point us in the wrong direction. But even if, for instance, r > ., we still have what I am calling an evincibility thesis: the relevant truths cannot be very unlikely on the evidence. In what follows, my concern will be with the evincibility of truths regarding the normative status of one’s doxastic states (or other mental states, or actions); in particular, the evincibility of rationality. Questions about the evincibility of doxastic states themselves, and about their rational status, interact in various ways. For instance, if the rationality of one’s states is evincible, then if one rationally believes p, it is fairly likely on one’s evidence that one believes p (since it is fairly likely that one rationally believes p). And of course, one kind of uncertainty about whether or not one rationally believes p derives from uncertainty about whether one believes p in the first place. In what follows, I will bracket uncertainty about what doxastic states one is in by assuming that one’s doxastic states are strongly evincible: truths regarding rational to do depends on one’s credences in claims about what it is morally right to do (e.g., Lockhart , Ross , Sepielli ). Gustafsson and Torpman () couch their view as concerning ‘what the morally conscientious agent chooses’. Guerrero () speaks of blameworthiness or moral culpability. However, several of the views that have been discussed are committed to something very much like higherorder defeat of moral rightness, or defeat of the status of being what one overall ought to do. For instance, Gracely () argues that one should act in accordance with the ethical theory most likely to be right. Harman () argues against the view that ‘an agent’s moral uncertainty (and specific moral credences) are crucially relevant to how the agent should act’, a view she attributes to various authors who put forth metanormative theories. Weatherson () argues against a view on which what is morally right (or permitted, or wrong) is affected by moral uncertainty: if, for instance, the Kantian view is indeed correct, then the fact that I become convinced that effective altruism is the way to go does not make it morally wrong for me to help my friend.

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whether or not one is in some doxastic state d are always certain on one’s evidence. While I think this assumption is false, it is helpful to set aside one kind of uncertainty in order to bring another kind into focus. It will be helpful to talk of conditions: a condition can obtain at one time, and not obtain at another. For instance, that I rationally believe that it is Monday might be true on Monday, but no longer true on Tuesday. We can distinguish between a positive (what I meant by ‘evincibility’ above) and negative evincibility thesis. For instance, if we assume that the relevant threshold value r > ., then according to the positive evincibility thesis, whenever a condition C obtains, it is likely that it obtains. According to the negative evincibility thesis, whenever C doesn’t obtain, it is likely that it doesn’t obtain. Together these entail that it is evincible whether C obtains: one’s evidence always points to the truth regarding whether C. Note that evincibility can be characterized in a way that abstracts away from evidentialist assumptions: we could characterize it directly in doxastic terms instead. The idea would be that if a condition C is evincible, then one is in a position to be rationally confident at least to a relevant degree r that C obtains. Indeed, in what follows I will assume evincibility to have such doxastic ramifications. Normative evincibility theses claim that truths concerning a relevant normative status of one’s doxastic states (actions, etc.) are evincible. If rationality is (positively) evincible, then if one rationally believes p, one is in a position to be rationally confident at least to degree r that one’s belief is rational—the condition that I rationally believe p is evincible. If rationality is both positively and negatively evincible, then one is always in a position to be reasonably confident about truths regarding the rational status of one’s own beliefs. According to a strong form of access internalism that most internalists these days shy away from, we always have a special kind of access to the normative (e.g., rational or justificatory status) of our doxastic states.¹⁸ Evincibility is a way of spelling out what such access might involve: for instance, it may be a matter of the relevant propositions always being likely on one’s evidence. According to Williamson’s () use, a luminous condition is such that a subject is always in a position to know that it obtains when it does. Evincibility and luminosity feature distinct kinds of epistemic access. A proposition might be true and likely on one’s evidence, even if one is in no position to know it because one is in a Gettier case. I also doubt whether being in a position to know a proposition p entails that p is likely on one’s evidence: there are cases in which a proposition is unlikely on one’s present total evidence, but one is nevertheless in a position to know it. (Perhaps this happens routinely when we form perceptual beliefs.) In any case, it should be clear that evincibility and luminosity make reference to different epistemic access conditions: that a proposition is likely at least to some degree r on the evidence is not tantamount to one being in a position to know it.¹⁹

¹⁸ See Chisholm (), Bonjour (), and for a more recent defence, Smithies (). ¹⁹ Berker () talks about lustrous conditions: a condition is lustrous iff whenever it obtains, one is in a position to justifiably believe that it obtains. The evincibility of a condition (given a way of specifying the threshold r) may amount to its lustrousness, assuming that one is in a position to justifiably believe a proposition p just in case it is likely to at least degree r on one’s evidence.

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At first sight, the evincibility of rationality might look to be at odds with the very considerations that motivate epistemologists to take higher-order evidence seriously. After all, examples of defeat involve even radically misleading evidence about the normative status of one’s beliefs. Indeed, I argue below that there is a serious tension between Acquisition and evincibility. But first I will argue that rather than contradicting evincibility, Higher-Order Defeat in fact entails some form of the evincibility of rationality. As many in the literature have realized, commitment to higher-order defeat is a commitment to some form of level-connecting principle. Indeed, many take commitment to such principles as a starting point, and argue for defeat on the grounds that they must be respected.²⁰ Christensen (: ) remarks that level-connecting principles formulate ‘ways of taking beliefs in general to be rationally constrained by beliefs about what beliefs are rational’. Sliwa and Horowitz () lay down the desideratum that ‘One’s rational first-order and higher-order doxastic attitudes should not be in tension’, an example of such a tension being believing a proposition, while being merely % confident that the proposition is supported by one’s evidence (and hence, being % confident that it is not). For instance, if Higher-Order Defeat states that evidence making it likely to some degree r or above that a state is irrational defeats its rationality, then HOD entails the following level-connection principle: Level-connection There is no case in which a relevant doxastic state d is rational but Pr(d is irrational)  r ²¹ Note that though views endorsing higher-order defeat are committed to some kind of level-connecting principle, the converse is certainly not true. Indeed, various authors have defended level-connection principles, while denying at least some form of Acquisition.²² Level-connection in turn entails an evincibility claim. It states that it is impossible for evidence to make it likely to degree r or above that a state d is irrational, if d is in fact rational. But then, if d is rational, and the evidence always makes claims about its rationality or irrationality likely to some degree, it must be likely to some degree below r that d is irrational. Insofar as a state is either rational or irrational, it follows that it must be likely to degree (—r) or above that d is rational. This is to claim that rationality is evincible; just how strong the claim is depends on exactly how

²⁰ E.g., Horowitz (), and Christensen in numerous writings, such as Christensen (). ²¹ Does this principle entail that it is irrational to believe p, while suspending judgement about whether one’s belief in p is irrational and, hence, that a ‘moderate’ form of akrasia is always irrational? This is far from clear. Suspension of judgement might be appropriate in situations in which one has no evidence whatsoever bearing on the question of whether one’s belief is irrational, situations in which the relevant evidential probabilities are imprecise or even undefined. Whether such ‘moderate’ forms of akrasia are irrational has been a source of debate. Hazlett (), for instance, argues that moderate akrasia is not irrational; others, like Feldman (), Huemer (), Smithies (), and Bergmann (), think that it is. ²² E.g., Titelbaum () and Littlejohn ().

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Higher-Order Defeat is fleshed out.²³ I argued above that a natural value for r, one that would explain rather standard verdicts in cases of peer disagreement, is .. Such a threshold would entail that rationality is evincible in the following sense: if a relevant doxastic state, such as the state of believing p, is rational, then it is likely (likely to some degree above .) that it is rational. Alternatively, one might think that only evidence making it likelier than not that a state d is irrational defeats the rationality of the state. In that case, rationality would be evincible in the following sense: if a state is rational, then it is likely to degree . or above that it is rational. That is, if a state is rational, it cannot be unlikely that it is rational. In what follows, I will assume this latter view as a default. This thesis builds in a substantial assumption about defeat. But the assumption is in fact rather conservative. Moreover, my discussion below does not essentially rely on this threshold. Thus construed, Higher-Order Defeat entails the following thesis: Evincibility of rationality For all relevant doxastic states d, if one is in d and d is rational, then Pr(d is rational)  . What the relevant doxastic states here are depends on one’s commitments about defeat. I will assume that Evincibility of rationality applies at least to the state of full belief. To block the inference from Higher-Order Defeat to evincibility, one could try to appeal to a distinction between two kinds of cases in which it is likely to degree . or above that one’s belief in some proposition p is irrational. In the first case, one simply lacks reasons or evidence for thinking that one’s belief in p is rational. In the second, one has positive reason to think that one’s belief is irrational. One might propose that defeat only occurs in the second kind of case and hence, that no general evincibility thesis can be inferred from a commitment to defeat. (Note that according to a possible view, when one has no evidence bearing on whether a state is rational or irrational, propositions about its rationality simply have no probability on the evidence. But that is not the view being proposed: the idea is that it might be likely to degree . or above that a belief is irrational simply because there is no evidence pointing to its rationality.) However, level-connecting principles are completely silent on such issues regarding the evidential basis of one’s attitudes, for they are structural in nature. If defeat is motivated by appeal to the idea that what doxastic states are rational is constrained by one’s opinions about what doxastic states are rational—as it often is— then there is no room for the kind of distinction being drawn. From this perspective, believing p while being confident that it is irrational for one to believe p, for instance, is always bad.²⁴

²³ Perhaps one could allow for propositions about such higher-order matters to have no probability on the evidence whatsoever; in that case, we could formulate a conditional principle: for instance, if one’s belief in a proposition p is rational, then insofar as the proposition my belief in p is rational has some probability on one’s evidence, its probability must be above .. ²⁴ Moreover, the strategy discussed assumes that there can be cases in which it is . or above likely on a subject’s evidence that a belief is irrational even though the subject lacks any evidence regarding the matter. On such a view there isn’t a presumption in favour of our own rationality built into the rational priors—in

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I have argued that higher-order defeat commits one to some form of the evincibility of rationality. At this point the following worry, already mentioned above, might resurface: aren’t views committed to defeat by higher-order evidence committed to both affirming and denying the evincibility of rationality? On the one hand, one can acquire deeply misleading higher-order evidence bearing on the rational status of one’s doxastic states (Acquisition). On the other, Higher-Order Defeat entails that the rational status of one’s doxastic states is evincible: if a belief is rational, then it cannot be too unlikely that it is rational (in fact, according to the thesis I dubbed Evincibility of rationality, if a state is rational, it cannot be unlikely to be rational). There is no blatant logical contradiction here—though, as I will argue, a deep tension remains. By Acquisition, one can acquire misleading evidence regarding the rational status of one’s beliefs, but assuming Higher-Order Defeat, the result of acquiring such evidence is not a situation in which one is (radically) misled about the rational status of one’s current beliefs. Consider again Rezi. At a time t, Rezi formed the belief that the appropriate treatment is a  mg dose of Wellstrol (pw). At a subsequent time t 0 , she received the testimony of the neuroscientists. Given this new evidence, it is likely on her total evidence at t 0 that her belief in pw was (and still is, assuming she holds it) irrational. However, if the rationality of the belief is defeated, then Rezi can no longer rationally believe pw. Instead, perhaps it is appropriate for Rezi to suspend judgement about the matter. Hence, while Rezi does not have access to the rational status of her belief at the earlier time t, the rational status of her current doxastic state (say her suspension of judgement) might be evincible.²⁵ In the rest of this chapter, I will argue that though there is no straightforward logical contradiction, there is nevertheless a serious tension between Acquisition and Evincibility of rationality. There is considerable pressure on those who are liberal about the possibility of misleading evidence regarding the rational status of one’s beliefs to give up on defeat by higher-order evidence.

. A tension The dialectic of this section will be as follows. There are two distinct (though not exclusive) ways of being misled about the normative status of one’s own beliefs and other doxastic states—and correspondingly, two different kinds of misleading HOE. Some HOE is misleading regarding what the correct theories or norms are; other HOE is misleading regarding contingent features of one’s epistemic situation fact, there might even be a presumption in favour of irrationality. But this raises tricky issues about how we could ever acquire evidence about our own rationality. According to a view able to avoid both scepticism and a kind of bootstrapping many would regard as dubious, we are entitled to assume our own faculties to function rationally, in the absence of evidence to the contrary. But if such views are right, then any case in which it is rational to be confident to some degree above . that a belief one holds is irrational is one in which there is some positive evidence for its irrationality. And then, there is no work left to do for the distinction appealed to. ²⁵ See also Skipper (forthcoming, sect. ), who distinguishes between higher-order defeat, which is a diachronic phenomenon, and ‘self-misleading evidence’, which is synchronic. Having evidence that is selfmisleading to a sufficiently high degree would be incompatible with evincibility, but as Skipper argues, defeat doesn’t entail the possibility of self-misleading evidence.

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(and some HOE is misleading regarding both). As an example of the former, one might acquire evidence calling into question an evidentialist norm stating that one is permitted to believe p when one’s evidence makes p likely; or, one might acquire evidence calling into question whether a determinate body of evidence E in fact makes p likely or not. As an example of the latter, one might acquire evidence calling into question exactly what one’s evidence consists in. Either kind of misleading evidence can threaten the evincibility of rationality. Focusing on these two distinct kinds of HOE creates two different contexts in which to investigate and test views committed to both Acquisition and Higher-Order Defeat. I will challenge the joint tenability of the two theses in both contexts. The methodological assumption will be that if neither context provides a fertile ground for the combination of views under discussion—views committed to a phenomenon of defeat by higher-order evidence—then neither will messier contexts involving HOE that fall under both kinds. Let me begin by distinguishing between these two different kinds of HOE. For any possible case, what I will call a complete theory tells one what doxastic states (if any) one is permitted or required to be in in that case. We could think of such complete theories as functions.²⁶ Let a total set of doxastic states be a set of doxastic states, each doxastic state (such as the state of believing p) in the set being a doxastic attitude toward a single proposition, such that one is in that total set of states just in case one is in each state in the set, and in no other doxastic states. A complete theory takes possible epistemic cases as inputs, and gives sets of total sets of doxastic states as outputs. T permits believing p in a case c just in case T takes c to a set of total sets of doxastic states such that believing p belongs to at least one of these total sets of doxastic states. T requires believing p in c just in case T takes c to a set of total sets of doxastic states such that believing p belongs to all of these total sets of doxastic states. (Indeed, if a theory permits only one total set of doxastic state in c, it will take c to a set with only that total set of states as a member.) T forbids believing p in c just in case T takes c to a set of total sets of doxastic states such that believing p belongs to none of these total sets of doxastic states. Theories are sensitive to certain (epistemically) relevant features of cases; just what those features are depends on the nature of the theory. For instance, an evidentialist theory might only be sensitive to what one’s total evidence in a case is. Other theories might be sensitive to other factors, like pragmatic stakes. We can derive norms from theories. For instance, if a theory requires one to believe p in all cases in which a condition C obtains, then the theory will entail the norm ‘If C, believe p!’ Conversely, we can infer facts about theories form the correct norms: if ‘If C, believe p!’ is a correct norm, then the correct theory will require believing p in all of the possible case in which C obtains.²⁷

²⁶ Such complete theories are discussed by Titelbaum (, p. ) and Skipper (forthcoming) in connection with their discussion of rational requirements. ²⁷ In what follows, I will assume that norms are informative generalizations. For instance, if a theory permits believing p in a particular case c (among many other cases), and condition C obtains only in that case, and in no others, then ‘If C, believe p!’ won’t be general enough to count as a norm in the intended sense.

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The first kind of misleading HOE is evidence that bears on what theories are correct in the first place. Assume that If C, believe p! is a correct epistemic norm, where C is a contingent condition that obtains in some cases and not in others. An example of this sort of evidence is evidence that this norm is incorrect: I might, for instance, acquire evidence that I am in fact forbidden to believe p when condition C obtains. Or, take the norm If your evidence makes p likely, believe p! Assume that I know all the contingent facts about my epistemic situation, including exactly what my evidence is. Nevertheless, I might acquire misleading evidence that E, my total body of evidence, does not make p likely. This is evidence that the correct theory does not permit believing p in situations in which E is my total evidence. By bearing on what complete theories are correct, the first kind of HOE bears on a domain of immutable and necessary normative truths. By contrast, the second kind of misleading HOE bears on contingent features of one’s situation, including facts about what doxastic states one is in, or how they came about. As a limiting case, an agent could be certain what the correct theory recommends in any possible case, but not know what the correct theory recommends for her, in virtue of being uncertain about epistemically relevant features of her situation. Given a norm of the form If C, believe p!, I will refer to C as its application condition. (Note that application conditions can concern the basing of one’s beliefs—and norms that concern doxastic justification or rationality standardly do.) As a general rule, misleading evidence concerning whether the application conditions of (correct) norms obtain falls into this second class.²⁸ The higher-order evidence in Resident is like this: for a wide range of candidate norms governing doxastically justified or rational belief, Rezi gets evidence that her belief that a  mg dose of Wellstrol is the appropriate treatment doesn’t satisfy the conditions specified by those norms. Her evidence makes it likely, for instance, that her belief isn’t appropriately based on sufficient reasons or evidence, that it isn’t the output of a reliable cognitive process, etc. I will now investigate the stability of views that endorse both Acquisition and Higher-Order Defeat in two different contexts, corresponding to these two different kinds of HOE that one might acquire. I will first bracket uncertainty about the kinds of necessary normative truths described above (e.g., truths concerning which theories are correct), focusing on uncertainty regarding contingent application conditions that arises from misleading HOE concerning one’s epistemic situation; I will then bracket uncertainty about application conditions, and more broadly, contingent features of one’s situation, focusing on uncertainty regarding what theories and norms are correct.

.. Uncertainty about application conditions of norms Focus first on misleading HOE that bears on contingent, epistemically relevant features of one’s situation. It is clear that the evincibility of rationality puts ²⁸ This is just a general rule. Consider the norm ‘Believe p just in case p is likely on your evidence!’ Evidence that is misleading regarding what evidence one has in the first place falls under the second kind of HOE concerning contingent matters. By contrast, evidence that is misleading regarding whether a determinate body of evidence E that one has makes p likely falls under the first kind of HOE: it is evidence regarding necessary truths concerning which theories are correct.

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constraints on the nature and strength of misleading evidence one could have regarding one’s current situation. To see this, consider the following kind of case. I know (and it is likely on my evidence) that Believe p if C! is a correct norm; similarly for Disbelieve p if C0 ! Assume that I in fact believe p, and that C obtains. However, my evidence regarding my situation is misleading: it is likely on my evidence that C0 obtains. Hence, it is likely that my belief is irrational—it is likely that I am in a situation in which norms I know to be correct recommend disbelief. Then, that my belief in p is rational is not evincible: it is false, but likely on my evidence to be true.²⁹ Demonstrating that some cases in which application conditions of norms fail to be evincible create failures of the evincibility of rationality isn’t, of course, to show that the evincibility of rationality entails that the application conditions of norms must be evincible. Assume, for instance, that the correct norms deem it rational to believe p if any of the conditions C₁, . . . , Cn obtain. The evincibility of rationality is compatible with, for example, C₁ obtaining, but having misleading evidence pointing to C₂. However, it is difficult to see a motivated way to allow having radically misleading evidence about central features of one’s epistemic situation, while nevertheless retaining the evincibility of rationality. If our evidence could even radically mislead us regarding epistemically central features of our situations, counterexamples to the evincibility of rationality are bound to crop up. In this section, then, I will consider the following kind of view: one can acquire even radically misleading evidence regarding whether the application conditions of a correct epistemic norm obtain, but one cannot have (very) misleading evidence regarding such matters. I will first discuss the worry that application conditions of correct epistemic norms are not evincible, and a failed but instructive attempt to construct evincible conditions out of non-evincible ones. I then look at the most plausible candidates for strongly evincible application conditions, conditions that concern a special class of internal mental states. I argue that views stipulating such states are hostile to higher-order defeat, and generalize the lesson: where appeal to a special class of internal mental states might help with the evincibility of rationality, it is at odds with instances of Acquisition relevant to putative cases of defeat by HOE. A first objection to any view entailing that the application conditions of correct epistemic norms are evincible is that the application conditions of correct epistemic norms are not evincible. For pretty much any condition, the objection goes, our evidence about whether it obtains can be radically misleading. Consider, for instance, an evidentialist norm that requires believing p when p is sufficiently likely on one’s evidence (possibly together with other conditions, such as that one has considered whether p, and/or that whether p is relevant in one’s situation). Assume that evidence is factive: for a proposition e to be part of one’s evidence, e must be true. Further, assume that for e to be part of one’s evidence, one must bear some suitable epistemic relation to

²⁹ The argument assumed that I know the correct norms. There is an ad hoc way to block the argument: even if I could have misleading evidence regarding application conditions of norms, perhaps in such circumstances I would also always have misleading evidence regarding the correct norms, and the two mistakes would cancel each other out. I will set such views aside.

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e (such as knowing e, or justifiably believing e). Such commitments make it implausible to think that conditions such as that p is likely on my evidence are evincible in any very interesting sense. First, some proposition e might fail to be part of my evidence due to being false, even if it is likely that e is part of my evidence—and hence, it is likely that my evidence supports p, in virtue of the fact that e strongly supports p. Any theory on which evidence consists exclusively of true propositions gives rise to the possibility of such case.³⁰ Second, some proposition e might be part of my evidence, even if it is likely that it is not: for instance, if there is such a thing as improbable knowing, I might in fact know e, even if it is very unlikely that I know e and hence, unlikely that e is in fact part of my evidence. In such a situation my evidence might in fact make e likely, even if it is highly improbable that it makes e likely. Such issues have had a decent bit of coverage in recent epistemology.³¹ The point is that as far as the present project goes, acknowledging that the application conditions of epistemic norms are not evincible means that we have not found a way to jointly accommodate Acquisition and Higher-Order Defeat, for we haven’t even found a way to accommodate Higher-Order Defeat. One possible reply to such worries is that though prima facie plausible candidates for application conditions of true epistemic norms are not evincible, we can manufacture evincible conditions from non-evincible ones by adding a further condition acting as a kind of no defeater clause. For instance, assume that evidence making it at least . likely that a doxastic state is not rational is sufficient to defeat the rationality of that state (the precise threshold won’t matter). The idea now is to construct application conditions that rule out having such defeating evidence. Here is a first stab. On one kind of view, some sort of evidentialist condition is necessary and sufficient for the rationality of a belief: for a belief to be rational, it must be sufficiently likely on the evidence, and perhaps based in the right kind of way on evidence making it likely (let this be condition C—the reader can plug in any non-evincible condition in place of C below). What we want is an additional necessary condition that takes care of higher-order defeat: that C itself be likely to obtain on one’s evidence. And hence, one might propose the following overall (necessary and sufficient) condition C* on rationality: C* = . C . It is likely to some degree above . that . Does this rule out the possibility of cases in which C* obtains, but it is likely to degree . or above that C* doesn’t obtain? It doesn’t. C* is equivalent to the conjunction of . and . Consider a case in which C itself is likely, but condition . is not—and hence, though C is likely, it is not likely to be likely. This would be a case in which C* obtains, even though it is likely to degree . or above that it does not and hence, that one’s belief is not rational. But then, we have a case in which HigherOrder Defeat fails.

³⁰ Cf. Weatherson (). ³¹ E.g., Christensen (b), Horowitz (), Lasonen-Aarnio (, forthcoming) Weatherson (), Williamson (, ), Worsnip (), Salow ().

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At this point one might propose the following fix: simply require that . itself is likely, and likely to be likely, etc. That is: C* = . C . It is likely to some degree above . that . (pr(C) > .) . It is likely to some degree above . that . (pr(pr(C) > .) > .) ⋮ In a case in which each of these conditions (of which there is a countable infinity) obtains, we might say that C is super-evincible. But we are assuming that C* constitutes the application condition of a correct epistemic norm. So it is C* itself, and not merely C, that needs to be evincible. But now we encounter essentially the same problem as before: even the obtaining of all of the above conditions ., , . . . does not entail that C* itself is evincible—it does not entail that C* is likely to some degree above .. C* is equivalent to the conjunction of its conditions; hence, C* is likely just in case the conjunction of its conditions is likely. Each condition n   specifies that condition n –  be likely. However, the fact that each conjunct individually is likely does not entail that the conjunction as a whole is likely. We have still not managed to specify an evincible condition.³² Lesson: It is not that easy to simply build up evincible conditions! Let me now explore a different tactic that does not give into anti-envincibility considerations in the first place. Views on which the application conditions of correct epistemic norms are evincible to begin with offer a more promising platform for Acquisition and Higher-Order Defeat. Some argue that we always have perfect access to our evidence: if e is not part of my evidence, it is certain that e is not part of my evidence; and if e is part of my evidence, it is certain that e is part of my evidence.³³ But to meet the present challenge, one must show not only that application conditions of correct norms are evincible, but also that Acquisition holds: one can acquire misleading evidence regarding these conditions. Those who think that we have perfect access to our evidence tend to think that evidence consists of a special class of internal mental ³² There are frames of probabilistic epistemic logic we can use to model such cases (for more on such frames, see Williamson ). Consider a probabilistic frame , where W is a finite set (informally, of worlds), R is diadic a relation over W (informally, a relation of accessibility between worlds, mapping a world x to all the worlds compatible with one’s evidence at x), and Pr is a prior probability distribution over subsets of W. Propositions can be modelled as subsets of W. Let W = {w₁, w₂, w₁}; R(w₁) = {w₁, w₂, w₃}, R(w₂) = {w₁, w₂, w₃}, and R(w₃) = {w₂, w₃}. Let Pr be uniform. A probability function Prx at world x results from conditionalizing Pr on the total evidence at that world— i.e., the set of accessible worlds. Note that in this frame Prw₁ = Prw₂ = Pr. Let p be {w₁, w₃}. Let ‘pr (p) > .’ stand for ‘the probability of p is above .’. (It is important that I am here using ‘pr (p)’ for a definite description). So, for instance, in the frame assumed the proposition pr(p) > . = {w₁, w₂}. Let p (already defined above), p0 , and p* be as follows: p ¼ fw1 ; w3 g: 0

p ¼ prðpÞ>:5 & prðprðpÞ>:5Þ>:5 & prðprðprðpÞ>:5Þ>:5Þ>:5::: ¼ fw1 ; w2 g: p* ¼ p & p0 ¼ fw1 g: At w, p is ‘super-evincible’ (it is likely, likely to be likely, etc), but Prw₁ (p*) = /. ³³ For an interesting recent defence of such views, see Salow ().

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states, or of propositions concerning such states. Let us see if appeal to such states would help. Assume that the correct epistemic norms make reference to a domain of accessible internal mental states. Perhaps, for instance, they make reference to seemings, a sui generis class of mental states the obtaining of which is always certain on one’s evidence—at least as long as one undergoes the relevant seeming. Assume that such seemings have a phenomenology with a special kind of epistemic glow: as long as one is actually in such a state, there can be no rational doubt that one is in it (compare such states with Descartes’s clear and distinct ideas). Or: whenever one is in the state, it is evincible in a very strong way that one is in it. As a result, my present, occurrent seemings are perfectly evincible, but not my past seemings, for I can rationally doubt what seemings I experienced just a moment ago.³⁴ Even when there is no forgetting, this might explain how I could rationally doubt my past seemings, but not my present seemings. And insofar as the application conditions of correct epistemic norms only make reference to seeming-like states, this would allow one to acquire misleading evidence regarding whether these conditions obtain. However, for the purposes of explaining higher-order defeat, such epistemic glow is too powerful. The problem is that acquiring misleading HOE doesn’t always march step in step with ceasing to be in the relevant seeming-states. Indeed, one of the peculiarities of defeat by HOE is that it calls for adjusting one’s beliefs—or so many have argued—even if one’s original evidence is still in place, and even if one correctly appreciates its force. So, for instance, it might still seem to me that p, even if I have evidence that it doesn’t in fact seem to me that p. But as long as it still seems to me that p, and my seemings glow, we don’t have a case in which it is rational for me to doubt whether it ever seemed to me that p in the first place. The problem is that the relevant application condition making reference to seemings might still obtain. Glow might explain evincibility, but doesn’t give the defeatist what she needs, for it is impossible to acquire misleading evidence about one’s seemings so long as one continues to experience them. (And of course, the strategy is hopeless if these special kinds of states that glow don’t exist in the first place.) The lesson can be generalized. Whether what is at issue is a very strong kind of evincibility on which certain conditions are always certain to obtain, or a weaker kind on which they cannot be likely not to obtain, the most plausible candidates for evincible conditions are conditions making reference to a special domain of internal mental states with a glow-like property. Glow guarantees that our overall evidence regarding these special states cannot be (too) misleading. Let C be a condition stating that one is in such an internal state, such as the condition that it seems to one that p. The problem now is that in order for one to acquire evidence making it likely that C doesn’t obtain, C has to cease to obtain. But if C concerns the obtaining of internal mental states, there is no reason to think that it always ceases to obtain when higherorder evidence that C doesn’t obtain comes in—and as a result, there is no reason to think that the obtaining of C marches in step with common verdicts about higherorder defeat.

³⁴ See Hawthorne and Lasonen-Aarnio (forthcoming) for a critical discussion of such views.

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I have not proved that it is impossible to accommodate systematic higher-order defeat—and in particular, its commitment to both the evincibility of rationality and Acquisition—within the present context that allows for misleading evidence regarding contingent features of one’s epistemic situation, but not regarding what theories or norms are correct. But I take the considerations given to at least give us strong prima facie reasons to be doubtful. What is needed is norms with application conditions that satisfy the following constraint: one can acquire even radically misleading evidence about whether the conditions obtain, but one cannot have evidence that is too misleading regarding whether they obtain. A first challenge for such views is to show that the application conditions of correct epistemic norms are evincible in the first place. But even setting this aside, a serious problem remains: where evincibility pushes one to think that the application conditions of correct epistemic norms make reference to a special class of internal states, Acquisition creates pressure in the opposite direction. Perhaps not surprisingly, several proponents of higher-order defeat and/or of level-connecting principles have explicitly acknowledged that the application conditions of norms are not evincible—and that their failure to be evincible creates seeming counterexamples to level-connecting principles, such as ones prohibiting a kind of epistemic akrasia.³⁵ Let me now investigate whether normative uncertainty creates a more fertile ground for higher-order defeat.

.. Normative uncertainty and evincibility Acquisition states that it is possible to acquire evidence making it rational to believe, or be confident, in falsehoods regarding the rational status of one’s doxastic states. Here is what acquiring such evidence would involve in the current context. Assume that I am in a case c, and that the correct theory permits d in c. Acquiring new HOE could then land me in a case c0 in which it is rational for me to believe (or be confident) in the falsehood I am forbidden to believe p in c. One way this could happen is if I acquire evidence making a false (complete) theory TFALSE likely, where TFALSE forbids believing p in c. Another, more realistic option is acquiring evidence making likely a possibly large disjunction of false theories, all of which forbid believing p in c. This might happen by acquiring evidence for false norms. I might, for instance, acquire evidence that d is only permitted when some condition C obtains, a condition I know not to obtain in c. The evincibility of rationality, however, is incompatible with having evidence making likely such false normative claims about my current situation: it cannot be likely on my evidence that being in some state d is forbidden in my current situation, if it is in fact permitted. To see this, consider the following kind of case. Assume that I am in fact required to disbelieve p (that’s what the correct theory/norms urge me to do), and I do so. However, I have misleading evidence regarding what the correct norms are. It is very likely on my evidence that I am required to believe p if C, for some condition C. Assume that now C obtains, and that I know (and it is likely on my

³⁵ E.g., Christensen (b), Titelbaum (), Horowitz (), Worsnip ().

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evidence) that C obtains. Then, the rationality of my state of disbelieving p is not evincible: it is by assumption rational, but it is likely on my evidence that I am rationally required to believe p instead and hence, that disbelieving p is rationally forbidden. More generally, given that we are bracketing possible uncertainty about contingent features of one’s epistemic situation, the evincibility of rationality will be threatened in any case c in which one is permissibly in some doxastic state d, but has evidence for a false theory or norm that forbids being in d in c. In the current context, higher-order defeat commits one to the view that while I can be even radically misled regarding a range of normative truths that are about situations other than the one I am currently in, I cannot be thus misled about my current situation. Here is a quick argument that such a view is a non-starter: If the rational status of my (current) beliefs is always evincible, then a range of normative truths must be sufficiently likely at all times and hence, evincible. But then, I cannot acquire misleading evidence regarding such truths, evidence that could trigger defeat by higher-order evidence. If the evincibility of rationality required the evincibility of the complete normative theory TTRUE, or the evincibility of all of the correct norms, then this reasoning would be correct. For then, that complete theory, or the total set of true norms, would have to be sufficiently likely in any possible case, and hence, evincible. However, someone might object that the evincibility of rationality does not require the evincibility of complete normative theories, or of all correct norms. To see this, assume that I am now in case c. Because rationality is evincible, at least theories or norms forbidding some of my perfectly rational doxastic states cannot be too likely on my evidence. A range of truths stating facts about what doxastic states the correct theories or norms require or permit in c must be sufficiently likely. It doesn’t follow from this, however, that the unique complete true theory TTRUE, or the total set of correct norms, must be sufficiently evincible. After all, a lot of the recommendations made by the complete theory apply to cases other than the one I am in. As I transition to another case c0 , I could acquire misleading evidence regarding normative facts pertaining to c, as long as I don’t have the kind of misleading evidence regarding what is now my actual case, c0 , that would threaten the evincibility of rationality. In principle both Acquisition and the evincibility of rationality could be retained, as long as my evidence bears on normative matters in the right kinds of ways as I transition from one epistemic situation to another. Nevertheless, such a view raises some serious explanatory challenges. If I always have a kind of access to facts about what the correct normative theory recommends in my current situation, why don’t I have access to facts about what it recommends in other situations? The purported asymmetry in one’s access to a range of normative truths regarding the situation one currently occupies, versus other situations, needs to be explained. Mike Titelbaum () has recently argued that this challenge cannot be met. We need some epistemic story of how it is that it can never be justified for one to both hold a doxastic state d, and to (falsely) believe that d is rationally forbidden. But, Titelbaum argues, any plausible story will generalize to a story banning such normative mistakes regarding other situations. For instance, an obvious explanation for why one could never have evidence making it likely that a

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doxastic state d is forbidden, when d is in fact permitted, is that ‘every agent possesses a priori, propositional justification for true beliefs about the requirements of rationality in her current situation’. But if one can gain true beliefs about what the correct theory requires or forbids in one’s current situation just by reflecting on features of what situation, then surely one can gain such true beliefs regarding other situations. Conversely, one could argue that a view explaining how it is that we don’t have access to what the correct theory recommends in situations other than our own will generalize to an explanation of why we don’t always have such access to our own situation. Here is another way to bring out the peculiar nature of the view under discussion. Assume that you are travelling with a testifier who makes claims about the local customs and habits at various places and times. The testifier occasionally lies. However, whenever she is lying about the place you are both currently in, she always gives out some telltale sign that she is lying. As a result, you are never misled about the customs and habits in your current location by such a testifier. Analogously, consider an epistemic theory that makes false recommendations in some situations. However, assume that in all of the situations in which the theory makes false recommendations, there are reasons to reject the theory anyway—the theory is unlikely on the overall evidence. Or, at the very least, in none of these situations is the theory likely. Call such a theory telltale false. There is no worry of being misled by such a theory into holding false beliefs about the rational status of one’s doxastic states, as the theory can never be likely in situations in which it strays from the right recommendations. In the current context, a commitment to higher-order defeat (Acquisition + Evincibility of rationality) incurs commitment to the following claim: all possible epistemic theories are false in a telltale way—at least when it comes to their falsely claiming of some permitted doxastic state that it is forbidden. That is, some possible epistemic theories falsely claim of some situations that doxastic states that are in fact permissibly held in those situations are forbidden, but in those situations those theories are never likely to be true. But why are only theories that are false in a telltale way even possible in the first place? Assume that it is likely on my evidence that certain doxastic states are forbidden in cases s₁, . . . , sn, even though those states are in fact permitted in those situations (I have acquired misleading HOE regarding those situations). There must be some guarantee that if I then transition to one of these epistemic situations, those normative claims cannot be likely. That is, it must either be impossible for me to come to be in one of these situations, or the relevant normative claims must automatically become unlikely once I come to be in them. But why might this be? Let me describe an example that brings out these challenges. Let TE be an evidentialist theory stating that one is rationally permitted to believe a proposition p whenever p is at least . likely on one’s evidence. TE + S is a theory that allows pragmatic stakes to affect just how likely a proposition must be on one’s evidence in order for one to be permitted to believe it. Assume that according to TE + S, when the stakes are high, p must be at least . likely in order for a belief in p to be rationally permitted; if p is only . likely, the rational attitude is suspension of judgement. (These are simplified toy theories, of course.) Assume for the sake of

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argument that TE + S is false, and that TE is true: stakes can have no such effect on what opinions are rational. Here is my situation at . p.m.: the stakes are low, and I rationally believe that there is a  p.m. train that will take me to the hospital to see my aunt (proposition train), on the grounds that it is a weekday, and I took the  p.m. train on a weekday a couple of weeks ago. Train is in fact at least . likely on my total evidence and, hence, my belief is rational. However, I have misleading philosophical evidence, and at . p.m. I am also rationally confident (perhaps I even rationally believe) that TE + S is true—that is, that stakes can affect what it is rational for me to believe. Nevertheless, at . p.m. I satisfy Evincibility of rationality, at least as far as my belief in train goes. As I am walking to the station, at . p.m. I receive a phone call: I am told that my aunt is in a critical condition. The stakes have now gone up, and I know it: according to TE + S, it is no longer rationally permitted for me to believe train. Assume that I nevertheless continue to believe it. Given that train is still at least . likely and the fact that TE is the true theory, my belief continues to be rational. However, the only way for me to continue to satisfy Evincibility of rationality is if I suddenly lose my confidence in TE + S. But there is nothing that would make this doxastic shift rational: I don’t acquire any new evidence bearing on such normative matters. Possible cases like this spell trouble for the evincibility of rationality. In the case described, I start out confident in a theory that makes a false recommendation in situations other than my own. But I then come to be in one of these situations: what starts out as normative uncertainty regarding other epistemic situations or cases becomes uncertainty about my current situation. TE + S is not false in a telltale way. But according to the present view, such theories are impossible. As far as I can see, there are two general, non ad hoc ways of block cases like the one I described from arising, and to meet the explanatory challenges raised above. The first is commitment to a picture of epistemic normativity entailing that if the stakessensitive theory TE + S is likely on my evidence, its recommendations are thereby correct. Hence, the stipulation that the purely evidentialist theory TE could still make a true recommendation in a situation in which TE + S is likely to be true is simply false. This is a deeply troubling picture of epistemic normativity. The second option posits epistemic dilemmas in all of the problem situations in which evincibility is threatened. In the above example, though TE could still be true, if a false theory TE + S making a different recommendation is likely, there is nothing I could do that is rational. Before I discuss these options, let me address the worry that I have taken uncertainty about necessary normative truths too seriously. Surely pretty much any view put forth for dealing with defeat by higher-order evidence has its restrictions, and is not tailored to accommodate just any evidence regarding normative matters. Consider the history of conciliatory views of peer disagreement. It was soon noticed that conciliatory views have untoward consequences if applied to disagreements concerning conciliationist views.³⁶ In response, Elga argues that the application of

³⁶ See Weatherson (), Elga ().

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conciliationism should be restricted in some motivated way. But rather than establishing the joint tenability of Acquisition and Higher-Order Defeat (and hence, evincibility), Elga’s restriction makes it impossible to acquire misleading evidence about normative matters. The kinds of complete theories I have discussed are what Lewis () calls inductive methods: systematic ways ‘of letting the available evidence govern your degree of belief in hypotheses’—or even more generally, systematic ways of letting relevant features of one’s epistemic situation govern one’s doxastic states. Following Field (), Elga argues that all acceptable inductive methods must be coherent, which requires that they never recommend other methods over themselves. Any candidate for a true theory T must always recommend being confident in T. Indeed, Field (: ) concludes on this basis that at least we must treat our basic inductive rules or methods as empirically indefeasible and hence, fully a priori. That is, we cannot rationally regard anything as evidence against them. While such a view is amenable to Evincibility of rationality, it is not amenable to the possibility of acquiring misleading higher-order evidence regarding what rationality consists in. Let me now discuss two responses to the challenges raised.

.. Reply 1: Epistemic Anarchy If I always have a kind of access to a range of facts about what the correct normative theory recommends in my current situation, why don’t I have access to facts about what it recommends in other situations? Relatedly, if a false theory TFALSE forbidding some doxastic states that are in fact permitted in some situations could be likely on my evidence, what is to prevent it from being likely in one of the situations in which it makes a false recommendation—in particular, in which it forbids being in doxastic states that are in fact permitted? That is, why are only theories that are false in a telltale way even possible? There is a story answering these question, albeit one that paints a deeply troubling picture of epistemic normativity. Consider a view on which the normative truths themselves are malleable, being shaped by one’s evidence or rational opinions regarding what the true theory or norms are. The idea is that if a theory is likely on my evidence, then its recommendations are thereby correct. Such a view explains why all possible false theories are false in a telltale way: as a theory becomes unlikely, its recommendations thereby cease to be correct. The view would also answer the explanatory challenge raised by Titelbaum: there is a constitutive kind of dependence between which normative claims are correct and which claims are likely on my evidence at a given time. The fact that such access is constitutive of their truth is what explains why I always have a kind of access to a range of normative truths that bear on what it is rational for me to believe in my current situation. Of course, given that I could in principle have evidence for just about any theory— at least any of the theories ever put forth by epistemologists—this leads to a view on which just about anything can be made to go, as long as one has evidence for a theory on which it is correct. Littlejohn () rightly calls the view a kind of epistemic anarchy. I won’t spend more time arguing against the view; I only note that it is an option, albeit one that very few would want to endorse.

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The other way of meeting the challenges presented above posits epistemic dilemmas in any case in which the evincibility of rationality (and the relevant levelconnecting principle) is threatened.

.. Reply 2: Dilemmas and evincibility Here is a procedure for turning any initial theory with the kinds of problems pointed to into one that both respects Evincibility of rationality and accommodates normative uncertainty. Consider all possible cases, and the recommendations made by a candidate true theory such as TE in those cases regarding a particular proposition p. We can split these into two classes. . First, there are cases in which TE permits or requires some doxastic attitude to p, and it is sufficiently likely on one’s evidence that this recommendation is correct (TE itself, some other theory making the same recommendation, or a disjunction of theories all making this recommendation, is sufficiently likely). . Second, there may be cases in which TE permits some doxastic attitude to p, but it is not sufficiently likely on one’s evidence that this recommendation is correct. The first kinds of cases are unproblematic from the perspective of Evincibility of rationality. Here is a strategy for dealing with cases of the second kind, which threaten evincibility: simply rule that in them no state is rational, for they constitute kinds of epistemic dilemmas. Why dilemmas? Because rationality is governed by two ideals that sometimes conflict. The first kinds of ideals are incorporated into our initial theory TE, which I have for the sake of argument assumed to be an evidentialist theory. The second kinds of ideals have to do with taking into account higher-order evidence; one might refer to them, as Christensen does, as ideals of epistemic modesty. It is this second kind of ideal which prohibits believing p when believing p is not likely to be rational. Situations in which the two ideals come into conflict are epistemic dilemmas: nothing the subject could do in them is rational.³⁷ Unlike the view just discussed on which anything goes (as long as it is likely to be right), the present view has no such implications. It does, however, have a similar negative implication: anything could be made not to go by evidence for a false theory. Are the consequences of this view plausible? Is it plausible, for instance, that just because a philosopher is unlucky enough to be inundated by evidence for an epistemological theory making a false recommendation in her current situation, she is doomed to irrationality, no matter what she does? Or, consider the analogous view in the moral case. Assume that the correct moral theory tells me to financially help my friend so she can be treated for cancer. I am, however, in the grips of consequentialist views that require me to help strangers in ways that are more effective, saving numerous lives with the same money. Assuming this recommendation is false, do we really want to say that by helping my friend I fail to act as

³⁷ Christensen (a, a) defends such dilemmas.

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I ought, since there is nothing I could do that would constitute acting in the morally right way? Of course, friends of higher-order defeat don’t merely make the negative claim that evidence that a belief is irrational defeats its rationality; they typically make positive recommendations about how one ought to take such evidence into account. For instance, Rezi should downgrade her confidence, or suspend judgement, in light of her HOE. This is even true according to Christensen, who advocates a version of the dilemma view. But a structure with conflicting ideals or norms cannot yield such overall recommendations: something more is needed, either a meta-rule, or some way of weighting the conflicting rule. My purpose here is not to argue against the dilemma view (see Lasonen-Aarnio ), but to outline the options. Note that appeal to such dilemmas is not a way of giving up commitment to evincibility: it is still true that any situation in which it is rational to believe p is one in which it is sufficiently likely on one’s evidence that it is rational to believe p. For any theory, the recommendations made by that theory can be correct only if it is sufficiently likely on one’s evidence that they are correct. So reasons to think that rationality is not evincible are reasons to think the dilemma view cannot be correct.

. Conclusions I applaud taking seriously the idea that there can be deep uncertainty about the normative status of our own beliefs, intentions, and actions. We are fallible, even regarding how rational our beliefs are, and how morally right our actions are. Not all cases of normative success are ones in which we have access to the fact that we succeeded, and not all cases of normative failure are ones in which we have access to the fact that we fail. But numerous authors, in both epistemology and ethics, have moved from this realization to an endorsement of a systematic kind of higher-order defeat: if I am sufficiently rationally confident that I fail by some normative standard, then I do in fact fail by that standard. Norms urging us to downgrade our opinions in light of evidence about our failure, or to split the difference with our epistemic peers, are often presented as requirements of a proper kind of epistemic modesty. I have argued that a commitment to systematic defeat by higher-order evidence amounts to a commitment to a kind of epistemic access that I have labelled evincibility. There is a sense in which such evincibility is not at all modest, as it denies that we could be deeply misled about how well we are doing when it comes to normative standards like rationality. Some authors like Smithies () and Titelbaum () happily endorse such access internalism at the outset. However, such views don’t sit well with those who admit the possibility of deeply misleading higher-order evidence about normative matters. I have argued that there is at least a deep prima facie tension between two theses that those who endorse a systematic phenomenon of defeat by higher-order evidence are committed to, Acquisition and Higher-Order Defeat. If this is right, then we are left with two stable views: we can either endorse Evincibility of rationality, or endorse Acquisition, and give up on the hope of accommodating a systematic kind of defeat, or of formulating a levelconnection principle not susceptible to counterexamples.

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To be sure, none of the arguments given above are conclusive. But at the very least, I hope to have identified a key challenge for those committed to defeat by higherorder evidence: explaining the kind of asymmetry such views are committed to when it comes to one’s epistemic access to a range of normative facts regarding one’s own epistemic situation, versus a lack of such access to a range of normative facts regarding situations other than one’s own.

Acknowledgements I am very grateful for written comments on an earlier draft of this chapter from the editors, a referee, Anna-Maria A. Eder, Mike Titelbaum, and Teru Thomas. Many thanks to discussions with Ville Aarnio, Jaakko Hirvelä, Mattias Skipper, and the audience at the Bochum Epistemology Seminar in June .

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Hawthorne, J. and A. Srinivasan (). “Disagreement Without Transparency—Some Bleak Thoughts.” In: D. Christensen and J. Lackey (eds), The Epistemology of Disagreement: New Essays, Oxford University Press, pp. –. Hazlett, A. (). “Higher-Order Epistemic Attitudes and Intellectual Humility.” In: Episteme (), pp. –. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs (), pp. –. Huemer, M. (). “The Puzzle of Metacoherence.” In: Philosophy and Phenomenological Research (), pp. –. Kelly, T. (). “Peer Disagreement and Higher-Order Evidence.” In: R. Feldman and T. Warfield, Disagreement, Clarendon Press. Lackey, J. (). “What Should We Do when We Disagree.” In: T. Gendler and J. Hawthorne (eds), Oxford Studies in Epistemology , Oxford University Press, pp. –. Lasonen-Aarnio, M. (). “Unreasonable Knowledge.” In: Philosophical Perspectives (), pp. –. Lasonen-Aarnio, M. (). “Disagreement and Evidential Attenuation.” In: Noûs (), pp. –. Lasonen-Aarnio, M. (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research (), pp. –. Lasonen-Aarnio, M. (). “New Rational Reflection and Internalism about Rationality.” In: Oxford Studies in Epistemology , pp. –. Lasonen-Aarnio, M. (forthcoming A). “Enkrasia or Evidentialism?” In: Philosophical Studies. Lasonen-Aarnio, M. (forthcoming B). “Dispositional Evaluations and Defeat.” In: J. Brown and M. Simion (eds.), Reasons, Justification and Defeat, Oxford University Press. Lewis, D. (). “Immodest Inductive Methods.” In: Philosophy of Science (), pp. –. Littlejohn, C. (). “Stop Making Sense? On a Puzzle about Rationality.” In: Philosophy and Phenomenological Research (), pp. –. Lockhart, T. (). Moral Uncertainty and its Consequences, Oxford University Press. MacAskill, W. (). Normative Uncertainty, D.Phil. Thesis, University of Oxford. Raz, J. (). Practical Reason and Norms, Oxford University Press. Ross, J. (). “Rejecting Ethical Deflationism.” In: Ethics , pp. –. Salow, B. (). “The Externalist’s Guide to Fishing Compliments.” In: Mind (), pp. – Schechter, J. (). “Rational Self-Doubt and the Failure of Closure.” In: Philosophical Studies (), pp. –. Sepielli, A. (). “What to Do When You Don’t Know What to Do.” In: Oxford Studies in Metaethics , pp. –. Skipper, M. (forthcoming). “Reconciling Enkrasia and Higher-Order Defeat.” In: Erkenntnis. Sliwa, P. and S. Horowitz (). “Respecting All the Evidence.” In: Philosophical Studies (), pp. –. Smithies, D. (). “Moore’s Paradox and the Accessibility of Justification.” In: Philosophy and Phenomenological Research (), pp. –. Titelbaum, M. (). “Rationality’s Fixed Point (Or: In Defense of Right Reason).” In: Oxford Studies in Epistemology , pp. –. Van Wietmarschen, H. (). “Peer Disagreement, Evidence, and Well-Groundedness.” In: The Philosophical Review (), pp. –. Weatherson, B. (). “Disagreements, Philosophical and Otherwise.” In: D. Christensen and J. Lackey (eds), The Epistemology of Disagreement: New Essays, Oxford University Press, pp. –. Weatherson, B. (). “Stalnaker on Sleeping Beauty.” In: Philosophical Studies , pp. –.

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Weatherson, B. (). “Running Risks Morally.” In: Philosophical Studies (), pp. –. Williamson, T. (). Knowledge and its Limits, Oxford University Press. Williamson, T. (). “Improbable Knowing.” In: T. Dougherty (ed.), Evidentialism and its Discontents, Oxford University Press. Williamson, T. (). “Very Improbable Knowing.” In: Erkenntnis , pp. –. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research (), pp. –.

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8 The Puzzles of Easy Knowledge and of Higher-Order Evidence A Unified Solution Ram Neta

The goal of this chapter is to provide a unified solution to two widely discussed epistemological puzzles. I will begin by setting out each of these two puzzles. I will then briefly survey some of the proposed solutions to each puzzle, none of which generalizes to the other. Finally, I will argue that the two puzzles arise because of a widespread confusion concerning the relation of substantive and structural constraints of rationality: or, in the epistemic domain, the relation of evidence and coherence. Clearing up this confusion allows us to clear up both puzzles at once. Before setting out the puzzles, I begin with some remarks about terminology. The puzzles that I will discuss here have been presented in many different ways, and under a variety of lexical guises. I present them both as puzzles about what it is rational for an agent to believe—whether or not the agent believes it. We can call this normative status “ex ante rationality,” to contrast with “ex post rationality,” the normative status that attaches to beliefs that the agent rationally holds. In speaking, as I did just now, of what it is rational for an agent to believe, I do not mean to be talking about what an agent is rationally committed to believing, given her current beliefs: an agent may be rationally committed to believing something that she is not ex ante rational to believe, or vice versa. For instance, suppose my evidence conclusively indicates that the Earth is smaller than the Sun, but I happen to believe both that the Sun is smaller than the Moon, and the Moon is smaller than the Earth. In that case, my evidence makes it ex ante rational for me to believe that the Earth is smaller than the Sun, but my actual beliefs make me rationally committed to believing that the Sun is smaller than the Earth. What it is ex ante rational for me to believe is a function of (at least) my evidence, but what I am rationally committed to believing is a function of what I currently believe, whether or not those beliefs are rational. To avoid confusing the normative status of being ex ante rational in believing something with what one is rationally committed to believing, some philosophers will speak of ex ante and ex post “justification” instead of “rationality.” But I am hoping that the present terminological remark is sufficient to enable me to get by without having to add “justification” to my lexicon. Ram Neta, The Puzzles of Easy Knowledge and of Higher-Order Evidence: A Unified Solution In: HigherOrder Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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Throughout this chapter, when I speak of what it is rational for an agent to believe, or what an agent is rational to believe, I mean to be talking about what it is ex ante rational for the agent to believe, and not what an agent is rationally committed to believing.

. The two puzzles Each of our puzzles can be concisely stated by considering a hypothetical case. Let’s begin with a case commonly used to frame the puzzle of easy knowledge. Before setting out the case, I should first mention that, although the puzzle is most commonly known as the puzzle of “easy knowledge,” that is a misnomer: fundamentally, the puzzle has to do not with knowledge specifically, but rather with ex ante rationality, or what an agent is rational in believing. This is because the same puzzle can arise no matter whether what an agent is rational in believing is true or false, and no matter whether she believes it or not. Now let’s consider this case: Red Wall. Abdullah clearly sees a red wall in normal lighting, and he has no reason to suspect that anything is unusual about his vision in this situation. Indeed, let’s fix the facts of the case so that Abdullah’s current visual experience makes it rational for him to believe that the wall he sees is red, but the remainder of Abdullah’s total evidence is evidentially irrelevant to the color of the wall. With respect to this case, each of the following three propositions is individually plausible: a. If it is rational for Abdullah to believe that the wall he sees is red, then it is also rational for him to believe that it’s not the case that the wall he sees is white but illuminated with red lights. b. It is not a priori rational for Abdullah to believe that it’s not the case the wall he sees is white but illuminated with red lights. c. Abdullah’s current visual experience does not make it rational for him to believe that it’s not the case that the wall he sees is not white but illuminated with red lights. But the plausibility of these three principles generates a puzzle: If (a) is true, then, given the stipulations of the case (viz., that Abdullah’s current visual experience makes it rational for him to believe that the wall he sees is red), it is rational for Abdullah to believe that the wall he sees is not white but illuminated with red lights. If (b) is true, it is not a priori rational for him to hold this belief. And if (c) is true then, given the stipulations of the case (viz., that the remainder of his total evidence is irrelevant to the color of the wall) it is not a posteriori rational for him to hold this belief. Thus: it must be rational for Abdullah to believe that the wall is not white but illuminated by red lights, but it cannot be either a priori or a posteriori rational for him to believe this. Since the distinction between a priori or a posteriori is logically exhaustive, it follows both that it must be rational for Abdullah to believe, but it cannot be rational for Abdullah to believe: contradiction. This case exemplifies the

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puzzle of easy knowledge. Exactly how this puzzle generalizes beyond this case will be considered in section . below. Some philosophers will try to solve this puzzle by denying (b): they claim that it is a priori rational for us to believe that our evidence is not misleading with respect to what it makes it rational for us to believe (Wright , White ). And still other philosophers deny (c), and claim that a particular perceptual experience can make it rational for us to deny a hypothesis that predicts that very experience (Pryor , Bergmann ). Finally, although I don’t know of any philosopher who denies (a), I suppose that such a denial is yet another way to try to solve this puzzle.¹ The costs of each of these solutions is already well documented. Now, let’s consider a case commonly used to frame the puzzle of higher-order evidence. Hypoxia. Yujia is flying a small plane to an airstrip just a few miles away, and wonders whether the plane has enough fuel to make it to the next airstrip, which is another  miles away. She checks all the gauges, and, as she sees, everything conclusively indicates that she has more than enough fuel. But she also notices that her altitude is , feet, and that conclusively indicates that she is probably suffering from hypoxia (a condition that, let’s stipulate, generates introspectively undetectable cognitive impairments). And if she were suffering from hypoxia, Yujia knows that her reasoning from the gauge readings would then be completely untrustworthy, and it would be irrational for her to believe the conclusion of such reasoning. Now, in such a case, each of the following three propositions is individually plausible: (a) Yujia should believe whatever propositions her evidence conclusively indicates to be true. (b) Yujia should not simultaneously believe that she has enough fuel, and also believe that it is irrational for her to believe that she has enough fuel. (c) It cannot be the case that, at one and the same time, Yujia should believe that she has enough fuel and also should refrain from believing that she has enough fuel. Again, the plausibility of these three principles generates a puzzle: if (d) is true, then, given the stipulations of the case (viz., that Yujia’s evidence conclusively indicates that she has enough fuel, and also conclusively indicates that she is probably hypoxic, and so it is irrational of her to believe that she has enough fuel) Yujia should believe that she has enough fuel, and Yujia should also believe that it is irrational of her to believe that she has enough fuel. If (e) is true, then she should not simultaneously believe that she has enough fuel while also believing that it is irrational of her to believe that she has enough fuel. But if (f) is true, then it can’t both be the case that she should believe that she has enough fuel and should also refrain from believing that she has enough fuel. If she should believe that she has enough fuel, then it’s not ¹ Although some philosophers have argued against closure principles for knowledge and for ex post rationality, it is uncommon for a philosopher to deny closure for ex ante rationality under known entailment.

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the case that she should refrain from believing that she has enough fuel. But then which is it? By the argument above, it cannot be either. This case exemplifies the puzzle of higher-order evidence. Exactly how this puzzle generalizes beyond this case will be considered in section . below. Some philosophers would solve this puzzle by denying (d): they claim that even if an agent’s evidence conclusively indicates some hypothesis to be true, that by itself does not always make it the case that the agent should believe that hypothesis (Worsnip ). Still other philosophers deny (e): they claim that it is sometimes rational both to believe a proposition while also believing that it is irrational to believe that very same proposition (Williamson , Lasonen-Aarnio ). And still others deny (f) and claim that such cases are epistemic dilemmas (Christensen ). But, as I’ll argue in section ., all of these denials face serious costs. Each of the puzzles that I’ve just illustrated is generated by combining the epistemic demands imposed by an agent’s evidence, on the one hand, with the structural features of epistemic demands in general, on the other. In the final sections of this chapter, I will propose a unified solution to both puzzles, by arguing that, just as the agent’s perceptible surroundings impose one kind of constraint on the contents of an agent’s evidence set at a given time, so too do the structural features of epistemic demands provide another kind of constraint on the contents of the agent’s evidence set. To telegraph my thesis here: the facts perceptible to an agent provide the matter of her evidence, while the structural features of epistemic demands provide the form of her evidence. And because of the way in which perceptible fact and structural feature conspire to determine the content of an agent’s evidence set at any given time, it will turn out that the stipulations of the cases given above—stipulations on which the demands imposed by her evidence conflict with each other, given the constraints imposed by the structural features of epistemic demands—are not jointly satisfiable. Thus, (a), (b), (c), (d), (e), and (f) are all true, and they are not in conflict with each other. They appear to conflict only if we assume that the stipulations of our cases are jointly satisfiable, and that is precisely the assumption I will criticize. Before I explain and defend this thesis, let me begin by presenting each puzzle in more general form, and then briefly reviewing alternative solutions to the puzzles.

. The puzzle of easy knowledge, and the positive constraints of coherence In presenting the puzzle of easy knowledge, I relied upon a case of perception of a particular object. But the puzzle has nothing specifically to do with perception: it will arise in any case in which an agent has some evidence E that makes it rational for her to believe some hypothesis H, but does not make it rational for her to believe H0 , which she knows to follow from H. Here are some non-perceptual examples with this structure.² ² See Neta () for the present treatment of these examples, along with an argument that the treatments proposed by Weisberg (), Titelbaum (), and Tucker () cannot provide a unified solution to the various problems of easy knowledge.

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Asking Directions. Fergus is walking around St Andrews looking for Market Street, and so he asks some passerby (who is utterly unknown to him) where Market Street is. She tells Fergus it is three blocks north, and that’s the end of his conversation with her. There is, let’s stipulate, nothing unusual about the situation, or about Fergus’s interlocutor. It seems the stranger’s testimony makes it rational for Fergus to believe that Market Street is three blocks north, but that same testimony doesn’t make it rational for him to believe that the stranger is not misrepresenting the location of Market Street. Leaky Memory Case. Deepa has heard that some people have unreliable memories, but she has no reason to suspect that she is one of these people. In fact, she clearly recalls learning (though she can’t now remember just how it is that she learned this) that her own memory is highly reliable, i.e., usually, when it seems to her as if she remembers that p, then it is true that p. Deepa’s apparent recall of learning about the reliability of her memory makes it rational for her to believe that she learned about the reliability of her memory, but it doesn’t make it rational for her to believe that this particular episode of apparent recall is not confabulation. Bootstrapping Case. Fara knows that the New York Times is a reliable newspaper, and so she believes what she reads in it. On Monday, she picks up the New York Times and sees that it asserts p. On this basis, she comes to believe that p is true, and she also comes to believe that the New York Times asserts p. On Tuesday, she picks up the New York Times again and sees that it asserts q. On this basis, she comes to believe that q is true, and she also comes to believe that the New York Times asserts q. On Wednesday, she picks up the New York Times again and sees that it asserts r. On this basis, she comes to believe that r is true, and she also comes to believe that the New York Times asserts r. She continues to do this every day for an arbitrarily large number of days, and this makes it rational for her to believe that the New York Times has asserted an arbitrarily long series of propositions p, q, r, . . . , and also makes it rational for her to believe that same arbitrarily long series of propositions p, q, r, . . . , but this does not make it rational for her to believe what she knows to be entailed by all of this, viz., that the New York Times did not falsely assert any of those same propositions. Each example has the same structure: an agent A is rational to believe one proposition X (e.g., that the wall is red, that Market Street is three blocks north, that her memory is reliable, that her vision is on the basis of some reason R). Since A is rational to believe X, then she must also be rational to believe any proposition that she knows to be entailed by X. But there is some proposition Y such that (i) A knows it to be entailed by X, and yet (ii) R cannot make it rational for A to believe Y. In such a case, whenever R makes it rational for A to believe X, there must be some other factor that makes it rational for A to believe Y. The puzzle arises in thinking about what that other factor could be. What could make it rational for Abdullah (in Red Wall) to believe that the light he sees is not white but illuminated by red lights, or

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make it rational for Fergus (in Asking Directions) to believe that the stranger told the truth? Of course, we might think that Abdullah and Fergus have other pieces of evidence to make it rational for them to hold these other beliefs—Abdullah might know that red illumination is very unusual, or Fergus might know that strangers seldom lie. But then how do they acquire this background knowledge? Here, we face a dilemma. Either they acquire this background knowledge a priori, or they acquire it a posteriori. But how could they acquire knowledge of such contingent and specific features of their environment a priori? What kind of reasoning or reflection could furnish them with knowledge that, say, red illumination is very unusual? And if they acquire such background knowledge a posteriori, then, once again, they must acquire it on the basis of other perceptual experiences, or other testimony. But if such perceptual experiences or testimony can make it rational for them to believe the content of that background knowledge, then what makes it rational for them to believe other, anti-skeptical propositions (e.g., it is not the case that there is a vast amount of well-hidden red illumination around here, it is not the case that there are lots of strangers who secretly coordinate their deception so as to avoid discovery) that they know to be entailed by the content of that background knowledge? The puzzle of easy knowledge is really the puzzle posed by this dilemma. Efforts to address the puzzle of easy knowledge have focused on embracing one or another horn of the dilemma, and trying to explain away its implausibility. Wright (), for instance, embraces the a priori horn, and argues that we have a priori reasons to accept certain generalizations to the effect that certain kinds of evidence provide a reliable guide to reality. Pryor () embraces the a posteriori horn, and argues that certain kinds of evidence, by their very nature, make it rational for us to deny skeptical hypotheses—even if those same skeptical hypotheses predict those very pieces of evidence. Both Wright’s view and Pryor’s have each been subject to criticisms that are by now well known, and proponents of each view are candid that they hold it because they see it as the only alternative to the other view. In sections . and . below, I will develop an alternative to both views: an alternative according to which we can be a priori rational to believe not that anti-skeptical hypotheses are false, but rather that our total body of empirical evidence makes it a posteriori rational for us to believe anti-skeptical hypotheses. For now, I want to note only this: the problem of easy knowledge arises because what it’s rational for us to believe is closed under obvious entailment. But the closure of ex ante rationality under obvious entailment is a positive requirement of coherence. That is to say, the requirement that we accept the obvious consequences of what we believe is a requirement our failure to comply with which generates a kind of incoherence. If I believe some proposition to be true, but then I fail to accept any of the obvious entailments of that belief, then I suffer from a certain kind of incoherence: I’m not accepting what my belief commits me to accepting. In order to achieve coherence, I must either accept the consequences of my belief, or else cease to hold that belief. I refer to the closure of ex ante rationality as a positive requirement of coherence because it is a requirement that takes current attitudes and generates rational commitment to adopt further attitudes. Such positive requirements of coherence contrast with negative requirements of coherence, which take current attitudes and generate rational commitment to avoid further attitudes.

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. The puzzle of higher-order evidence, and the negative constraints of coherence I used the Hypoxia case to present the puzzle of higher-order evidence: in that case, Yujia has evidence that makes it rational for her to believe that she is flying at an altitude of , feet, but that very same evidence also makes it rational for her to believe that she probably has hypoxia, and so it is irrational for her to believe the conclusions that she draws from the fuel gauge readings. The puzzle of higher-order evidence arises in any case in which there is some factor that would make it rational for an agent to adopt a particular credal attitude towards a proposition, but there is also something that would make it rational for that agent to regard the first factor as not making it rational for her to adopt that credal attitude. The first factor in the Hypoxia case is perceptual evidence from the fuel indicator, along with reasoning from that perceptual evidence to the conclusion about the fuel level. But perceptual evidence from the altimeter, along with other reasoning to the conclusion about hypoxia, provides reason to distrust that first factor. The same kind of puzzle, however, can arise in cases involving testimony, and also in cases involving known vagueness in one’s evidence. Splitting the Bill. Umut and Umit are finishing their dinner at a restaurant, and the server brings the check. They each look at the check and carefully calculate how much they owe, and then put the amount of cash that they think they owe into the middle of the table. But after both putting this cash down, they realize that they are short: thus, they each come to know that at least one of them must have made a mistake in figuring out what portion of the check they owe. Although each of them performed careful calculations that made it rational for them to think that they owed a particular amount, and they both know that their owing that particular amount, in conjunction with the fact that the total cash that they put in the middle of the table wasn’t enough, entails that the other one of them must have made an error in their calculations, those very same careful calculations do not make it rational for them to believe that the other one of them must have made an error in their calculations. Sleepy Detective. Sam is a police detective, working to identify a jewel thief. He knows he has good evidence—out of the many suspects, it will strongly support one of them. Late one night, after hours of cracking codes and scrutinizing photographs and letters, he finally comes to the conclusion that the thief was Lucy. Sam is quite confident that his evidence points to Lucy’s guilt, and he is quite confident that Lucy committed the crime. In fact, he has accommodated his evidence correctly, and his belief is rational. He calls his partner, Alex. “I’ve gone through all the evidence,” Sam says, “and it all points to one person! I’ve found the thief!” But Alex is unimpressed. She replies: “I can tell you’ve been up all night working on this. Nine times out of last ten, your late-night reasoning has been quite sloppy. You’re always very confident that you’ve found the culprit, but you’re almost always wrong about what the evidence supports. So your evidence probably doesn’t support Lucy in this case.” Though Sam hadn’t attended to his track record before, he rationally trusts Alex and believes that she is right—that

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he is usually wrong about what the evidence supports on occasions similar to this one. Dartboard. You have a large, blank dartboard. When you throw a dart at the board, it can only land at grid points, which are spaced one inch apart along the horizontal and vertical axes. (It can only land at grid points because the dartboard is magnetic, and it’s only magnetized at those points.) Although you are pretty good at picking out whether the dart has landed, you are rationally highly confident that your discrimination is not perfect: in particular, you are confident that when you judge where the dart has landed, you might mistake its position for one of the points an inch away (i.e., directly above, below, to the left, or to the right). You are also confident that, wherever the dart lands, you will know that it has not landed at any point farther away than one of those four. You throw a dart, and it lands on . You should then be certain—or at least nearly certain— that the dart landed on one of the following five points: , , , , or . But now consider the proposition RING that the dart landed on one of the four of those five points other than : you should be nearly % confident in that proposition, since it mentions four of the five points on which the dart could have landed. But this level of confidence in RING is rational only if the dart landed at ; if it landed anywhere else, then you should be able to rule out some of the four locations mentioned in RING as locations on which the dart could have landed, and so should be less confident in RING. But this last conditional is one you can deduce, and so come to know, simply given the stipulations of the case. Furthermore, given those same stipulations, you should be no more than % confident that the dart landed at . Thus, you should be highly confident in RING, while also having no more than % confidence that your evidence supports RING. Each example has the same structure: an agent A is rational to believe one proposition X (e.g., that she has enough fuel, that she owes a particular amount of cash for the dinner bill) on the basis of some factor F: F can be some evidence for X, or it can be some reasoning that leads to the conclusion X, or it can be the appearance of X’s truth. Furthermore, A knows that nothing other than F makes it rational for her to believe X. But A is also rational to believe (falsely) that F is somehow misleading or unreliable, and that therefore F cannot make it rational for her to believe X, and that therefore (given her knowledge that nothing else makes it rational for her to believe X if F doesn’t) that she is not rational to believe X. Thus, A is rational to believe both that X is true, and that she is not rational to believe that X is true. But for A to be rational in holding both beliefs is for A to be rational in holding beliefs that fail to cohere with each other: it is for A to be rational in being “epistemically akratic.” And this is puzzling: how can an agent be rational in holding both of two beliefs that fail to cohere with each other? The puzzle of higher-order evidence is the puzzle of either answering this question, or showing how at least one of its presuppositions is false. Some philosophers (Williamson , Lasonen-Aarnio ) accept the presuppositions of the question and therefore insist that epistemic akrasia can be rational, that is, we can be rational in holding both of two beliefs that fail to cohere with each other. But none of these philosophers provides a convincing diagnosis of the

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implausibility of this view: rather, they each attempt to demonstrate that the view must be true because it follows from the presuppositions we’ve outlined (or from other more contentious principles). Other philosophers (Titelbaum , Smithies , Worsnip ) have argued that epistemic akrasia can never be rational, and so have concluded that one or another of the presuppositions of our question must be false. But they too have failed to provide a convincing diagnosis of the plausibility of all of those presuppositions. In sections . and . below, I will sketch a solution to the puzzle of higher-order evidence according to which it is a condition of the possibility of an agent being rational to believe some proposition X on the basis of some factor F that the same agent at that time not also be rational to believe that F is misleading with respect to X.³ For now, I want to note only this: the puzzle of higher-order evidence arises because of the negative requirements of coherence, viz., requirements that we not simultaneously believe things that are in certain kinds of rational tension with each other.

. The sources of ex ante rationality When an agent is ex ante rational in holding some attitude, there is something or other that makes her so rational. Facts about who is ex ante rational in holding which attitudes are not explanatorily primitive: for each such fact, something or other must explain why it obtains. (I don’t assume that the explainers must all be of some particular kind.). Whatever factors explain why an agent is ex ante rational in holding an attitude, let’s refer to these factors as the “ex ante rationalizers” of that agent’s holding that attitude. Whatever else these ex ante rationalizers include, they will at least include all of the evidence that an agent possesses at that time. But what evidence does an agent possess at a given time? This question has received considerable discussion over the past two decades. Williamson argues that the evidence an agent possesses at any given time is all and only the facts that the agent knows at that time. Goldman argues that the evidence an agent possesses at any given time is all and only those propositions that the agent is non-inferentially rational to believe at that time. And I have argued elsewhere that the evidence an agent possesses at any given time is simply all and only the facts that the agent is in a position to know non-inferentially at that time. Perhaps none of these views is exactly correct, but what they all have in common is this: for F to be an element of an agent’s evidence set at a given time, that agent must bear some epistemic relation to F at that time, whether the relation is that of knowledge, or of being in a position to know noninferentially, or of being non-inferentially rational to believe. Of course, something can be evidence for a hypothesis even if it is evidence that does not belong to any agent: there is the evidence that no one has yet come across. But for something to be a particular agent’s evidence—for it to affect what it is rational for that particular agent to believe on its basis—it must stand in some epistemic relation or other to that ³ See Neta () for an earlier version of my solution to the puzzle of higher-order evidence. I failed to recognize there how that solution rests on the same basis as my solution to the puzzle of easy knowledge.

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agent. For present purposes, we needn’t worry about precisely what epistemic relation this must be. It should not be surprising that an agent must stand in some epistemic relation to F if F is to be part of her evidence. Part of what it is for F to be part of your evidence is that you are entitled to use F as a ground for believing something or other. But you are not entitled to use just any old proposition or fact as a ground for belief. There are lots of propositions that you have no business believing, and you are clearly not entitled to use any of them as grounds for believing anything else. And there are lots of facts of which you have no inkling whatsoever, and again you are clearly not entitled to use any of them as grounds for believing anything else. If you are entitled to use F as a ground for believing something else, you must stand in some epistemic relation to F. Precisely what epistemic relation that is may be a matter of dispute—as it is between Williamson and Goldman and me—but it cannot be a matter of serious dispute that you must stand in some such relation. So, for F to be part of your evidence, you must stand in some epistemic relation to F. But whatever epistemic relation that is, it is a relation the obtaining of which depends in part upon what other things you stand to in that same epistemic relation. For instance, if the epistemic relation is one of being rational to believe F, then it is an epistemic relation that you can stand in to F only if you are not also rational to believe something incompatible with F: you cannot be rational (or at least not fully rational) in believing each of two contradictory propositions. And the same sort of constraint holds whatever other epistemic relation you might stand in to F: standing in that relation to F precludes your simultaneously standing in that same relation to not-F. Whatever epistemic relation is relevant here, if an agent must stand in that epistemic relation to F in order for F to be evidence that the agent possesses, this imposes a structural constraint on the agent’s total evidence at any given time. Precisely what structural constraint it imposes may depend upon precisely what epistemic relation is relevant. But, without getting into the details, we can at least see that this structural constraint will involve certain kinds of coherence in the agent’s total evidence. For instance, an agent’s total evidence at a time cannot include both F and not-F. Thus, if every element of an agent’s total evidence is something that the agent is entitled to use as a ground for belief, there will need to be coherence constraints on an agent’s total evidence at a time, and those constraints are going to include at least the constraints of logical consistency. But, as I will now argue, the coherence constraints on an agent’s total evidence will have to include more than just logical consistency. That’s because, in addition to playing the role of grounds that the agent is entitled to use in forming her beliefs, the agent’s total evidence must also play the role of selecting, from among the infinitely many coherent credence functions (viz., functions from propositions in an algebra to degrees of confidence) that an agent can have at any given time, the credence functions that are rational for her to have at that time. For instance, consider the algebra of propositions built around the single propositional atom “Red lights are illuminating the wall in front of me”: this algebra includes “it is not the case that red lights are illuminating the wall in front of me,” “Red lights are illuminating the wall in front of me and it is not the case that red lights are illuminating the wall in front of me,” “Red lights are illuminating the wall in front

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of me or it is not the case that red lights are illuminating the wall in front of me.” (Note: I am using this simple algebra merely in order to make a point that applies to the more complicated algebras of propositions over which normal human beings normally distribute their credences. I am not assuming that it is possible for any epistemic agent to distribute their credence over an algebra of propositions as simple as this one: just as it is a condition of the possibility of having any beliefs that an agent has lots of beliefs, it may also be a condition of the possibility of investing any credence in any proposition that an agent distribute her credence over lots of propositions.) Because “Red lights are illuminating the wall in front of me and it is not the case that red lights are illuminating the wall in front of me” is a contradiction, it must be assigned credence , and because “Red lights are illuminating the wall in front of me or it is not the case that red lights are illuminating the wall in front of me” is a tautology, it must be assigned credence . But what credence is it rational for you at this moment to assign to the proposition “Red lights are illuminating the wall in front of me”? Let’s assume that you are in a situation in which the wall in front of you is the familiar north wall of your bedroom, and this wall has appeared white to you for the many years that this has been your bedroom, and now you bring a big red lightbulb into your room and turn it on so that it illuminates the wall, and the wall appears red. We may add that other people in the room with you at the time you turn on the red light make such remarks as “what a beautiful shade of red that light is!” “how interesting the wall looks when illuminated by that light,” and so on. We may also add that your situation contains no specific evidence against the hypothesis that red lights are illuminating the wall in front of you. In such a situation, it will be rational for you to assign a very high credence— something close to —to the proposition “Red lights are illuminating the wall in front of me.” And it will be irrational for you to assign a middling, let alone a low, credence to that proposition. None of this is to say that rationality requires that your credence in the proposition has a precise point value—I leave open that rationality permits your credence to have an interval value. It is also not to say that there is a single credence that it is rational for you to have in that proposition in such a situation— perhaps there are different credences that you could have, all of which are equally rational. But even if your rational credence is an interval, or even if there are multiple equally rational credences that you could have, all of those rational credences will be very high: the situation is one in which rationality demands of you that you invest a high credence in the proposition “Red lights are illuminating the wall in front of me.” But notice that none of this is determined by the constraints of credal coherence: coherence alone does not require assigning any particular credence to “Red lights are illuminating the wall in front of me,” and so the rational constraint that you have a high credence in “Red lights are illuminating the wall in front of me” is a constraint that does not derive from credal coherence. Your evidence is at least part of what imposes that further rational constraint on your credal function. The example I’ve chosen was representative. In general, then, your total evidence at a given time helps to constrain the range of coherent credence functions that it is rational for you to have at that time. For your total evidence to play this role, it must be not only logically consistent (and so not entail everything), but it must also systematically provide more normative support to some propositions than to others.

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To illustrate what kind of systematicity is demanded here, consider what would happen if your total evidence were normatively to support the proposition “Red lights are illuminating the wall in front of me” to a very high (nearly maximal) degree, but were also normatively to support the proposition “It is not the case that red lights are illuminating the wall in front of me” to a middling degree. What sort of constraint could such a body of total evidence place on which of the infinitely many coherent credence functions you may rationally have at that time? Can you rationally have credence . that you are human and credence . that you are not human? Any such credence function would be ruled out by giving too low a credence to “It is not the case that red lights are illuminating the wall in front of me,” given your evidence. Could you rationally have credence . that you are human and credence . that you are not human? Such a credence function would be ruled out by giving too low a credence to “Red lights are illuminating the wall in front of me” and also by giving too low a credence to “It is not the case that red lights are illuminating the wall in front of me,” given your evidence. In fact, there is no coherent credence function that would be permissible if your evidence were to normatively support “Red lights are illuminating the wall in front of me” to a nearly maximal degree and also normatively support “It is not the case that red lights are illuminating the wall in front of me” to a middling degree. But this is an absurd result: if your total evidence at a given moment is going to do the job of selecting which among the infinitely many coherent credence functions it is rational for you to have at that moment, then your total evidence at that moment cannot simultaneously provide nearly maximal normative support to “Red lights are illuminating the wall in front of me” while also providing middling normative support to “It is not the case that red lights are illuminating the wall in front of me.” Your total evidence must be so structured that if it provides a very high degree of normative support for one proposition, then it provides a complementarily low degree of normative support for its negation. And, for similar reasons, your total evidence must be so structured that the degrees of normative support that it provides to different elements of your algebra of propositions make it rational for you to distribute your credence over the elements of that algebra in a way that is coherent. Any totality of facts or propositions that don’t provide normative support in this systematic way cannot do the job that an agent’s total evidence is supposed to do, and so cannot be identical to any agent’s total evidence. To sum up: not just any totality of facts or propositions can be an agent’s total evidence at a given moment. The problem is not simply that, for each fact or proposition that is going to be part of the agent’s total evidence at a given moment, the agent must bear a specific kind of epistemic relation to it. The problem is also that there is a structural constraint on the body of facts or propositions that can fit together to collectively form an agent’s total evidence. Perhaps the structural constraint derives from the kind of epistemic relation that the agent must bear to each element in her total evidence, or perhaps it does not: we don’t need to settle that issue now. Our point here is simply that, if an agent’s body of total evidence is going to do the two kinds of normative work that it functions to perform—namely, the work of providing grounds for belief, and also of selecting rational credence functions—then that body of total evidence must fit together in a way that allows it to do that work.

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Any set of facts or propositions that can be an agent’s total evidence at a given time must itself be, in some way that we have not yet specified, coherent. This conclusion, though not spelled out with any precision, is nonetheless sufficient to point the way to a unified solution to both of our puzzles.⁴

. How the structural constraints on an agent’s total evidence can provide a unified solution to both of our puzzles Recall the structure of each of our puzzles. In the puzzle of easy knowledge, there is some reason R that makes it rational for an agent to believe X, and there is some further proposition Y that the agent knows to be entailed by X, but R cannot make it rational for the agent to believe Y. Since A is rational to believe X, she must also be rational to believe Y, but R can’t be what makes it rational for her to believe Y, so it must be some other factor that makes it rational for her to do so. The puzzle arises in thinking about what that other factor could be. What could make it rational for Abdullah (in Red Wall) to believe that the light he sees is not white but illuminated by red lights, or make it rational for Fergus (in Asking Directions) to believe that the stranger told the truth? In the puzzle of higher-order evidence, an agent A is rational to believe one proposition X (e.g., that she has enough fuel, that she owes a particular amount of cash for the dinner bill) on the basis of some factor F. But A is also rational to believe (falsely) that F is somehow misleading or unreliable, and therefore that F cannot make it rational for her to believe X, and therefore (given her knowledge that nothing else makes it rational for her to believe X if F doesn’t) that she is not rational to believe X. Thus, A is rational to believe both that X is true, and that she is not rational to believe that X is true. But how can an agent be rational in holding both of two beliefs that fail to cohere with each other? Each puzzle arises when the epistemic demands imposed by an agent’s evidence seem to conflict with each other, given the structural features of epistemic demands. But are such cases so much as possible? The puzzle arises if we assume that such cases are possible: but the argument of section . should lead us to call that assumption into question. Let me indicate how this would go for some of the hypothetical cases described above. Recall the stipulations of Red Wall: Abdullah clearly sees a red wall in normal lighting, and he has no reason to suspect that anything is unusual about his vision in this situation. Indeed, let’s fix the facts of the case so that Abdullah’s current visual experience makes it rational for him to believe that the wall he sees is red, but the remainder of Abdullah’s total evidence is evidentially irrelevant to whether the wall he sees now is red. But are these stipulations jointly satisfiable? Abdullah’s current visual experience can make it rational for him to believe that the wall he sees is red only under certain conditions: for instance, it cannot make it rational for him to believe that the wall he sees is red if the rest of Abdullah’s total evidence provides ⁴ See Neta () for an account of those structural constraints.

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 

strong grounds for him to believe that his current visual experience is not a good indication of the color of the wall. Thus, Abdullah’s current visual experience can bear a rationalizing relation to the proposition that the wall is red only if the rest of his total evidence satisfies certain constraints. Now, what if we allow that one such constraint is that the rest of his total evidence must support the hypothesis that his current visual experience indicates the actual color of the wall? On one reading, this is consistent with the stipulation that the rest of Abdullah’s total evidence (besides his current visual experience) is evidentially irrelevant to whether the wall is now red. But on another reading it is not consistent with that stipulation: the sense in which the rest of his total evidence is evidentially relevant to whether the wall is red is that his current visual experience can support the proposition that the wall is red only if the rest of his total evidence supports the proposition that his current visual experience accurately indicates the color of the wall. If this latter constraint is determined by a priori coherence constraints on agent’s body of total evidence, then it will be a priori that, when an agent is rational to believe some proposition P on the basis of some bit of evidence E she has, the rest of her total evidence supports the proposition that E is a reliable indicator of the truth of P. Now recall the stipulations of Hypoxia: Yujia is flying a small plane to an airstrip just a few miles away, and wonders whether the plane has enough fuel to make it to the next airstrip, which is another  miles away. She checks all the gauges, and, as she sees, everything conclusively indicates that she has more than enough fuel. But she also notices that her altitude is , feet, and that conclusively indicates that there is at least a significant chance that she is suffering from hypoxia (a condition that, let’s stipulate, generates introspectively undetectable cognitive impairments). And if she were suffering from hypoxia, her reasoning from the gauge readings would be completely untrustworthy. Are these stipulations jointly satisfiable? Of course, whether or not Yujia has hypoxia, she might nonetheless be able to see all the gauges, and the gauges can conclusively indicate that she has more than enough fuel, but the question is whether her total body of evidence can include the gauge readings that she sees. Of course, if Yujia is sufficiently cognitively impaired, then the mere fact that she sees something or other will not suffice to put her in the epistemic relation to that thing for it to be evidence that she has. But Yujia doesn’t know if she actually does suffer from hypoxia: she is rational to believe only that there is a significant chance that she suffers from it. But if there’s a significant chance that she suffers from hypoxia, then, even if she does see the fuel gauge point to F, this perceptual state may still not suffice to make it the case that she is entitled to use the fact or proposition that the fuel gauge points to F as a ground for believing such conclusions as “I have enough fuel.” When Yujia is rational to believe that she may very well be suffering from hypoxia, this can considerably narrow the range of things that she is entitled to use as a ground for further beliefs. Of course, if Yujia is entitled to use the fact or proposition that the fuel gauge points to F as a ground for belief, then her evidence will make it rational for her to believe that she has enough fuel. But if Yujia is instead entitled to use merely the fact or proposition that she has a visual experience as of the fuel gauge pointing to F as a ground for belief, then, especially if she is also rational to believe that she may very well have hypoxia, this narrower body of evidence will not make it rational for her to believe that she has enough fuel. Thus, whether Yujia’s

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evidence makes it rational for her to believe that she has enough fuel may depend on whether her evidence makes it rational for her to believe that she may very well have hypoxia: if her evidence makes it rational for her to believe the latter, then that can ensure that her evidence set only includes facts or propositions that are jointly insufficient to make it rational for her to believe the former. I have argued that the structural constraints on an agent’s body of total evidence at a time—constraints that derive from the normative role of evidence—may make it impossible for the stipulations of each of our puzzle cases to be jointly satisfied. I have not argued here that these structural constraints will make it impossible, and that’s because I have not tried to specify those structural constraints sufficiently finely here: I have tried to remain neutral on various disputed issues concerning what those constraints are. So what I’ve argued does not prove that neither puzzle can be coherently formulated, but it does prove that neither puzzle has yet been formulated in a way that ensures the joint satisfiability of its stipulations. If this is correct, then we have not yet been given a good reason to accept any of the controversial epistemological views that have grown out of efforts to address one or the other of these two puzzles.

Acknowledgements Thanks to Mattias Skipper and Alex Worsnip for helpful comments on an earlier draft of this chapter.

Bibliography Bergmann, M. (). “Easy Knowledge: Malignant and Benign.” In: Philosophy and Phenomenological Research , pp. –. Christensen, D. (). “Higher-Order Evidence.” In: Philosophy and Phenomenological Research , pp. –. Goldman, A. (). “Williamson on Knowledge and Evidence.” In: P. Greenough and D. Pritchard (eds), Williamson on Knowledge, Oxford University Press, pp. –. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs , pp. –. Lasonen-Aarnio, M. (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research , pp. –. Neta, R. (). “What Evidence Do You Have?” In: British Journal for the Philosophy of Science , pp. –. Neta, R. (). “Easy Knowledge, Transmission Failure, and Empiricism.” In: Oxford Studies in Epistemology , pp. –. Neta, R. (). “Evidence, Coherence, and Epistemic Akrasia.” In: Episteme (), pp. –. Pryor, J. (). “The Skeptic and the Domatist.” In: Noûs , pp. –. Pryor, J. (). “What’s Wrong with Moore’s Argument?” In: Philosophical Issues , pp. –. Smithies, D. (). “Ideal Rationality and Logical Omniscience.” In: Synthese , pp. –. Titelbaum, M. (). “Tell Me You Love Me: Bootstrapping, Externalism, and No-Lose Epistemology.” In: Philosophical Studies , pp. –. Titelbaum, M. (). “Rationality’s Fixed Point (Or: In Defense of Right Reason).” In: Oxford Studies in Epistemology , pp. –.

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Tucker, C. (). “When Transmission Fails.” In: The Philosophical Review , pp. –. Weisberg, J. (). “Bootstrapping in General.” In: Philosophy and Phenomenological Research , pp. –. White, R. (). “Problems for Dogmatism.” In: Philosophical Studies , pp. –. Williamson, T. (). “Knowledge as Evidence.” In: Mind , pp. –. Williamson, T. (). “Very Improbable Knowing.” In: Erkenntnis , pp. –. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research , pp. –. Wright, C. (). “Warrant for Nothing (and Foundations for Free?)” In: Aristotelian Society Supplement (), pp. –.

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9 Higher-Order Defeat and the Impossibility of Self-Misleading Evidence Mattias Skipper

. Introduction Evidentialism is the thesis, roughly, that one’s beliefs should fit one’s evidence. The enkratic principle is the thesis, roughly, that one’s beliefs should “line up” with one’s beliefs about which beliefs one ought to have. Both theses have seemed attractive to many philosophers. However, they jointly imply a controversial conclusion, namely that a certain kind of self-misleading evidence is impossible. That is to say, if evidentialism and the enkratic principle are both true, it is impossible for one’s evidence to support certain false beliefs about what one’s evidence supports. Recently, a number of authors have argued that self-misleading evidence is possible on the grounds that misleading higher-order evidence does not have the kind of strong and systematic defeating force that would be needed to rule out the possibility of such self-misleading evidence. If they are right, we are left with a seemingly unattractive choice between sacrificing the enkratic principle on the altar of evidentialism, or vice versa. Put differently, anyone who wants to save both evidentialism and the enkratic principle faces a challenge of explaining why cases of misleading higher-order evidence are, in fact, not cases of self-misleading evidence. The aim of this chapter is to propose a view of higher-order evidence that does indeed render self-misleading evidence impossible. Central to the view is the idea that higher-order evidence acquires its normative significance by influencing which conditional beliefs it is rational to have. I will say more to clarify and motivate this idea as we proceed. But what I hope will emerge is an independently plausible view of higher-order evidence, which has the additional benefit the it allows us to retain both evidentialism and the enkratic principle. The rest of the chapter is structured as follows. In section ., I offer some more precise formulations of evidentialism and the enkratic principle and explain why they jointly imply that self-misleading evidence is impossible. In section ., I then review what I take to be the most serious challenge, due to Worsnip (), against the view that self-misleading evidence is impossible, and I argue that the challenge Mattias Skipper, Higher-Order Defeat and the Impossibility of Self-Misleading Evidence In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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is unconvincing as it stands. In section ., I go on to propose a novel view of higherorder evidence, which not only offers a diagnosis of where Worsnip’s challenge goes wrong, but also lends positive support to the view that self-misleading evidence is impossible. In section ., I provide some further motivation of the view by showing how it can help us to make sense of a number of peculiar features of higher-order evidence that are otherwise difficult to understand. Section . is a brief summary.

. Evidentialism, the enkratic principle, and self-misleading evidence Let me begin with a few remarks on terminology and notation. As usual, I will write “Bp” to say that the proposition p is believed by the agent under consideration. In addition, if a is a doxastic attitude towards some proposition, I will use the following shorthands to talk about epistemic rationality and evidential support: ra: a is rationally permitted Ra: a is rationally required

ea: a is sufficiently supported by the evidence Ea: a is decisively supported by the evidence

If a is not rationally permitted, I will say that a is rationally forbidden. The operators R and E are treated as duals of r and e in the usual way: a is rationally required if and only if ~a is not rationally permitted, and a is decisively supported by the evidence if and only if ~a is not sufficiently supported by the evidence. I will not rely on any particular semantics of these four operators. But formally inclined readers may think of them as pairs of possibility and necessity operators from standard epistemic and deontic logic (cf. Hintikka ; ). On my usage of the term “evidential support,” it is doxastic attitudes towards propositions rather than propositions themselves that are said to be (or not to be) supported by the evidence. For ease of exposition, I will also sometimes say about a proposition that it is supported by the evidence, but this should be taken to mean that the relevant belief in that proposition is supported by the evidence. For present purposes, doxastic attitudes are individuated in a coarse-grained manner. That is, I will be talking about belief, disbelief, and suspension of judgment, but not degrees of belief or credences. The reason is that the enkratic principle is most often understood in terms of binary attitudes rather than graded ones. Graded attitudes like credences can obviously be akratic as well. But to keep matters relatively simple, I will focus on the binary case here. Consider, then, the following statement of the evidentialist thesis: Evidentialism: Necessarily, (i) ra $ ea; and (ii) Ra $ Ea. According to this thesis, a doxastic attitude is rationally permitted if and only if it is sufficiently supported by the evidence, and it is rationally required if and only if it is decisively supported by the evidence. Three comments about Evidentialism are in order. First, when I say that “the evidence” supports this-or-that doxastic attitude, I always have the total evidence in

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mind. Otherwise, Evidentialism would clearly be false, since different parts one’s total evidence might pull in different directions with respect to the same proposition. Second, note that Evidentialism is a “substantive” requirement of rationality in the sense that it constrains which individual doxastic attitudes it can be rational to have. By comparison, the requirement to avoid contradictory beliefs is a “structural” requirement of rationality, because it constrains which combinations of doxastic attitudes it can be rational to have. Finally, Evidentialism is surely not the only evidentialist thesis that one might care about. For example, some authors hold that even the most decisive evidence only ever makes it rationally permissible (not required) to adopt this-or-that doxastic attitude. If they are right, we should only accept the first clause in Evidentialism.¹ However, since both clauses will be needed to establish the conclusion that self-misleading evidence is impossible, I will, for dialectical reasons, focus on this relatively strong version of Evidentialism. Is Evidentialism a plausible thesis? On the face of it, it seems clear that a person’s evidence somehow plays a role in determining which doxastic attitudes it is rational for the person to adopt. If we are asked to judge whether an agent’s beliefs on some matter are rational, it seems natural to consider whether the agent has responded in a reasonable manner to the available evidence. But despite its prima facie appeal, Evidentialism is subject to ongoing debate. For present purposes, I do not want to enter a detailed discussion of the connection between evidential support and epistemic rationality. I will simply take Evidentialism for granted to let the challenge, to which I aim to respond, arise. Consider, next, the following statement of the enkratic principle: Enkratic Principle: Necessarily, (i) RðBRa ! aÞ; and (ii) RðBR a ! aÞ. e e This thesis says, roughly, that one’s doxastic attitudes should “line up” with one’s beliefs about which doxastic attitudes one ought to have. More precisely, according to the first clause, one is never permitted to believe that one is required to have a doxastic attitude that one does not have. For example, I’m never permitted to believe that “I should believe that it’s raining” while failing to believe that “it’s raining.” Conversely, according to the second clause, one is never permitted to have a doxastic attitude that one believes one is not permitted to have. For example, I’m never permitted to believe that “I shouldn’t believe that it’s raining” while believing that “it’s raining.” In contrast to Evidentialism, the Enkratic Principle is a structural requirement of rationality, because it constrains which combinations of doxastic attitudes it can be rational to have, but says nothing about which particular doxastic attitudes it can be rational to have. All the Enkratic Principle asks is that agents maintain a certain coherence between their doxastic attitudes and their beliefs about which doxastic attitudes they ought to have.

¹ See, e.g., Conee and Feldman.

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Is the Enkratic Principle a plausible thesis? The thesis might seem almost selfevident at first blush. Just as it seems manifestly irrational to believe Moorean propositions like “it’s raining, but I don’t believe that it’s raining” so it seems manifestly irrational to believe akratic propositions like “it’s raining, but I shouldn’t believe that it’s raining.” Yet, But despite its prima facie appeal, the Enkratic Principle is subject to ongoing debate.² Again, however, I will not enter a general discussion of whether epistemic akrasia can be rational. For present purposes, I will simply take the Enkratic Principle onboard alongside with Evidentialism. A number of authors have recently observed that Evidentialism conflicts with the Enkratic Principle in cases where an agent’s total evidence misleads about what it itself supports.³ That is to say, Evidentialism and the Enkratic Principle jointly imply that a certain kind of self-misleading evidence is impossible. More precisely, Evidentialism and the Enkratic Principle imply the following thesis (a proof can be found in the Appendix): Impossibility of Self-Misleading Evidence: Necessarily, (i) Ea ! eB ea; and e e (ii) eea ! eeBEa. This thesis says, roughly, that there are certain false beliefs about what one’s evidence supports that cannot be supported by one’s evidence. More precisely, according to the first clause, if one’s evidence decisively supports a given doxastic attitude, it cannot sufficiently support believing that it does not sufficiently support that attitude. For example, if my evidence decisively supports believing that “it’s raining,” it cannot sufficiently support the false belief that “my evidence doesn’t sufficiently support believing that it’s raining.” Conversely, according to the second clause, if one’s evidence does not sufficiently support a doxastic attitude, it cannot sufficiently support believing that it decisively supports that attitude. For example, if my evidence does not sufficiently support believing that “it’s raining,” it cannot sufficiently support the false belief that “my evidence decisively supports believing that it’s raining.” Thus, the Impossibility of Self-Misleading Evidence rules out two kinds of selfmisleading evidence as impossible: evidence that sufficiently supports believing that it does not sufficiently support a doxastic attitude that it in fact decisively supports; and evidence that sufficiently supports believing that it decisively supports a doxastic attitude that it in fact does not sufficiently support. Let me pause to explain, in informal terms, why Evidentialism and the Enkratic Principle imply the Impossibility of Self-Misleading Evidence. We begin by supposing that an agent’s total evidence at once decisively supports a doxastic attitude a and sufficiently supports believing that it does not sufficiently support a. This supposition

² For recent discussions of epistemic akrasia, see Christensen (), Coates (), Dorst (this volume), Greco (), Horowitz (), Lasonen-Aarnio (forthcoming), Littlejohn (), Skipper (forthcoming), and Titelbaum (). ³ Most notably Worsnip (; this volume) and Lasonen-Aarnio (forthcoming; this volume). Dorst (this volume), Titelbaum (), and Williamson (; ) also come close to the same point.

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amounts to denying the first clause in the Impossibility of Self-Misleading Evidence. Assuming that Evidentialism is true, it then follows that the agent is at once rationally required to adopt a and rationally permitted to believe that a is not sufficiently supported by her evidence. Hence, she is at once rationally required to adopt a and rationally permitted to believe that she is not rationally permitted to adopt a.⁴ But this contradicts the second clause in the Enkratic Principle. Thus, if Evidentialism and the second clause in the Enkratic Principle are both true, the first clause in the Impossibility of Self-Misleading Evidence must be true as well. The second clause in the Impossibility in Self-Misleading Evidence can be established in a similar way using the first clause in the Enkratic Principle. While Evidentialism and the Enkratic Principle rule out certain sorts of selfmisleading evidence, they do not rule out all sorts of self-misleading evidence. More specifically, they are compatible with the following thesis: Possibility of Weakly Self-Misleading Evidence: Possibly, (i) ea&eB ea; or e (ii) ea&eBea. e The first clause says that one’s evidence can at once sufficiently support a doxastic attitude a and sufficiently support believing that it does not sufficiently support a. For example, my evidence might sufficiently support believing that “it’s raining” while sufficiently supporting the false belief that “my evidence doesn’t sufficiently support believing that it’s raining.” The second clause says that one’s evidence can at once not sufficiently support a, but sufficiently support believing that it does sufficiently support a. For example, my evidence might not sufficiently support believing that “it’s raining,” but sufficiently support believing that “my evidence does sufficiently support believing that it’s raining.” We can sum up the foregoing observations by saying that Evidentialism and the Enkratic Principle are compatible with weakly self-misleading evidence, but incompatible with radically self-misleading evidence. Of course, it might seem arbitrary or ad hoc to maintain that weakly self-misleading evidence is possible, whereas radically self-misleading evidence is not. But it is worth being clear about what follows directly from Evidentialism and the Enkratic Principle, and what does not. The connection between Evidentialism, the Enkratic Principle, and the Impossibility of Self-Misleading Evidence will constitute an important “dialectical fixedpoint” in what follows: anyone who accepts Evidentialism and the Enkratic Principle must accept the Impossibility of Self-Misleading Evidence as well, and anyone who rejects the Impossibility of Self-Misleading Evidence must reject either Evidentialism or the Enkratic Principle. Eventually, I will defend the former option: there is, indeed, good reason to think that self-misleading evidence is impossible. But first, let us consider why some philosophers have thought otherwise.

⁴ This step of the argument requires an auxiliary assumption to the effect that one is rationally permitted to believe that a isn’t sufficiently supported by one’s evidence only if one is rationally permitted to believe that one isn’t rationally permitted to adopt a. See the Appendix for the details.

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. Putative cases of self-misleading evidence Consider the following two stories: Driver’s Bias: John believes himself to be above-average at driving. His belief is strongly supported by the evidence: he has many years of driving experience, and he has a better crash record than most of his acquaintances. But in reading today’s newspaper, John learns about the well-documented driver’s bias: the tendency, especially among male subjects, to overestimate their own driving skills. Poor Logic: Sophie and her classmates are asked to prove, independently of each other, whether a logical formula T is tautological or not. As it happens, Sophie makes a few errors and draws the wrong conclusion that T is nontautological. All of Sophie’s classmates happen to reach the same wrong conclusion. Sophie is aware of their agreement. What is more, their otherwise competent logic professor makes an occasional blunder and assures the students that they have reached the right conclusion. What does John’s total evidence support after he has learned about the driver’s bias? And what does Sophie’s total evidence support after she has learned that her logic professor and classmates unanimously agree with her? Let us take a closer look at the evidence in each case. In Driver’s Bias, John starts out with a body of evidence about his own driving history, which strongly indicates that his driving skills are above average. Evidence of this sort is often said to be of “first order,” because it directly concerns the question of whether John is better than average at driving. By assumption, then, John’s first-order evidence strongly supports that he is better than average at driving. However, John then receives some additional evidence about the driver’s bias, which indicates that he has overestimated his own driving skills. Evidence of this sort is said to be of “higher order,” because it only indirectly concerns the question of whether John’s driving skills are above average. I will later (in section ..) offer a more precise characterization of this intuitive distinction between first-order and higher-order evidence. For now, it will suffice to have a rough understanding of the distinction as it applies in the cases above. By contrast, in Poor Logic, Sophie starts out with a body of first-order evidence consisting of her (flawed) proof, which does not support that T is non-tautological. She then receives a body of higher-order evidence consisting of the unanimous agreement of her classmates and logic professor, which indicates that her proof is indeed correct. That is, while her first-order evidence does not support that T is nontautological, her higher-order evidence indicates that her first-order evidence does support that T is non-tautological. What Driver’s Bias and Poor Logic have in common is that the higher-order evidence in each case misleads about what the first-order evidence supports. In Driver’s Bias, the first-order evidence in fact supports that John is better than average at driving, but the higher-order evidence suggests otherwise. In Poor Logic,

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the first-order evidence in fact does not support that T is non-tautological, but the higher-order evidence suggests that it does.⁵ How strong are these seemingly opposing evidential relations in Driver’s Bias and Poor Logic? Given the present dialectical context, we are looking to create the most hospitable environment for counterexamples to the Impossibility of Self-Misleading Evidence to arise. Let us therefore suppose that John’s first-order evidence is strong enough to decisively support the belief that he is better than average at driving, and let us suppose that his higher-order evidence is strong enough to sufficiently support believing that the first-order evidence does not sufficiently support believing that he is better than average at driving. Similarly, let us suppose that Sophie’s first-order evidence does not sufficiently support the belief that T is non-tautological, and that the higher-order evidence sufficiently supports believing that the first-order evidence decisively supports believing that T is non-tautological. We can state these four stipulative assumptions about the evidential relations in Driver’s Bias and Poor Logic in a more convenient way by introducing the following shorthands for the propositions that feature in the two cases (where “F” stands for first-order, and “H” stands for higher-order): PF: John is better than average at driving. PH: John’s first-order evidence sufficiently supports BPF. QF: The logical formula T is not tautological. QH: Sophie’s first-order evidence decisively supports BQF. Given this, the four stipulations become: John’s first-order evidence decisively supports believing PF; John’s higher-order evidence sufficiently supports believing ~PH; Sophie’s first-order evidence does not sufficiently support believing QF; and Sophie’s higher-order evidence sufficiently supports believing QH. Let us now return to the question of what the total evidence in each case supports. Given the above stipulations, one might be tempted to reason as follows: Naïve Argument Driver’s Bias: () John’s first-order evidence decisively supports BPF. () John’s higher-order evidence sufficiently supports B~PH. () So, John’s total evidence decisively supports BPF and sufficiently supports B~PH. Poor Logic: () Sophie’s first-order evidence does not sufficiently support BQF. () Sophie’s higher-order evidence sufficiently supports BQH. () So, Sophie’s total evidence does not sufficiently support BQF and sufficiently supports BQH. Each conclusion, if true, constitutes a counterexample to the Impossibility of SelfMisleading Evidence: the first conclusion violates the first clause in the Impossibility of Self-Misleading Evidence in virtue of saying that John’s total evidence at once decisively supports believing PF and sufficiently supports believing that it does not ⁵ Similar cases of misleading higher-order evidence can be found in Christensen (), Elga (), Horowitz and Sliwa (), Schoenfield (; ), and several chapters in this volume.

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sufficiently support believing PF; and the second conclusion violates the second clause in the Impossibility of Self-Misleading Evidence in virtue of saying that Sophie’s total evidence at once does not sufficiently support believing QF and sufficiently supports believing that it decisively supports believing QF. Thus, if the Naïve Argument is sound, we are forced to give up either Evidentialism or the Enkratic Principle. However, the Naïve Argument ignores the fact that evidential relations are not in general monotonic (hence the pejorative label “naive”). That is to say, a body of evidence need not support a doxastic attitude just because a subset of the evidence does. Many examples of this familiar phenomenon can be found in the literature on epistemic defeat, and I shall not extend the list here.⁶ The initial observation I want to make at this point is just that it would be too hasty to conclude that Driver’s Bias and Poor Logic are cases of self-misleading evidence on the basis of considerations about what the first-order and higher-order evidence supports when taken separately. What matters is what the first-order and higher-order evidence supports when taken together. This much should be uncontroversial. In a recent paper, Worsnip () offers what may be seen as a refined version of the Naïve Argument, which purports to show that, even if we take into account potential defeat relations in cases like Driver’s Bias and Poor Logic, they still constitute genuine counterexamples to the Impossibility of Self-Misleading Evidence (at least on some ways of filling in the details of such cases). Here is a simple reconstruction of Worsnip’s main argument, as it applies to Driver’s Bias:⁷ Asymmetry Argument W₁ W₂ W₃

John’s first-order evidence supports PF more strongly than his higher-order evidence supports ~PF. John’s higher-order evidence supports ~PH more strongly than his firstorder evidence supports PH. So, John’s total evidence supports both PF and ~PH.

Two preliminary remarks about this argument are in order. First, much of Worsnip’s own exposition is devoted to establishing two asymmetry claims that are distinct from W₁ and W₂: first, the claim that the higher-order evidence bears more strongly on PH than on PF; second, the claim that the first-order evidence (if anything) bears more strongly on PF than on PH. But note that these two asymmetry claims have no straightforward bearing on the conclusion, W₃, that Worsnip ultimately wants to establish. Thus, I take it that Worsnip wants (or, in any case, needs) to establish two asymmetry claims akin to W₁ and W₂ that do bear on W₃.

⁶ For overviews of different issues related to epistemic defeat, see Kelly () and Koons (). ⁷ Worsnip’s own discussion centers on a case that is structurally similar to Poor Logic, that is, one where the first-order evidence by itself does not support the relevant first-order proposition (see Worsnip , §.b). However, I take his argument to apply mutatis mutandis to Driver’s Bias as well.

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Second, note that W₃ is not a clear-cut counterexample to the Impossibility of SelfMisleading Evidence, since it says nothing about how strongly John’s total evidence supports PF and ~PH respectively. However, I do not want to resist the Asymmetry Argument on the grounds that the total evidence in cases like Driver’s Bias and Poor Logic is only ever weakly self-misleading. So, for present purposes, I will simply grant that W₃, if true, indeed constitutes a genuine counterexample to the Impossibility of Self-Misleading Evidence. For the same reason, I will also simply concede the two asymmetry claims W₁ and W₂. Even so, I find the Asymmetry argument unconvincing as it stands. The problem is that W₁ and W₂ are claims about how the first-order and higher-order evidence bear on PF and PH when taken separately, whereas W₃ is a claim about how the first-order evidence and higher-order evidence bear on PF and PH when taken in conjunction. This raises much the same worry that led us to reject the Naïve Argument in the first place. Just as the Naïve Argument draws a conclusion about what the total evidence supports from premises about what the first-order and higher-order evidence supports when taken separately, so the Asymmetry Argument draws a conclusion about what the total evidence supports from premises about what the first-order and higher-order evidence supports when taken separately. Is there a way to modify the Asymmetry Argument so as to avoid this problem? One could introduce certain additional premises that would ensure that the presence of the first-order evidence does not undermine the bearing of the higher-order evidence on PF and PH, and that the presence of the higher-order evidence does not undermine the bearing of the first-order evidence on PF and PH. That is, one could try to amend the Asymmetry Argument with the following premises: No Bottom-Up Undercutting: The first-order evidence does not undercut the support relation between the higher-order evidence and ~PH. No Top-Down Undercutting: The higher-order evidence does not undercut the support relation between the first-order evidence and PF. The term “undercutting” is used in the standard way (cf. Pollock ): roughly, if a body of evidence supports a given doxastic attitude, an undercutting defeater is any body of evidence in light of which the original body of evidence no longer supports that attitude. Thus, No Bottom-Up Undercutting amounts to the claim that the higher-order evidence continues to speak against PH in the presence of the first-order evidence, and No Top-Down Undercutting amounts to the claim that the first-order evidence continues to speak in favor of PF in the presence of the higher-order evidence. By combining W₁ and W₂ with No Bottom-Up Undercutting and No Top-Down Undercutting, we have what looks like a strong argument for W₃. However, there is strong independent reason to reject No Top-Down Undercutting; or so I will argue. Of course, even if No Top-Down Undercutting is false, one might still try to weaken No Top-Down Undercutting in various ways by allowing for undercutting defeat to occur as long as the defeat is sufficiently weak (so as to not undermine W₃). But the considerations I will offer against No Top-Down Undercutting are going to speak equally against such weakened versions of that principle.

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Hence, if the considerations below are correct, the strategy of weakening No TopDown Undercutting to save W₃ is not going to succeed.

. Higher-order defeat and the Impossibility of Self-Misleading Evidence So far, I have deliberately avoided all talk about what the total evidence in Driver’s Bias and Poor Logic seems to support from an intuitive or pre-theoretical standpoint. The reason is dialectical: I think Worsnip’s challenge to the Impossibility of SelfMisleading Evidence can be resisted without appealing to such intuitive judgments. Yet, I take it to be a widely shared intuition that John’s total evidence in fact does not sufficiently support believing that his driving skills are above. Likewise, I take it to be intuitive that Sophie’s total evidence does sufficiently support believing that T is non-tautological. Thus, for what they are worth, these intuitive verdicts suggest that Driver’s Bias and Poor Logic are not cases of self-misleading evidence. More importantly, however, I think we can make good theoretical sense of these intuitions. Indeed, as I shall argue below, there is an independently plausible view of higher-order evidence available that not only vindicates our intuitive verdicts in cases like Driver’s Bias and Poor Logic, but also allows us to maintain that self-misleading evidence is impossible. As a first step, I want to review what I take to be a compelling diagnosis, due to Christensen (; ), of what gives rise to our intuitive judgments in cases like Driver’s Bias and Poor Logic. Christensen’s central observation is that agents who possess strong misleading higher-order evidence seem forced into a kind of dogmatic or question-begging reasoning, if they maintain their original beliefs in light of the higher-order evidence. Take John as an example: if he, after having received the higher-order evidence, continues to maintain that his first-order evidence supports PF, it must be because he takes the higher-order evidence to be misleading. After all, if the higher-order evidence had not been misleading, the first-order evidence would not have supported PF. Yet, in assuming that the higher-order evidence is misleading, John seems to beg the question in much the same way as someone who disregards a body of evidence merely on the grounds that it opposes his or her prior opinions. Thus, to avoid this sort of dogmatic or question-begging reasoning, John cannot continue to maintain that his first-order evidence supports PF. Much the same goes for Sophie: just as John cannot reasonably continue to maintain that his first-order evidence supports PF, she can reasonably begin to maintain that her first-order evidence supports QF in light of the higher-order evidence. We can unpack Christensen’s diagnosis a bit further by asking how likely John and Sophie should consider PF and QF to be on their respective bodies of first-order evidence before versus after having received their respective bodies of higherorder evidence. By assumption, John should consider PF to be quite likely on the first-order evidence before having received the higher-order evidence. However, the foregoing considerations suggest that he shouldn’t consider PF to be likely on the first-order evidence after having received the higher-order evidence. If he did, he would disregard the higher-order evidence in what looks like a dogmatic manner.

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Much the same goes for Sophie: she should not consider QF to be likely on the firstorder evidence before having received the higher-order evidence, but she should consider QF to be likely on the first-order evidence after having received the higher-order evidence. This seems to suggest that No Top-Down Undercutting must be rejected. Recall that, according to No Top-Down Undercutting, the evidential relation between John’s first-order evidence and PF cannot be undermined by his higher-order evidence, and the evidential relation between Sophie’s first-order evidence and QF can likewise not be undermined by her higher-order evidence. Yet, this is precisely what we have just denied on the grounds that John and Sophie would otherwise fall prey to a kind of dogmatic or question-begging reasoning. Thus, we also have a diagnosis of where, exactly, the Asymmetry Argument goes wrong: as explained in section ., the inference from W₁ and W₂ to W₃ presupposes No Top-Down Undercutting and No Bottom-Up Undercutting. So, if No Top-Down Undercutting is false, the Asymmetry Argument does not go through. The rejection of No Top-Down Undercutting not only allows us to resist the Asymmetry Argument, but also lends more direct support to the view that selfmisleading evidence is impossible. To see why, let us reformulate the previous claims about the normative impact of John and Sophie’s respective bodies of higher-order evidence in terms of the changes that it should make to their conditional doxastic attitudes towards PF and QF. We can think of an agent’s conditional doxastic attitudes as reflecting the way in which the agent takes different bodies of evidence to bear on different propositions. For example, my doxastic attitude towards the proposition “it has recently been raining” conditional on “the streets are wet” is belief, since I take the fact that the streets are wet to be strong evidence that it has recently been raining. By contrast, my doxastic attitude towards “it has recently been raining” conditional on “the streets are dry” is disbelief, since I take the fact that the streets are dry to be strong evidence against recent rain. Finally, my doxastic attitude towards “it has recently been raining” conditional on “Paris is the capital of France” is suspension of judgment, since I take the fact that Paris is the capital of France to have no significant bearing on whether or not it has recently been raining.⁸ The notion of a conditional all-or-nothing belief plays much the same role in the present context as the notion of conditional credences plays in a Bayesian context. Just as conditional credences reflect the way in which Bayesian agents take different bodies of evidence to bear on different propositions, so conditional beliefs reflect the way in which “non-Bayesian” agents take different bodies of evidence to bear on different propositions.⁹

⁸ Of course, these conditional beliefs will be contingent on the possessed background information, just as an agent’s conditional credences are contingent on background information in a Bayesian framework. ⁹ This obviously raises some further questions about how an agent’s conditional beliefs should relate to the agent’s unconditional beliefs. However, we need not settle this matter here. For present purposes, it suffices to have at least an intuitive grasp of the notion of a conditional belief.

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We can then ask: how, if at all, should John and Sophie revise their conditional doxastic attitudes towards PF and QF in light of their respective bodies of higherorder evidence? We have said that before John receives the higher-order evidence, his doxastic attitude towards PF conditional on the first-order evidence is belief. By assumption, this conditional belief is rational, since his first-order evidence is assumed to support PF. However, after having received the higher-order evidence, it is no longer rational for him to hold this conditional belief towards PF, since the higher-order evidence indicates that his first-order evidence does not support PF. Thus there is a rational pressure for John to revise his conditional doxastic attitude towards PF (from belief to suspension of judgment, or perhaps even to disbelief, depending on how we fill in the details of the case). A similar story can be told about Sophie: before she receives the higher-order evidence, her doxastic attitude towards QF conditional on the first-order evidence is belief. By assumption, this conditional belief is irrational, since her first-order evidence is assumed not to support QF. However, after having received the higherorder evidence, it becomes rational for Sophie to hold this conditional belief towards QF, because the higher-order evidence strongly indicates that her first-order evidence indeed supports QF. Thus, while John’s higher-order evidence creates a rational pressure to give up his conditional belief, Sophie’s higher-order evidence alleviates an existing rational pressure to give up her conditional belief. As flagged in the introduction, the lesson I want to draw from these considerations is that higher-order evidence acquires its normative significance by influencing which conditional doxastic attitudes it is rational to have. In Driver’s Bias, the higher-order evidence gets to have a normative impact on John’s doxastic attitude towards PF, because it requires him to revise his conditional belief towards PF. In Poor Logic, the higher-order evidence gets to have a normative impact on Sophie’s doxastic attitude towards QF, because it removes an existing requirement to revise her conditional belief towards QF. This also means that we have a way of vindicating our intuitive verdicts in Driver’s Bias and Poor Logic: John’s total evidence sufficiently (if not decisively) supports believing ~PH, and does not sufficiently support believing PF. Sophie’s total evidence sufficiently (if not decisively) supports believing QH, and sufficiently supports believing QF. Neither total body of evidence is self-misleading. Hence, Driver’s Bias and Poor Logic do not constitute counterexamples to the Impossibility of Self-Misleading Evidence. Obviously, even if cases like Driver’s Bias and Poor Logic are not examples of selfmisleading evidence, there might be other such examples. In particular, I have said nothing to sway those philosophers who want to reject the Impossibility of SelfMisleading Evidence on the grounds that one can lack access to what one’s evidence is (even if one cannot be misled about what one’s evidence supports).¹⁰ The possibility of such access failures is firmly rooted in the ongoing dispute between internalists ¹⁰ Authors who have pushed this line of argument include Lasonen-Aarnio (forthcoming) and Worsnip (, §.a), both of whom rely on anti-luminosity considerations, which have originally been used by Williamson (, ch. ; ) to defend a broadly externalist stance on epistemic notions like knowledge, evidence, and epistemic justification.

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and externalists about epistemic rationality, which lies well beyond the scope of this chapter. But I hope to have offered some reasons to think that the Impossibility of Self-Misleading Evidence should at least not be rejected on the grounds that one can receive strong misleading higher-order evidence about what one’s first-order evidence supports. Before we proceed, I should note that the idea that agents sometimes ought to revise their conditional beliefs is not a new one. For instance, Lange () has argued that a number of central problems in Bayesian confirmation theory can be resolved if we allow for certain credal changes to come about as the result of revising our conditional credences (or “confirmation commitments” as he follows Levi () in calling them). Along similar lines, Brössel and Eder () and Rosenkranz and Schulz () have suggested that certain types of peer disagreement should lead the disagreeing parties to revise their credences in the disputed proposition conditional on the shared evidence. The idea, then, that certain types of doxastic changes should come about as the result of revising one’s conditional doxastic attitudes has already been put to use in various contexts. What I hope to have shown here is that the same idea can serve as the basis of a quite general view about the normative significance of higher-order evidence, which allows us to reconcile Evidentialism and the Enkratic Principle.

. Further explanatory attractions With the core proposal on the table, I now want to provide some further motivation of the proposal by showing how it can help us to understand various peculiar features of higher-order evidence that are otherwise difficult to make sense of doing so will also give me an opportunity to clarify and elaborate some aspects of the proposed view.

.. The retrospectivity of higher-order evidence The first property I want to focus on is the retrospective aspect of higher-order evidence. As a number of authors have pointed out, someone who receives a higher-order defeater thereby acquires a reason to think that his or her doxastic state was irrational even before receiving the higher-order defeater.¹¹ Take again John as an example: when he receives the information about the driver’s bias, he thereby gets a reason to think that his belief was never supported by his first-order evidence in the first place. By contrast, first-order evidence does not display this sort of retrospectivity. If, for example, I believe that it is raining outside based on the testimony of a reliable friend, but I then look out the window and sees that it does not, in fact, rain, I do not thereby get a reason to think that it was irrational of me to believe as I did before looking out the window. Likewise, if I believe that the wall in front of me is

¹¹ See, e.g., Christensen (), Lasonen-Aarnio (), and DiPaolo ().

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red based on its reddish appearance, but I then learn that the wall is merely lit up by a red spotlight, I do not thereby get a reason to think that it was irrational of me to believe as I did before learning about the red spotlight. Why do higher-order defeaters display this kind of retrospectivity, when firstorder defeaters do not? The proposed view offers a simple explanation: the reason why higher-order defeaters have a retrospective aspect is that they work by indicating that one’s original conditional doxastic attitude (that is, the conditional doxastic attitude that one had before receiving the higher-order defeater) was irrational all along. By contrast, and the reason why first-order defeaters do not have a retrospective aspect is that they do not work by indicating that one’s original conditional doxastic attitude was irrational to begin with. On the present picture, this explains why only higher-order evidence has a distinctive retrospective character.

.. The agent-relativity of higher-order evidence The second peculiar feature of higher-order evidence is that its normative significance in many cases depends on who possesses it. To illustrate this sort of agentrelativity, consider a case adapted from Christensen (, p. ): Arithmetic on Drugs: You and I decide to the square root of  independently of each other. Unbeknownst to us, we both settle on the answer “” and hence form the belief that “√ = .” However, upon having performed our respective calculations, we both learn that I have been given a reason-distorting drug that subtly, but significantly, impairs my ability to perform even simple arithmetic calculations. Here it seems that the higher-order evidence (that is, the information that I have been given a reason-distorting drug) has very different normative implications for you and me: while I should lose confidence in my belief, at least to some extent, there is no apparent reason for you to do the same. After all, the fact that someone else than yourself seems utterly irrelevant to the question of whether the square root of  equals . As Christensen () and Kelly () have pointed out, this sort of agentrelativity is at least initially a bit puzzling. We are used to think that the question of how an agent should respond to a given body of evidence does not depend for its answer on who the agent is. Of course, the answer might depend on the agent’s background information. But the kind of agent-relativity at play in Arithmetic on Drugs remains even if we assume that you and I have exactly the same background information. Thus, higher-order evidence seems to give rise to a kind of agentrelativity that differs importantly from the kind of relativity to background information with which we are familiar. What explains this distinct sort of agent-relativity? Again, the proposed view offers a simple answer: in Arithmetic on Drugs, the difference between you and me is that I get a reason to doubt that my first-order evidence supports that “√ = ,” whereas you do not get a reason to doubt that your first-order evidence

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supports that “√ = .” This means that I should revise my conditional belief towards “√ = ,” whereas you should not. On the present picture, this is why the same body of higher-order evidence may have very different normative implications for different agents.

.. The insignificance of hypothetical higher-order evidence The third property that I want to discuss is that the normative significance of a body of higher-order evidence sometimes depends on whether it is regarded as actually obtaining or as merely hypothetically obtaining. To illustrate the phenomenon, consider another case adapted from Christensen (, §): Cognitive Impairment: Brenda is a formidable scientist who is going to carry out an experimental test of a hypothesis h next week. The experiment has two possible outcomes, o₁ and o₂. While o₁ would strongly confirm h, o₂ would strongly disconfirm h. Being a formidable scientist, Brenda is well aware of these evidential relations. But she is also aware that she will suffer from a cognitive impairment next week that will make her unable to give an accurate assessment of how the experimental results bear on h. Following Christensen, let us ask two questions about Brenda. First, how confident should Brenda now be in h conditional on o₁ and the fact that she will be cognitively impaired next week? Intuitively, very confident! After all, the fact that she will be cognitively impaired next week seems utterly irrelevant to the question of whether o₁ supports h. Second, if Brenda actually learns o₁ next week, how confident should she then be that h is true? Intuitively, not very confident! After all, she will then be aware that she is cognitively impaired and thus unable to give an accurate assessment of the experimental results. If we put these two verdicts together, we end up saying that Brenda’s doxastic attitude towards h next week should not match her current doxastic attitude towards h conditional on o₁ and the fact that she will be cognitively impaired next week. This is a striking result. We are used to think that an agent’s credence in a proposition p after having acquired a body of evidence should match the agent’s prior credence in p conditional on that evidence. In other words, we are used to think that the normative significance of a body of evidence does not depend on whether it is regarded as actually obtaining or as merely hypothetically obtaining. As Zhao et al. () point out, this idea lies at the heart of orthodox Bayesianism, since the rule of Updating by Conditionalization effectively amounts to the claim that one’s credences after having learnt that such-and-such is the case should match one’s prior credences on the supposition that such-and-such were the case. We can illustrate this familiar Bayesian idea with a standard case of first-order defeat: Defective Experiment: Joe is a formidable scientist who is going to carry out an experimental test of a hypothesis h next week. The experiment has two possible outcomes, o₁ and o₂. While o₁ would strongly confirm h, o₂ would strongly

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disconfirm h. Being a formidable scientist, Joe is well aware of these evidential relations. However, while he is conducting the experiment, Joe learns that one of the key measurement devices in the experimental setup is defective. Let us ask the same two questions about Joe. First, how confident should Joe now be in h conditional on o₁ and the fact that one of the key measurement devices will be defective next week? Intuitively, not very confident! After all, the defective instrument would render the entire experiment unreliable. Second, if Joe actually learns that the outcome of the experiment is o₁, and that one of the measurement devices is defective, how confident should he then be in h? Intuitively, not very confident! It seems, then, that Joe’s doxastic attitude towards h after having conducted the experiment should, indeed, match his current doxastic attitude towards h conditional on o₁ and the fact that one of the key measurement devices will be defective next week. Hence, If so the normative significance of the evidence about the defective measurement device does not depend on whether it is learnt or supposed. What explains this difference between first-order evidence and higher-order evidence? The proposed view offers the following explanation: when Brenda learns that she is drugged, she should not simply respond to this higher-order evidence by conditionalizing on it. She should also revise her conditional doxastic attitude towards h (conditional, that is, on o_). Naturally, then, her resulting doxastic attitude towards h will not match her prior doxastic attitude towards h conditional on o₁ and the fact that she will be cognitively impaired next week. By contrast, when Joe learns that one of the key measurement devices is defective, he should simply respond to this first-order evidence by conditionalizing on it, which means that his resulting doxastic attitude towards h will match his prior doxastic attitude towards h conditional on o₁ in conjunction with and the fact that one of the key measurement devices is defective next week. On the present picture, this is why higher-order evidence may depend for its normative significance on whether it is regarded as learnt or supposed, whereas first-order evidence does not.

.. The indirectness of higher-order evidence The final property I want to focus on goes back to our initial characterization of the distinction between first-order evidence and higher-order evidence in section .. The idea there was that first-order evidence bears directly on the proposition at hand in a way that higher-order evidence does not. But what, exactly, does this talk of “direct” and “indirect” evidence amount to? The proposed view suggests the following answer: higher-order evidence is “indirect” in the sense that it influences which doxastic attitudes it is rational to have by influencing which conditional doxastic attitudes it is rational to have (conditional, that is, on the relevant first-order evidence). For example, in John’s case, the evidence about the driver’s bias bears indirectly on PF by requiring him to revise his doxastic attitude towards PF conditional on the evidence about his driving history.

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By contrast, first-order evidence is “direct” in the sense that it influences which doxastic attitudes it is rational to have simply by requiring one to conditionalize on it. For example, the evidence about John’s driving history bears directly on PF by of requiring him to revise his doxastic attitude towards PF as a result of conditionalizing on this evidence. On the present picture, this is what the intuitive distinction between “direct” and “indirect” evidence amounts to.

. Concluding remarks I have argued in this chapter that higher-order evidence acquires its normative force by influencing which conditional doxastic attitudes it is rational to have. If this is correct, we can maintain that cases of misleading higher-order evidence are not cases of self-misleading evidence. This would be an important result, since Evidentialism and the Enkratic Principle jointly imply that self-misleading evidence is impossible. The considerations put forth in this chapter thus give us a way of reconciling Evidentialism with the Enkratic Principle. None of this shows that these principles are true. Even if what I have said is basically correct, there might obviously be independent reasons to reject Evidentialism or the Enkratic Principle. But at the very least, I hope to have shown that Evidentialism, the Enkratic Principle, and the proposed view of higher-order evidence form an attractive package. The present chapter is clearly just a first step towards a complete theory of the normative significance of higher-order evidence. In particular, I have yet to give a detailed and precise account of how, exactly, a given body of higher-order evidence should influence an agent’s conditional doxastic attitudes. I suspect that the various peculiar features of higher-order evidence discussed in section . may inspire such an account. But the finer details are left for future work.

Appendix This appendix gives a formal derivation of the Impossibility of Self-Misleading Evidence from Evidentialism and the Enkratic Principle. Let ‘☐’ represent conceptual necessity, and let ‘◊’ be its dual. The following auxiliary assumption is needed for the derivation to go through: rB-closure : ☐ ð☐ ½p ! q ! ½rBp ! rBqÞ: According to rB-closure, one is always rationally permitted to believe the necessary consequences of what one is rationally permitted to believe. This is obviously not an uncontroversial assumption. But given the dialectics of the present chapter, I will simply grant the assumption in order to allow for the derivation to go through.¹²

¹² For background on various epistemic closure principles, see Klein (; ) and Schechter ().

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Lemma . The Enkratic Principle (ii), Evidentialism, and rB-closure jointly entail the Impossibility of Self-Misleading Evidence (i). Proof. 1: 2: 3: 4: 5: 6: 7:

☐ RðBRea ! e aÞ ◊eðEa ! e eB e eaÞ ◊ðEa & eB e eaÞ ðRa & rB eraÞ ◊rða & B e raÞ ◊ eRðB e ra ! eaÞ ☐ ðEa ! eeB e eaÞ

ðEnkratic Principle ðiiÞÞ ðfor reductioÞ ð2Þ ð3; Evidentialism; rBclosureÞ ð4Þ ð5Þ ðreductio from 1; 2; and 6Þ

Lemma . Evidentialism, the Enkratic Principle (i), and rB-closure jointly entail the Impossibility of Self-Misleading Evidence (ii). Proof. Similar to that of Lemma . Theorem . Evidentialism, the Enkratic Principle, and rB-closure jointly entail the Impossibility of Self-Misleading Evidence. Proof. Immediate from Lemmas  and .

Acknowledgements An earlier version of this chapter was presented at University of Cologne. I would like to thank the audience on that occasion for very helpful feedback. Thanks also to Alex Worsnip, Asbjørn Steglich-Petersen, and Jens Christian Bjerring for helpful comments and criticism.

References Brössel, P. and A.-M. Eder (). “How to Resolve Doxastic Disagreement.” In: Synthese , pp. –. Christensen, D. (). “Higher-Order Evidence.” In: Philosophy and Phenomenological Research , pp. –. Christensen, D. (). “Disagreement, Question-Begging, and Epistemic Self-Criticism.” In: Philosophers’ Imprint . Christensen, D. (). “Disagreement, Drugs, Etc.: From Accuracy to Akrasia.” In: Episteme , pp. –. Coates, A. (). “Rational Epistemic Akrasia.” In: American Philosophical Quarterly (), pp. –. Conee, E. and R. Feldman (). “Evidentialism.” In: Philosophical Studies , pp. –. Conee, E. and R. Feldman (). Evidentialism, Oxford University Press. DiPaolo, J. (). “Higher-Order Defeat is Object-Independent”. In: Pacific Philosophical Quarterly. Early view. Dorst, K. (this volume). “Higher-Order Uncertainty.” In: M. Skipper and A. Steglich-Petersen (eds), Higher-Order Evidence: New Essays, Oxford University Press. Elga, A. (). “The Puzzle of the Unmarked Clock and the New Rational Reflection Principle.” In: Philosophical Studies (), pp. –. Greco, D. (). “A Puzzle about Epistemic Akrasia.” In: Philosophical Studies , pp. –.

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Greco, D. and B. Hedden (). “Uniqueness and Metaepistemology.” In: The Journal of Philosophy , pp. –. Hintikka, J. (). Knowledge and Belief, Cornell University Press. Hintikka, J. (). “Some Main Problems of Deontic Logic.” In: R. Hilpinen (ed.), Deontic Logic: Introductory and Systematic Readings, Studies in Epistemology, Logic, Methodology, and Philosophy of Science , Springer Dordrecht, pp. –. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs , pp. –. Horowitz, S. and P. Sliwa (). “Respecting All the Evidence.” In: Philosophical Studies , pp. –. Kelly, T. (). “The Epistemic Significance of Disagreement.” In: J. Hawthorne and T. Gendler (eds), Oxford Studies in Epistemology , Oxford University Press, pp. –. Kelly, T. (). “Evidence.” In: E. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Winter  Edition). Klein, P. (). Certainty: A Refutation of Skepticism, University of Minnesota Press. Klein, P. (). “Skepticism and Closure: Why the Evil Genius Argument Fails.” In: Philosophical Topics , pp. –. Koons, R. (). “Defeasible Reasoning.” In: E. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Summer  edition). Lange, M. (). “Calibration and the Epistemological Role of Bayesian Conditionalization.” In: The Journal of Philosophy , pp. –. Lasonen-Aarnio, M. (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research , pp. –. Lasonen-Aarnio, M. (forthcoming). “Enkrasia or Evidentialism? Learning to Love Mismatch” In: Philosophical Studies. Lasonen-Aarnio, M. (this volume). “Higher-Order Defeat and Evincibility.” In: M. Skipper and A. Steglich-Petersen (eds), Higher-Order Evidence: New Essays, Oxford University Press. Levi, I. (). The Enterprise of Knowledge, MIT Press. Littlejohn, C. (). Justification and the Truth-Connection, Cambridge University Press. Littlejohn, C. (). “Stop Making Sense? On a Puzzle about Rationality.” In: Philosophy and Phenomenological Research. Online First. Pollock, J. (). Knowledge and Justification, Princeton University Press. Rosenkranz, S. and M. Schulz (). “Peer Disagreement: A Call for the Revision of Prior Probabilities.” In: Dialectica , pp. –. Schechter, J. (). “Rational Self-Doubt and the Failure of Closure.” In: Philosophical Studies , pp. –. Schoenfield, M. (). “A Dilemma for Calibrationism.” In: Philosophy and Phenomenological Research , pp. –. Schoenfield, M. (). “An Accuracy Based Approach to Higher Order Evidence.” In: Philosophy and Phenomenological Research (), pp. –. Shah, N. (). “A New Argument for Evidentialism.” In: The Philosophical Quarterly , pp. –. Silva, P. (). “How Doxastic Justification Helps Us Solve the Problem of Misleading Higher-Order Evidence.” In: Pacific Philosophical Quarterly (), pp. –. Skipper, M. (forthcoming). “Reconciling Enkrasia and Higher-Order Defeat.” In: Erkenntnis. Steglich-Petersen, A. (). “Epistemic Instrumentalism, Permissibility, and Reasons for Belief.” In: Conor McHugh, J. Way, and D. Whiting (eds), Normativity: Epistemic and Practical, Oxford University Press, pp. –. Steglich-Petersen, A. (this volume). “Higher-Order Defeat and Doxastic Resilience.” In M. Skipper and A. Steglich-Petersen (eds), Higher-Order Evidence: New Essays, Oxford University Press.

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Titelbaum, M. (). “Rationality’s Fixed Point (Or: In Defense of Right Reason).” In: T. Gendler and J. Hawthorne (eds.), Oxford Studies in Epistemology , Oxford University Press, pp. –. Williamson, T. (). Knowledge and its Limits. Oxford University Press. Williamson, T. (). “Improbable Knowing.” In: T. Dougherty (ed.), Evidentialism and its Discontents, Oxford University Press. Williamson, T. (). “Very Improbable Knowing.” In: Erkenntnis (), pp. –. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research (), pp. –. Worsnip, A. (this volume). “Can Your Total Evidence Mislead About Itself?” In M. Skipper and A. Steglich-Petersen (eds), Higher-Order Evidence: New Essays, Oxford University Press. Zhao, J., V. Crupi, K. Tentori, B. Fitelson, and D. Osherson (). “Updating: Learning versus Supposing.” In: Cognition , pp. –.

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10 Higher-Order Defeat and Doxastic Resilience Asbjørn Steglich-Petersen

. Introduction It seems obvious that when higher-order evidence makes it rational for one to doubt that one’s own belief or credence on some matter is rational, this in itself can undermine the rationality of that belief or credence. For example, evidence that I suffer from the common self-enhancement bias undermines the rationality of my belief that I am a better than average cook. This phenomenon is known as higherorder defeat. However, despite its intuitive plausibility, it has proved puzzling how higher-order defeat works, exactly. To highlight two prominent sources of puzzlement, higher-order defeat seems to defy being understood in terms of conditionalization, since higher-order evidence does not affect the probability of the contents of the beliefs it undermines. And higher-order defeat can place agents in what seem like epistemic dilemmas, when the first- and higher-order evidence pull in different directions. In this chapter, I try to make progress on these issues by drawing attention to an overlooked aspect of higher-order defeat, namely that it can undermine the resilience of one’s beliefs. This aspect has been noted briefly by Andy Egan and Adam Elga (), but has not yet received systematic treatment.¹ The notion of resilience was originally devised to understand how one should reflect the ‘weight’ of one’s evidence in one’s beliefs. But I argue that it can also be applied to understand how one should reflect one’s higher-order evidence. The idea is particularly useful for understanding cases where one’s higher-order evidence indicates that one has failed in correctly assessing the evidence, without indicating in what direction one has erred, that is, whether one has over- or underestimated the degree of support for a proposition from one’s evidence. But as I shall argue, it is exactly in such cases that the puzzles of higher-order defeat seem most compelling.

¹ Roger White () and Alex Worsnip () both discuss the significance of resilience for how to respond to epistemic disagreement. However, while they focus on the resilience of one’s estimate of one’s own (and one’s interlocutor’s) reliability, and the significance of this for epistemic disagreement, I am interested in the impact of higher-order evidence on the resilience of one’s first-order credences. Asbjørn Steglich-Petersen, Higher-Order Defeat and Doxastic Resilience In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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Here is how I will proceed. In section ., I propose that in some cases it can be rational to maintain one’s level of credence despite rational doubt that the credence is rational. In section ., I argue that in such cases, one should respond by lowering the resilience of one’s credence, rather than its level, and sketch how the relevant changes in resilience can be understood in terms of changes in an agent’s higherorder credences, which are directly constrained by one’s higher-order evidence. In section ., I discuss why higher-order doubt undermines credences and categorical beliefs in importantly different ways, and how the idea of resilience-defeat can explain this. In section ., I show how the proposed account can help resolve the puzzles concerning conditionalization and epistemic dilemmas. Section . is a brief summary.

. Rational credence in the face of rational doubt I will begin by considering some cases where it seems rationally permissible for one to maintain a certain level of credence in a proposition, even in light of rational doubt that this is the right level of credence to hold in one’s epistemic situation. Consider the following case, described by David Christensen (a, p. ): Doubtful Ava: ‘[ . . . ] Ava is considering the possibility that (D) the next U.S. President will be a Democrat. She gives D some particular credence, say .; this reflects a great deal of her general knowledge, her feel for public opinion, her knowledge of possible candidates, etc. But given the possibility that her credence is affected by wishful thinking, protective pessimism, or just failure to focus on and perfectly integrate an unruly mass of evidence, Ava very much doubts that her credence is exactly what her evidence supports.’ As Christensen notes, it seems ‘eminently reasonable’ for Ava to doubt that her level of credence is exactly right, but this in itself doesn’t seem to undermine the rationality of her maintaining that level of credence. That is not to say, of course, that Ava’s credence could not be irrational for other reasons. Perhaps she really did misinterpret her evidence, thus making her credence irrational in light of her first-order evidence alone. But the fact that she harbours rational doubt that her credence is rational in light of the evidence should not in itself move her to revise it. Similar examples are easy to come by. Consider the following case, inspired by Kahneman and Tverski’s famous discussion of the base rate fallacy (): Doubtful Bob: A taxicab has been involved in a hit and run accident at night, and Bob has been called to serve on a jury in court, where a driver from Blue Cabs stands accused. There are just two taxi companies in town, Green Cabs being the dominant one. Only % of the taxis in town are blue, % are green. A witness saw the incident, and identified the cab as blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colours % of the time, and failed % of the time. Bob is carefully considering how likely it is that the cab was blue rather than green in light of the available evidence, and ends

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up giving it credence .. However, Bob always found probabilities tricky, and his grasp of Bayes’ Rule is pretty hazy. He thus severely doubts that his credence is right given the evidence. In fact, he would not be very surprised if he is quite far off the mark. Again, given Bob’s insight into his own imperfect understanding of probabilities, it seems reasonable for Bob to doubt the correctness of his credence. But again, this does not seem by itself to make it irrational for Bob to maintain this credence. He has considered his evidence carefully, applied principles of probabilistic reasoning to the best of his ability, and can’t make the result come out any other way. So if his doubt is to have an impact on his credence, it is unclear in what direction he should revise it. It therefore doesn’t seem as if his doubt by itself should make Bob adopt a different level of credence. Not all cases of rational higher-order doubt allow one to maintain one’s credence. Indeed, the kind of cases that epistemologists have tended to focus on do not seem to allow this. Immediately after presenting the case I’ve called Doubtful Ava, Christensen goes on to present a case of higher-order doubt that clearly does require a revision of credence (a, p. ). Here, Brayden, a staunch Republican, also initially gives credence . to D, but then receives compelling evidence of his own tendency to become irrationally confident of unpleasant possibilities, and to never underestimate them. This makes it rational for Brayden to believe that his credence is irrationally high, which plausibly means that he should lower his credence in D. The difference between this case and Doubtful Ava, Christensen notes, is that while Ava’s higher-order evidence of her own fallibility does not ‘lopsidedly’ suggest that her credence deviates from what is rational in one particular direction, Brayden’s does. Most cases of higher-order defeat discussed in the literature are lopsided in this way. Cases of revealed peer-disagreement, for example, where two peers discover that they differ in the level of credence they have adopted in response to the same evidence, seem to have this property. By contrast, I want to focus, at least initially, on the ‘non-lopsided’ cases, like Doubtful Ava and Doubtful Bob. As will become clear, I think that understanding the kind of defeat involved in these cases is helpful for understanding higher-order defeat in general. The plausibility of such cases does not depend on exactly how much doubt in the correctness of their own credence it is rational for Ava and Bob to have. The cases are not very specific in this regard, and it seems that a relatively generous range of doubt, from slight to quite severe, leaves the intuition that it doesn’t require a revision of credence intact. Ava ‘very much doubts’ that her credence is right, yet it is rationally permissible for her to maintain it. The same is the case for Bob. This leaves it possible there is an upper limit to how much doubt in one’s own credence rationality allows. I will return to that question later. But rationality allows at least a relatively high degree of doubt. However, even if Ava and Bob aren’t rationally required to revise their level of credence, it seems that their doubt ought to be reflected somehow in their credences. But how? The proposal that I want to explore in the following is that while Ava and Bob may maintain their level of credence, their credences should become less resilient as a result of their rational doubt. I now turn to introduce that idea.

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

ø -

. Doxastic resilience and higher-order doubt The notion of resilience was originally introduced to explain how evidential weight should be reflected in one’s credences. Keynes introduces the notion of evidential weight as follows: As the relevant evidence [for a hypothesis] at our disposal increases, the magnitude of [its] probability may either decrease or increase, according as the new knowledge strengthens the unfavourable or favourable evidence; but something seems to have increased in either case—we have a more substantial basis on which to rest our conclusion. [ . . . ] New evidence will sometimes decrease the probability of [the hypothesis] but will always increase its ‘weight’. (Keynes , p. )

To illustrate this idea, suppose that you have found a coin left behind by a deceased cardsharp.² You have some reason to suspect that the coin is biased, but you are not sure of this, and have no idea to what side it is biased, if it is. You can now imagine tossing the coin a number of times, and consider what credence you should adopt that the th toss will yield heads, given various series of prior results. Before tossing the coin at all, you have very little information, but the little you have points equally strongly towards heads and tails on the th toss. It thus seems that in that situation, you should adopt credence . in heads. Suppose now that you have tossed the coin  times, and that the results are divided evenly between heads and tails. Still, it seems that you should have a credence of . in the th toss landing heads. However, something has obviously changed: you now have a much weightier basis for your credence. How should this increased weight be reflected in your credence, if not in its level? One plausible answer is that it should be reflected in what Brian Skyrms () and others have called its degree of ‘resilience’, which can be understood as measuring how much the level of credence should change in the face of additional data. Imagine, for example, that we add a series of five heads to the two evidential situations above. Adding this series to the situation where you haven’t yet observed a single toss should clearly have a large impact on your credence that the th toss will land heads. You should move from . to something much closer to . But adding it to the situation where you have already observed  tosses should make a relatively modest impact. Your level of credence in this latter situation is thus much more resilient than in the first.³ Although the idea of resilience was originally introduced to explain how one should reflect the weight of one’s evidence, it seems that other factors besides weight should have a similar effect. Egan and Elga (, pp. –) observe that when one’s credence is based on how reliable one takes some channel of information to be, changes in this reliability should be reflected not only in the level of one’s credence, but also in its resilience. Suppose that you seem to remember that the person across the room is called Sarah. If you rationally regard your memory for names as %

² This example originates in Popper’s discussion of the paradox of ideal evidence (); this particular version is inspired by one found in Hansson (). ³ For a helpful discussion of the impact of evidential weight on doxastic resilience, see Joyce ().

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reliable, you should adopt a credence of . that the person’s name is indeed Sarah. If it becomes rational for you to regard your memory as less reliable, you should adjust your credence in its deliverances accordingly. For example, were you to become rationally convinced that your memory is only % reliable, your credence should be ., etc. But this reduction of trust in your own memory should have a further effect: it should make your memory-based credences less resilient. To show this, Egan and Elga consider what would happen if you overheard someone else calling the person across the room ‘Kate’ instead of ‘Sarah’, and you rationally regard this person as % reliable in remembering names. In the situation where you regard yourself as % reliable, this experience should make you adjust your credence only moderately, to .. But in the situation where you regard yourself as % reliable, the reduction you should make is rather drastic, namely to ..⁴ So reduced trust in your channel of information should affect both your level of credence, and the sensitivity of this credence to new information, that is, its resilience. This indicates that resilience is a property that should be affected by several different factors, evidential weight and reliability of one’s informational source being two examples. I now want to argue that changes in rational doubt about the correctness of one’s level of credence in light of the evidence should have a similar effect on the resilience of one’s credence. To give some initial motivation for this idea, consider the following variation of Doubtful Ava: Confident Eve: Eve is considering the possibility that (D) the next US President will be a Democrat. She gives D some particular credence, say .; this reflects a great deal of her general knowledge, her feel for public opinion, her knowledge of possible candidates, etc. As Eve is well aware, she is even-minded and very experienced in integrating unruly bodies of evidence, and she is therefore rationally highly confident that her level of credence reflects what her evidence supports. While it seems rationally permissible for both Ava and Eve to maintain credence . in D, it seems plausible that Ava’s credence should be less resilient than Eve’s in the light of new evidence that they might receive. Suppose, for example, that Ava and Eve both receive some new evidence which clearly speaks against D, but not decisively so, namely that a heightened threat of terrorist attacks has moved some voters to prefer a more hawkish national security policy, traditionally associated with the Republicans. For simplicity we can assume that both Ava and Eve correctly understand the significance of this new evidence, and that they are both rationally confident of what this is. How should their level of credence in D change once they add the new evidence? Here it seems plausible that Doubtful Ava should move to a lower credence than Confident Eve. Ava, after all, was unsure, and was reasonable to be unsure, that she was correct in taking the original evidence to speak in favour of D as much as her credence reflected. So when she becomes rationally convinced that the new evidence

⁴ As they note, this assumes that independently of your memory impressions, you regard the two names as being equally likely to be correct (Egan & Elga , p. ).

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ø -

speaks against D, this should pull her a fair bit in the other direction. Eve, on the other hand, was rationally convinced that she was correct in taking the old evidence to support her credence, so she can place more weight on the old evidence when taking the new evidence into account. A similar verdict seems plausible if we compare Doubtful Bob to a more confident juror, Bill: Confident Bill: A taxicab has been involved in a hit and run accident at night, and Bill has been called to serve on a jury in court, where a driver from Blue Cabs stands accused. There are just two taxi companies in town, Green Cabs being the dominant one. Only % of the taxis in town are blue, % are green. A witness saw the incident, and identified the cab as blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colours % of the time, and failed % of the time. Bill is carefully considering how likely it is that the cab was blue rather than green in light of the available evidence, and ends up giving it credence .. Bill always loved probability puzzles, and is a long time student of Bayesian reasoning. He is thus rationally very confident that his credence is right given the evidence. Again, although their different levels of rational doubt allow both Bob and Bill to maintain the same level of credence, it seems plausible that their credences should differ in sensitivity to new evidence. Suppose that the prosecutor introduces a new piece of evidence: a blue paint trace from the crime scene. The paint is of a common kind, and it cannot be dated with accuracy, so the new evidence is not conclusive. But it certainly does support to some degree that the guilty cab was blue. Suppose that both Bob and Bill become rationally convinced of the significance of this new evidence. How should they react? It seems plausible that Bob should revise his credence more than Bill should. Prior to receiving the new evidence, they both had a relatively low credence that the cab was blue, but only Bill was rationally confident that this low credence adequately reflected his evidence. Bill should thus place more weight on this prior evidence, than Bob should, and thus be moved less in the direction suggested by the new evidence. In these examples, the difference in resilience showed up in how much the relevant credences should be affected by new evidence bearing directly on the object-level proposition, that is, that the next president will be a Democrat, and that the cab at the crime scene was blue. But it seems that there should be a similar difference in resilience in the face of new higher-order evidence, for example in the form of revealed disagreement. Suppose, for example, that Ava and Eve both meet another person, Adam, that after careful consideration of their evidence judges it to support a credence in D of .. Insofar as Ava and Eve place at least some confidence in Adam’s ability to assess the evidence, they should both reduce their level of credence to some degree. But again, it seems that Ava should move her credence more than Eve in the direction of Adam’s. So the lower resilience of Ava’s credence should manifest itself in sensitivity to new evidence of both first- and higher-order (more on such cases below). Obviously, if differences in how rationally confident we should be in our own assessments of the evidence make a difference to how resilient our credences ought to

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be, higher-order evidence bearing directly on how confident we should be in our own assessments will normatively constrain the resilience of our credences. If Doubtful Ava acquires compelling evidence that she is actually less prone to bias and wishful thinking than she thought, she should become more confident in her assessment of the evidence, and her credence should thus become more resilient. And if Confident Eve acquires compelling evidence that she is actually more prone to bias and wishful thinking than she thought, she should lower her confidence in her own assessment of the evidence, and her credence should thus become correspondingly less resilient. If we think of higher-order evidence as evidence that concerns how rationally one has responded to one’s first-order evidence, this suggests a new normative role for higher-order evidence, namely as something that should affect the resilience of one’s credences. More specifically, our higher-order evidence constrains how confident we ought to be that our first-order credences correctly reflect our evidence. It constrains, in other words, one’s ‘higher-order credences’ about the correctness of one’s first-order credences. And how confident we ought to be that our first-order credences correctly reflect our evidence in turn constrains how resilient those firstorder credences ought to be. How exactly do rational higher-order credences constrain the resilience of the first-order credences they take as their object? This can be seen by reflecting further on cases of epistemic disagreement.⁵ As mentioned above, the resilience of a firstorder credence that p can be a matter of sensitivity to both first-order evidence bearing directly on p, and higher-order evidence indicating that a level of credence different from one’s own better reflects the first-order evidence. Begin with the latter kind, that is, sensitivity of a first-order credence in p to new higher-order evidence concerning the correctness of that credence. This sort of case is familiar from the literature on epistemic disagreement. Suppose that you have adopted some particular level of credence in response to some evidence, and then meet someone who has adopted a different level of credence in response to the same evidence. You thus disagree about what credence your shared evidence supports. How should you react? That depends on how reliable it is rational for you to regard yourself and your friend as in assessing the evidence, that is, how likely you are rational to think it is for each of you to have assessed the evidence correctly. Other things being equal, the more likely it is for you that you have evaluated the evidence correctly, the less you should revise your credence in the direction of your friend’s, and vice versa. So rational higher-order credences about how well your first-order credences reflect your evidence affect how sensitive those credences should be to new higher-order evidence, that is, their resilience in the face of such evidence. But how can sensitivity to new first-order evidence bearing directly on the objectlevel proposition be understood in terms of higher-order credences? Again, cases of disagreement provide a useful model, although this time it requires taking a step back. Philosophers have tended to focus on cases where the disagreeing parties have responded differently to the same shared body of evidence, but it seems that parties to ⁵ I assume here a broadly conciliatory approach to disagreement, such as that defended by, e.g., Christensen (). For a precise probabilistic account of how peer-disagreement can be understood in terms of higher-order evidence, see Rasmussen et al. ().

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ø -

a disagreement should respond in more or less the same way if their evidence isn’t identical, but merely on par. Christensen () considers a case where I have good reason to believe that my friend’s evidence with respect to p, although different from mine, is just as good. But whereas his evidence supports his credence in p of ., my evidence supports my credence of .. For simplicity, let us assume that the two bodies of evidence are on par with respect to evidential weight. The effect on resilience generated by weight should thus be the same for both bodies of evidence. As Christensen observes, it seems that even if we have responded to different bodies of evidence, we should both revise our credences in the direction of the other. Furthermore, in such cases, changes in cognitive parity should affect what the disagreeing parties should do, in the same way as in cases where the parties have the same evidence. If I discover that I am in fact my peer’s cognitive superior, I should revise less in his direction, and vice versa, even if we have responded to different bodies of evidence. In fact, this seems to hold even when the evidence possessed by the disagreeing parties is not only non-identical, but also not on a par. When my evidence with respect to p is better than my disagreeing friend’s evidence, I have some reason in virtue of that to favour my own level of credence in p over my friend’s. But I still cannot completely disregard my friend’s assessment of his evidence, and should thus revise in his direction, albeit less so than if his evidence had been as good as mine. Conversely, when my friend’s evidence is better than mine, I have reason to favour his level of credence, but needn’t completely disregard my own. And again, even in situations such as these, changes in cognitive parity should matter to how much I should revise. Even in cases where my evidence is better or worse than my friend’s, I should revise less if I discover that I am his cognitive superior, and more if I discover that he is cognitively superior to me. With this in mind, return now to the question of sensitivity to new first-order evidence. Suppose that I have adopted credence . in p in response to one body of evidence E, and that I rationally regard myself as highly reliable in evaluating E. I now get some new evidence bearing on p, E, which is on a par with E in terms of evidential weight. Again, I rationally regard myself as highly reliable in evaluating E, and judge that this evidence, in isolation, supports a credence of . in p. What credence in p should I adopt in light of my total evidence? The conflicting bodies of evidence are on a par, and I rationally regard myself as equally reliable in evaluating both. So it seems that by reasoning parallel to that used in cases of epistemic disagreement, I should adopt a credence roughly in the middle, that is, ..⁶ But what if I rationally regard myself as being less reliable in evaluating E than E, perhaps because of the mathematics involved in that body of evidence, and thus rationally regard it as less likely that I have evaluated E correctly than E? I would then seem to find myself in a situation parallel to the case of disagreement with evidential parity but cognitive non-parity. In that case, I should thus adopt a credence closer to that which I judge to be supported by E alone, than that which I judge to be supported by E alone. On the other hand, if I rationally regard myself as more

⁶ This assumes, of course, that the two bodies of evidence are independent, and that there are no undercutting defeating relations between them.

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reliable in evaluating E than E, I should end up with a credence to closer to that which I judge to be supported by E alone. Again, as in the case of disagreement, this can be generalized beyond cases with parity between conflicting bodies of evidence. Regardless of whether E and E are on a par as evidential bodies, ceteris paribus I should end up with a credence closer to, or farther from, that which I judge to be supported by E alone, depending on how reliable I rationally think I am at assessing the import of E. So it seems that cases of epistemic disagreement with non-identical evidence provide a model for how to understand the resilience of first-order credences in the face of new object-level evidence, in a way that relies on rational higher-order credences about one’s reliability in assessing the respective bodies of evidence. The resilience of a first-order credence in the face of new first-order evidence is affected by how reliable one regards oneself as being in assessing the new and old evidence, in roughly the same way as the sensitivity of one’s credence in cases of disagreement depends on how reliable one thinks that oneself and the disagreeing friend are. The resulting picture is that higher-order evidence of one’s own rational failure undermines the resilience of one’s first-order credences, by undermining how likely it is rational for one to think that those credences correctly reflect the evidence on which they are based. This in turn makes those credences more sensitive to both higher-order evidence that a different credence level is correct, as in cases of disagreement, and to new first-order evidence supporting a different credence level; in other words, it makes those credences less resilient. So far, I have focused on cases of higher-order defeat which allow one to maintain one’s level of credence. I have argued that in such cases, the defeat should be understood as undermining the resilience of one’s credence, instead of its level. But not all cases of higher-order defeat are like that. In some cases, the evidence not only lowers the probability that one’s credence correctly reflects the evidence, but also ‘lopsidedly’ indicates in what direction one has erred. This is the case in Christensen’s example of Brayden the Republican, and in cases of epistemic disagreement. Both kinds of higher-order defeat can be understood in terms of the effect they have on one’s higher-order credences about what first-order credence best reflects the evidence. In lopsided cases, the higher-order evidence makes it more likely for one that another credence is correct in light of one’s evidence, which means that one must revise one’s level of credence. How much one should revise depends on how probable it is for one that one’s original level of credence was correct, as well as how probable the higher-order evidence makes it that another level is correct.⁷ In non-lopsided cases, defeating higher-order evidence makes it less likely for one that one’s credence is right in light of one’s first-order evidence, but does not make another level of credence more likely to be correct than the original one. It ‘spreads out’ or ‘flattens’ one’s higher-order probability distribution, without pulling its highest point in a particular direction. This means that what’s defeated is not the level of one’s first-order credence, but only its resilience. In general, then, higher-order defeat works by affecting one’s higher-order probabilities, but only in some cases does this require a change in credence.

⁷ For a detailed account of this, see Rasmussen et al. ().

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. Resilience defeat and categorical belief In my discussion so far, I have focused on the effects of higher-order doubt and evidence on credences or graded beliefs. One might reasonably wonder how well the above account applies to categorical beliefs. On the face of it, it may seem as if rather little applies. Consider the kind of cases where it seemed plausible that some particular level of credence in a proposition could be rationally combined with a fairly high degree of doubt that this level of credence was correct in light of the evidence. Could similar cases be thought of with categorical beliefs? That is, could it be rational to believe that p outright while harbouring a high degree of rational doubt that it is rational or justified to believe that p in light of one’s evidence? That seems much less plausible. It may well be possible to rationally believe that p while having some relatively low degree of rational doubt about this belief, but the room for this seems much more limited. This difference ought to strike us as puzzling. Both credences and categorical beliefs must be based on adequate evidence in order to be rational or justified. So why can it be rationally permissible to maintain a credence in the face of rational doubt about its rationality, when this isn’t permissible with categorical beliefs? In addition to being a puzzling difference in itself, something that we ought to try to explain, the difference might also be seen to undermine the picture that I have proposed of higher-order defeat. This is because there is usually thought to be a tight connection between one’s beliefs and credences. On one popular account of the relation, categorical belief simply is a suitably high degree of credence.⁸ The picture suggested above allows one to maintain one’s credences in the face of rational doubt about those very credences. So if categorical belief is simply a high degree of credence, my proposal would implausibly allow categorical belief in the face of higher-order doubts too.⁹ Fortunately, there is a plausible alternative theory of the relation between credence and categorical belief that allows my account to both explain the puzzling difference, and defuse the challenge it threatens to raise. This is the ‘stability theory’ of belief, developed in recent writings by Hannes Leitgeb (; ; ), and attributed by him and others to David Hume, as an early proponent.¹⁰ According to the stability theory, a high credence in p is necessary but not sufficient for belief that p. In order to count as a belief, the high credence must also be suitably ‘stable’, which enables belief to ‘play its characteristic functional role in decision-making, reasoning, and asserting [ . . . ] in the course of processes such as perception, supposition and communication’ ⁸ This is what Richard Foley identifies as the ‘Lockean Thesis’ (, ch. ). For further discussion of this and other accounts of the relation between credence and categorical belief, see Keith Frankish (). ⁹ The tension between the present theory of higher-order defeat and the Lockean thesis may be less severe than this suggests, for two reasons. First, if categorical belief requires a high degree of credence, it may be hard to find realistic cases where resilience is lowered dramatically while leaving the credence level intact or above the level required for belief. This is because rational doubt that a high credence level is correct leaves more room for error in the ‘too high’ direction than in the ‘too low’ direction. Second, in some cases where it is rational for to doubt that one’s credence level is correct, the margin of error may be too small to reach below the threshold for belief. I am grateful to David Christensen for these points. ¹⁰ For this interpretation of Hume’s account of belief, see in particular Louis Loeb (). For Hume’s own presentation, see his (–: Bk I, Pt III, §).

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(Leitgeb , p. ). Leitgeb understands the stability of a credence as a matter of the credence being preserved under conditionalization, that is, under the supposition of new evidence or information (, p. ). In other words, stability is a matter of resilience, in roughly the same sense as I have used that term above. I will not specify or motivate the stability theory of belief further here.¹¹ What interests me is how the theory combines with the above theory of higher-order defeat. The difference between high credence and categorical belief implies that the two states are subject to different rationality constraints. While it can be rational to have a high credence without it being rational for this credence to be very resilient, the rationality of categorical belief requires the rationality of both a high credence and a high degree of resilience. This is what explains why a high credence in a proposition is rationally compatible with a high degree of rational doubt in that credence being correct, while this is not possible for categorical belief. A high degree of rational doubt about the correctness of one’s own credence would exclude the rationality of the credence being sufficiently stable, and would thereby exclude the rationality of what is necessary for outright belief, namely stable high credence. This also defuses the challenge raised by the puzzling difference between credences and beliefs, for my account of higher-order defeat. Indeed, if rational belief requires rational stability, while rational high credence does not, and higher-order evidence sometimes undermines rational stability without undermining a high degree of credence, we should expect this apparently puzzling difference. Rather than undermining it, the difference between credences and beliefs in sensitivity to higher-order doubt thus becomes a point in favour of the proposed account.

. Applications Thinking of higher-order evidence and defeat as something that undermines the resilience of one’s first-order credences can help explain a number of puzzling features of higher-order evidence and defeat. Here, I want to focus on two such puzzling features, concerning (i) belief revision by conditionalization, and (ii) how one should respond to misleading higher-order evidence.

.. Higher-order evidence and conditionalization The first problematic feature of higher-order evidence is, in short, that it seems to give rise to mismatches between the level of confidence one should adopt in a proposition on the supposition that one acquires certain evidence and the confidence one should adopt in the proposition if one actually acquires that evidence. Consider the following example adapted from Christensen (b). A scientist is contemplating the confirmation that some currently unrealized experimental result E would provide for a hypothesis H, and judges correctly that it would provide a high degree of confirmation. He thus regards it as highly probable that H is true on the supposition that E. This makes him judge that if he actually ¹¹ I see no reason to suppose that the stability theory is incompatible with the teleological account of belief that I have defended elsewhere (e.g., Steglich-Petersen ; ).

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

ø -

learns that E, he should become highly confident that H is true. However, the scientist also considers how probable H would be on the supposition of E along with a further factor D: that he has been given a powerful reason-distorting drug prior to assessing his experimental results. Reasonably, the scientist realizes that this addition should have no effect whatsoever on the probability of H. E confirms H to the same degree, whether or not he, or anyone else for that matter, happens to be unable to rationally assess this. So he regards it as highly probable that H is true on the supposition of E&D. But here’s the catch: even though the scientist is convinced of this conditional probability, he should not resolve to become highly confident that H is true if he actually learns E&D. In fact, if he were to learn E&D, he should not be very confident of H at all. So we now appear to have a puzzling mismatch between the scientist’s prior conditional probability, and how confident he should become upon acquiring the evidence, thus contradicting the standard updating model. By understanding higher-order defeat as something that, at least in non-lopsided cases, undermines the resilience of a credence, rather than its level, we can uphold the standard updating model. Suppose that the scientist assesses the prior probability of H given E to be .. If he were to learn E, he should thus adopt credence . in H. As we have supposed, the scientist ascribes an identical probability to H given E&D. He should thus also adopt credence . in H if he were to learn E&D. But if he were to learn E&D, he should become much less confident that . is the right level of credence. As we have already seen, this is not necessarily irrational. Having a particular level of credence can be rationally combined with a high degree of doubt in that level being correct. This, however, should make the credence less resilient, that is, more susceptible to being revised in the face of new evidence. The higher-order evidence is thus allowed to make its mark on the scientist’s credence without giving up conditionalization. If we understand the reduced confidence in one’s credence being correct in terms of higher-order credences, we can understand that too in terms of conditionalization. But what is the relevant prior conditional probability? As we have already seen, D does not affect the probability of H. Does D somehow affect the prior conditional probability of H given E? That is, does the fact that the scientist is drugged somehow affect how probable the hypothesis is in light of the experimental result, for example by making it less likely that that conditional probability is of a certain level? Again, the answer is clearly no. But something that D clearly does affect is the probability of the scientist correctly assessing the support for H given by E. More specifically, the scientist should regard it as less probable that the assessment he reaches of H given E is correct on the supposition of D, compared to the probability of that on the supposition of not-D. So now we can explain both the credence in H that the scientist should adopt, and the low resilience that credence should have, in terms of conditionalization. Consider first the scientist’s credence towards H. Prior to receiving the experimental result E, he holds the probability of H given E to be .. He also holds the probability of H given E&D to be .. So when he learns E&D he should adopt credence . in H. So how does learning D impact his credence? Call the proposition that the scientist will correctly assess the support for H given by E ‘C’. Prior to learning D, the scientist ascribes some relatively low probability to C given D, say .. So, when the scientist learns that D, he

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

rationally adopts credence . in C, which, given his actual assessment of H given E, amounts to having a credence of . that his credence of . in H is correct. As long as he regards a credence of . in H as being more likely to be correct than any other level of credence, this can be a rational combination of credences. It just means that his credence is H should not be very resilient. The result is that the scientist has revised by conditionalization in response to both E and D, without letting D affect the probability of H, while allowing D to make its mark on the credence in H by reducing its resilience. Here’s a possible objection. Above I characterized the problem as that of explaining the mismatch between how probable one should regard a proposition on the supposition of some evidence, and the ‘level of confidence’ one should adopt in the proposition if one actually acquires that evidence. But does the above really explain that? After all, if we by ‘level of confidence’ mean level of credence, my proposal does not deliver this result. Indeed, on my proposal, one’s posterior credence should match the prior conditional probability. However, I think that the intuition that one should become less ‘confident’ upon acquiring the higher-order evidence trades on an ambiguity in that notion. Having a high credence in a proposition is the typical understanding of what it means to be highly confident in that proposition. But if one is highly unsure about one’s level of credence being right, thus making the credence easy to affect by new evidence, it seems wrong to characterize having such a credence in a proposition as being ‘highly confident’ that the proposition is true, even if the level of credence is high. If so, my proposal does deliver the correct verdict. What about lopsided cases? If the challenge to the standard updating model persists for such cases, the above solution might not be much of a solution at all. For example, suppose that the reason-distorting drug is designed specifically to dispose people to overestimate the evidence and adopt a higher credence than the evidence actually warrants. Surely, learning that one has been given such a drug should lead one to reduce one’s level of credence. Yet, shouldn’t one’s prior conditional probability of H given E&D still match that of H given E alone? After all, whether one has been drugged has no bearing on the actual evidential bearing on E for H. If so, the mismatch between the prior conditional probability and the credence one should adopt upon acquiring the evidence seems to persist. However, it is not obvious to me that one’s prior conditional probability of H given E&D should be the same as that of H given E in lopsided cases. Consider the following lopsided case, where this seems particularly dubious. Instead of considering the probability of H given the experiment and being drugged, the scientist now considers the probability of H given the experiment and being told by his colleague Stephen Hawking that the probability of H given E is ., and not . as the scientist is independently inclined to judge. If learning this, the scientist should clearly adopt a credence a fair bit below .. But it seems equally clear that his prior conditional probability given this should be below . as well, thus retaining the match between the two probabilities. The scientist should clearly not be very confident that if he were to learn of the experiment and of Stephen Hawking’s testimony, then H would be true, despite his actual confidence in the support for H provided by E. But why think, then, that the above case with the overestimation-drug is any different? After all, Stephen Hawking’s testimony is also evidence that the scientist’s initial judgement overestimates the evidential import of the experiment.

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ø -

This suggests that what’s driving the intuition in Christensen’s original case of the reason-distorting drug is that in that case, the drug is not stipulated to have a lopsided effect. It is this fact that makes it plausible that having taken the drug doesn’t affect the conditional probability of H given E, while at the same time requiring a lowered confidence upon learning that one has taken it. But as argued above, this lowered confidence can be understood in terms of resilience, rather than credence level. If this is plausible, neither lopsided nor non-lopsided cases exhibit mismatches between the prior conditional probability and the level of credence one should adopt upon acquiring the evidence.

.. Misleading higher-order evidence and akrasia Another major source of interest in the nature of higher-order evidence stems from cases where it appears to undermine the otherwise highly plausible idea that it cannot be rational to hold a belief that one believes to be irrational, that is, that epistemic akrasia is always irrational. Despite the intuitive appeal of this, a number of authors have argued that in situations with sufficiently compelling misleading higher-order evidence, it can in fact be rational to hold an akratic combination of beliefs. For those who want to resist that conclusion, the challenge is to explain why akratic combinations of beliefs are not the correct way to reflect one’s evidence, but this has proven difficult. In this final section, I argue that resilience offers an attractive explanation. I will focus on a case of apparently rational epistemic akrasia developed by Allen Coates (), in which it is especially clear how resilience helps resolve the problem: Watson is an apprentice of Holmes, a master sleuth. As part of his training, Watson will often accompany Holmes to crime scenes and other locations, size up the evidence as best he can, and tell Holmes what conclusion he has drawn, and how he has drawn it. Holmes will then assess Watson’s conclusion as rational or irrational, though not as true or false. Of course, this assessment is based in part on whether Holmes thinks the evidence supports the conclusion. But just as a logic student may use invalid steps to arrive at a conclusion that follows validly from the premises, so too Watson may use poor reasoning to arrive at a conclusion that is nevertheless supported by the evidence. In such a case, Watson would be irrational in holding his conclusion, and Holmes will assess it accordingly. Thus, it is possible for Holmes to arrive at the same conclusion from the same evidence as Watson, and still claim that Watson’s belief is irrational. Watson is aware of this, and so he cannot infer from such a claim that Holmes thinks his conclusion is false. In fact, he cannot even infer that Holmes thinks that the evidence does not support his conclusion. All he can infer is that Holmes thinks that he has arrived at his conclusion irrationally. This is by design: Watson is to consider the evidence on his own until he arrives at a conclusion rationally, and his knowing Holmes’ own conclusion would interfere with this. Now suppose that Holmes brings Watson to a crime scene, that the evidence indicates that the butler is guilty, and that Watson uses good reasoning to arrive at that conclusion. In short, Watson rationally believes that the butler did it. But when he tells Holmes of his conclusion and how he arrived at it, Holmes’ only response is, ‘Your conclusion is irrational.’ Since Holmes is a master sleuth, Watson is justified in believing Holmes to be correct: Holmes’s testimony on these matters is very authoritative. But authoritative though he is, he is not infallible, and this is one of the rare occasions in which he is wrong. So when Watson accepts Holmes’ assessment, he accepts a falsehood. Watson, then, may reasonably but

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wrongly judge that his conclusion is irrational. Therefore, if he nevertheless maintains his belief in the butler’s guilt, both it and his epistemic judgment of it are rational. Yet, in holding them both, he is akratic. (Coates , p. )

Since both of Watson’s beliefs are rational in light of his evidence, this seems like a prima facie case of rational epistemic akrasia. And several authors do in fact accept this possibility, Coates included.¹² This, however, is highly counterintuitive, and leads to other problems concerning how one should reason from the akratic beliefs, what actions they rationalize, and more.¹³ But the alternatives to allowing rational akrasia are problematic as well. If Watson adopts the ‘steadfast’ approach, as some philosophers recommend, and believes that the butler did it and that his evidence supports this, Watson fails to acknowledge his higher-order evidence. But if Watson instead adopts the ‘conciliatory’ approach recommended by others, and becomes less confident that the butler did it and that his evidence supports this, he will fail to acknowledge his first-order evidence. So either way, Watson will fail to respect some of his evidence.¹⁴ How does resilience help us understand cases such as this? A crucial feature of the above case is that Holmes’s testimony isn’t evidence of the butler’s innocence. Holmes may judge Watson’s belief to be irrational, even if he thinks that it is true. As Coates observes, without this feature it would clearly not be rational of Watson to continue to believe in the butler’s guilt. The significance of this becomes clear when we consider Watson’s beliefs in terms of the underlying credences. Suppose that the evidence at the crime scene supports believing that the butler did it partly in virtue of supporting a credence in this above the threshold for categorical belief. After correctly assessing the evidence, Watson thus adopts the belief that the butler did it, partly in virtue of adopting a high credence in the butler’s guilt. When Watson hears Holmes’s testimony, this makes it rational for Watson to believe that his belief in the butler’s guilt is irrational. Again, it is natural to interpret this in terms of credence, namely as the belief that his high credence in the butler’s guilt is irrational. But since Holmes’s testimony isn’t evidence of the butler’s innocence, it doesn’t by itself support moving to a lower credence in the butler’s guilt. Recall that as the case is set up, Watson’s credence may be correct in light of the evidence, even if it is irrational. This makes it plausible that Watson may maintain his high credence, even after hearing Holmes’s testimony. However, Watson’s belief that his high credence is irrational should have the effect of lowering the resilience of that credence. It should dispose him to revise his credence more readily in light of new first-order evidence about the culprit, and in light of new higher-order evidence that, contrary to Holmes’s testimony, does suggest that a different level of credence is correct. But in that case, Watson’s credence will no longer qualify as a categorical belief, since this requires a certain level of resilience (assuming the stability theory of belief ).

¹² For other defenders, see, e.g., Wedgwood (), Lasonen-Aarnio (), and Williamson (). ¹³ For criticism, see, e.g., Horowitz (). ¹⁴ For the ‘steadfast’ approach to misleading higher-order evidence, see e.g. Titelbaum (). For the ‘conciliatory’ approach, see, e.g., Feldman (), Christensen (b), and Horowitz ().

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

ø -

In this way, Watson can respect both his first- and higher-order evidence, without it being akratic. His first-order evidence is correctly reflected in his high credence in the butler’s guilt. His higher-order evidence is correctly reflected in his low confidence in his own credence being rational in light of the evidence, and in the correspondingly low resilience of that credence. And it is not a case of akrasia because, while believing p is rationally incompatible with believing that believing p is irrational, it is not necessarily irrational to hold some level of credence while being doubtful that that level is rational. If Watson responds to his evidence as he ought to do, he is therefore not epistemically akratic. As in the discussion of conditionalization, this solution faces the potential objection that, while it may work in cases where the higher-order evidence doesn’t suggest that a different credence level is correct, it doesn’t work when this condition isn’t met, that is, in lopsided cases. But it is far from clear that such cases are problematic to begin with. Coates clearly thinks that his verdict that Watson’s akratic beliefs are rational depends on Watson not being able to infer from Holmes’s testimony that the butler is innocent. Had he been able to infer this, his belief that the butler is guilty would not be rational, and hence there wouldn’t be a puzzling case of rational akrasia. At least some other cases of supposedly rational epistemic akrasia in the literature appear to be non-lopsided in this way as well.¹⁵ But I shall have to leave detailed discussion of this objection for another occasion.

. Conclusion I have argued that higher-order defeat should sometimes be understood as undermining the resilience of one’s credences rather than their level. I showed how this integrates with a general picture of higher-order evidence as something that constrains one’s higher-order credences, and outlined how the proposal helps explain two puzzling features of higher-order defeat. Obviously much more can and must be said to develop and motivate this account. But I hope to have shown that the account is sufficiently promising to warrant further work.

Acknowledgements This chapter has been presented in Edinburgh, Aarhus, and Roskilde. I am grateful to those present for fruitful discussion. Thanks in particular to David Christensen, Mattias Skipper, and Jens Christian Bjerring, who read earlier drafts and provided helpful comments.

References Christensen, D. (). “Epistemology of Disagreement: The Good News.” In: The Philosophical Review (), pp. –. Christensen, D. (a). “Rational Reflection.” In: Philosophical Perspectives (), pp. –. ¹⁵ For example, Wedgwood () discusses higher-order evidence pointing to one’s own irrationality or incompetence, but not evidence supporting that a different inference or conclusion is in fact correct.

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Christensen, D. (b). “Higher-Order Evidence.” In: Philosophy and Phenomenological Research (), pp. –. Coates, A. (). “Rational Epistemic Akrasia.” In: American Philosophical Quarterly (), pp. –. Egan, A. and A. Elga (). “I Can’t Believe I’m Stupid.” In: Philosophical Perspectives (), pp. –. Feldman, R. (). “Respecting the Evidence.” In: Philosophical Perspectives (), pp. –. Foley, R. (). Working Without a Net, Oxford University Press. Frankish, K. (). “Partial Belief and Flat-Out Belief.” In: F. Huber and C. Schmidt-Petri (eds), Degrees of Belief, Springer, pp. –. Hansson, S. O. (). “Do We Need Second-Order Probabilities?” In: Dialectica (), pp. –. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs (), pp. –. Hume, D. (–/). A Treatise of Human Nature, L.A. Selby-Bigge (ed.), nd ed. revised by P.H. Nidditch, Clarendon Press. Joyce, J. (). “How Probabilities Reflect Evidence.” In: Philosophical Perspectives (), pp. –. Keynes, J. (). A Treatise on Probability, Macmillan. Lasonen-Aarnio, M. (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research (), pp. –. Leitgeb, H. (). “The Stability Theory of Belief.” In: The Philosophical Review (), pp. –. Leitgeb, H. (). “I—The Humean Thesis on Belief.” In: Aristotelian Society Supplementary Volume (), pp. –. Leitgeb, H. (). The Stability of Belief. How Rational Belief Coheres with Probability, Oxford University Press. Loeb, L. (). Stability and Justification in Hume’s Treatise, Oxford University Press. Popper, K. (). The Logic of Scientific Discovery, Hutchinson. Rasmussen, M. S., A. Steglich-Petersen, and J. C. Bjerring (). “A Higher-Order Approach to Disagreement.” In: Episteme , pp. –. Skyrms, B. (). “Resiliency, Propensities, and Causal Necessity.” In: The Journal of Philosophy (), pp. –. Steglich-Petersen, A. (). “No Norm Needed: On the Aim of Belief.” In: The Philosophical Quarterly (), pp. –. Steglich-Petersen, A. (). “Weighing the Aim of Belief.” In: Philosophical Studies (), pp. –. Titelbaum, M. (). “Rationality’s Fixed Point (Or: In Defense of Right Reason).” In: Oxford Studies in Epistemology , pp. –. Wedgwood, R. (). “Justified Inference.” In: Synthese (), pp. –. White, R. (). “On Treating Oneself and Others as Thermometers.” In: Episteme , pp. –. Williamson, T. (). “Very Improbable Knowing.” In: Erkenntnis (), pp. –. Worsnip, A. (). “Disagreement about Disagreement? What Disagreement about Disagreement?” In: Philosophers’ Imprint , pp. –.

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11 Return to Reason Michael G. Titelbaum

The argument of my “Rationality’s Fixed Point (or: In Defense of Right Reason)” () began with the premise that akrasia is irrational. From there I argued to a thesis that can be stated in slogan form as follows: Fixed Point Thesis (rough): mistakes of rationality.

Mistakes about the requirements of rationality are

The basic idea of the Fixed Point Thesis is that an agent who forms a false belief about what rationality requires thereby makes a rational error. I then applied the Fixed Point Thesis to cases of peer disagreement—cases in which two agents of equal reasoning abilities reason from the same evidence to opposite conclusions. I argued that if one of the agents has drawn the rationally required conclusion from that common evidence, it would be a rational mistake for her to withdraw that conclusion upon discovering the disagreement with her peer. The premise of “Right Reason” ’s argument, the thesis to which it leads, and the position on peer disagreement that follows have all been subsequently challenged in a number of ways. This chapter responds to many of those challenges. Section . clarifies how I understand rationality, and describes the Akratic Principle, according to which akratic states are rationally forbidden. It then explains more fully the argument for this principle I sketched in “Right Reason.” This indicates my response to those who would set aside the Akratic Principle in order to avoid its consequences. Section . provides a high-level gloss of my argument from the Akratic Principle to the Fixed Point Thesis. The discussion reveals the intuitive problem at the core of that argument, and highlights the argument’s generality. That generality becomes important in section ., which responds to authors who try to mitigate the argument’s consequences by distinguishing ideal rationality from everyday rationality, or rationality from reasonableness, or structural norms from substantive. Section . also takes up the suggestion that authoritative evidence for falsehoods about rationality creates a rational dilemma. Section . addresses the charge that my notion of rationality must be externalist or objectivist, because any internalist account of rationality would excuse an agent who’s unable to figure out what rationality requires. I show that this charge is misaimed against the Fixed Point Thesis, which is perfectly compatible with subjectivist/internalist accounts of rationality (including my own). A similar mistake occurs Michael G. Titelbaum, Return to Reason In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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

in peer disagreement debates, when critics of my position wonder how the agent who drew the rationally required conclusion before meeting her peer can determine that she was the one who got things right. The short answer is that she’s able to figure out what was rationally required after interacting with her peer because she was able to figure it out beforehand. The best response I’m aware of to this point can be reconstructed from Declan Smithies’s work on logical omniscience. The basic idea is that responding to a disagreeing peer brings new reasoning dispositions of the agent’s to bear, and those dispositions’ unreliability defeats the agent’s doxastic justification for her initial conclusion. In section . I carefully reconstruct this response, then offer some responses of my own. Ultimately, I find these challenges to the “Right Reason” position on peer disagreement unconvincing. Nevertheless, I believe that position requires revising. In “Right Reason,” I assumed that if peer disagreement could rationally change an agent’s attitude toward her original conclusion, it would have to do so by altering her stance on what the initial evidence required. I now see that, contrary to a common assumption in the literature, peer disagreement can rationally affect an agent’s opinions without providing any higher-order evidence. Section . provides some examples, and indicates how my position on peer disagreement must change as a result. Before we begin, I should admit that while the argumentation in “Right Reason” is slow, careful, and detailed, my approach here will be much more brisk and at times hand-wavy. I will not pause over details and caveats I dwelt on in the earlier piece. I hope that if you’ve read the other essay, this one will deepen your understanding, fill in some gaps, and improve the view. If you’re reading this chapter first, I hope it will encourage you to work through the fine print elsewhere.

. Rationality and the Akratic Principle The main premise of my argument is the Akratic Principle: No situation rationally permits any overall state containing both an attitude A and the belief that A is rationally forbidden in one’s situation. This requires a bit of unpacking. As I understand it, rationality involves an agent’s attitudes’ making sense from her own point of view. In this chapter, I will use “rational” as an evaluative term, applied primarily to an agent’s total set of attitudes (beliefs, credences, intentions, etc.) at a given time. I’ll call that set of attitudes the agent’s “overall state.” Whether a particular set of attitudes makes sense for an agent at a given time may depend on her circumstances at that time. The aspects of an agent’s circumstances relevant to the rationality of her overall state constitute what I’ll call her “situation.” One important way for an agent’s attitudes to fail to make sense is for them to stand in tension either with each other or with her situation. Such tensions or conflicts constitute rational flaws. When I say that an agent’s situation at a particular time rationally permits a particular overall state, this entails that if the agent possessed that overall state at that time it would contain no rational flaws. A situation requires a state when that state is the only rationally permitted state.

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An individual attitude is rationally permitted when at least one permitted state contains that attitude; an attitude is required when all permitted states contain it. The Akratic Principle says that no matter an agent’s situation, if she possesses both some attitude and the belief that that attitude is rationally forbidden, then her overall state is rationally flawed. Notice that there is no evaluation of the agent here. The question of whether an agent herself is rational or irrational is connected to the question of whether her set of attitudes is rationally flawless, but the connection is complex and indirect. An agent’s overall state may contain rational flaws without the agent’s thereby being criticizable or blameworthy. A great deal of interesting work (Smithies ; Greco ; Horowitz ; Littlejohn ; Worsnip ) has been done in recent years on why akratic states are rationally flawed. In “Right Reason,” I offered the following terse and obscure explanation: The Akratic Principle is deeply rooted in our understanding of rational consistency and our understanding of what it is for a concept to be normative. Just as part of the content of the concept bachelor makes it irrational to believe of a confirmed bachelor that he’s married, the normative element in our concept of rationality makes it irrational to believe an attitude is rationally forbidden and still maintain that attitude. The rational failure in each case stems from some attitudes’ not being appropriately responsive to the contents of others. (p. , emphases in original)

The argument I meant to gesture at is fairly simple, and not original to me.¹ But it may help to elaborate here. As I said a moment ago, a rational overall state lacks internal conflicts assessable from the agent’s own point of view. I also consider rationality a genuinely normative category: when an overall state violates rational requirements, there is something genuinely wrong with that state. (Here I invoke the familiar distinction between evaluations, prescriptions, etc. that are genuinely normative and those—like, say, the rules of Twister—that are norms in some sense but do not have deep normative force. If I fail to get my left hand on yellow, I have violated a norm of Twister but have not done anything genuinely wrong.) Now consider an agent who possesses both an attitude A and the belief that A is rationally forbidden to her. (I’ll refer to the content of that belief as “B.”) Since rationality is a genuinely normative category, B entails that there is something wrong with possessing A. So there is an internal tension in the agent’s overall state, and that state is rationally flawed. As I said, it’s a simple argument. It can’t be denied by disputing my position that internal tensions undermine a state’s rationality; that’s just how I use the term “rationality.” One could deny that the requirements of rationality so understood are genuinely normative. That would lead to some broad questions in the theory of normativity to which I’m assuming a particular answer here. A more interesting response would deny that co-presence of the attitude A and the belief that B generates any tension in the agent’s overall state. Compare the canonical rational tension generated by an agent’s believing both p and ~p. Those beliefs have logically inconsistent contents, and therefore clearly stand in conflict. The same is ¹ Compare, e.g., Wedgwood (, p. ).

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true of the belief that someone’s a bachelor and the belief that he’s married, which unfortunately makes my example from “Right Reason” a bit inapt. For there’s no conflict between the contents of the attitude A and the belief that B; B states that A is rationally forbidden, while A might be an attitude with virtually any content at all. But not every conflict involves inconsistent contents, especially when it comes to the normative. Clinton Castro suggested to me the example of a person smoking in front of a “No Smoking” sign. There’s clearly a tension in this tableau—between the content of the sign and the state of the person. Similarly, akrasia involves a tension between the content B of an agent’s belief (that attitude A is forbidden) and that agent’s state of possessing attitude A. This tension is not exactly the same as the tension between two contradictory beliefs, but it still constitutes a rational flaw.

. From the Akratic Principle to the Fixed Point Thesis Suppose we grant, either for these reasons or for reasons offered by the other authors I mentioned, that akratic overall states are rationally flawed. Why does it follow that it’s a rational mistake to have false beliefs about what rationality requires? Because rationality either requires things, or it doesn’t. Given the Akratic Principle—and perhaps even without it, as we’ll see in section .—the entire Fixed Point debate comes down to that. What might rationality require? I’ve already suggested that rationality requires avoiding akrasia; perhaps it also requires avoiding contradictory beliefs; perhaps it also requires maximizing expected utility. Opponents of the Fixed Point Thesis typically suggest that rationality requires respecting evidence from certain types of sources. Testimony from a privileged class of agents is the most frequently invoked such source, but intuition, reasoning, or even perception might play the role as well. It’s crucial, though, that Fixed Point opponents characterize these authoritative sources in terms independent of what rationality requires, so as not to beg any questions about whether rationality always requires that they be respected. Supposed counterexamples to the Fixed Point Thesis arise when an authoritative source provides false information about what rationality requires. Suppose, for instance, that an authoritative source says maximizing expected utility is rationally forbidden, when in fact utility maximization is rationally required. Intuitively, the agent is at least rationally permitted to believe the authority. But if she does, we have a problem. When the agent, having formed that belief, confronts a choice between two acts and recognizes that one of them maximizes expected utility, what intention should she adopt? If she intends to perform the maximal act, she violates the Akratic Principle, because she believes that maximizing expected utility is rationally forbidden. But if she fails to so intend, she will violate the rational requirement to maximize expected utility. Fixed Point opponents will reply that rationality isn’t so simple. Rationality requires the agent to maximize expected utility (say) unless an authoritative source convinces her it’s forbidden. Maximizing expected utility full-stop isn’t really what rationality requires. Fine. Complicate the requirements of rationality all you want. If

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you like, make them conditional, or pro tanto, or prima facie. Whatever the requirements are, and whatever logical form they take, they must ultimately combine to yield some all-things-considered facts of the form “When the agent’s situation is like this, these overall states are rationally permitted,” “When the agent’s situation is like that, those states are rationally permitted,” etc. Rationality must require something. Now hone in on one of those requirement facts, of the form “When the situation is S, states X, Y, and Z are rationally permitted.” In particular, choose a case in which S includes an authoritative source saying that in situation S, the relevant states X, Y, and Z are all rationally forbidden. If the agent is permitted to believe this authoritative source, then there’s a permitted state in situation S containing the belief that X, Y, and Z are forbidden. But since X, Y, and Z are the only permitted states in situation S, that means one permitted state contains the belief that it itself is forbidden—which violates the Akratic Principle. There are only three ways out of this problem: () insist that no situation S involves an authoritative source ruling out as forbidden all of the permitted states; () maintain that when an authoritative source says such things, rationality permits and requires the agent not to believe them; () hold that such authoritative pronouncements generate rational dilemmas, in which no permitted state is available. On each of these options, the case changes so that it no longer involves evidence from an authoritative source making it permissible for the agent to have a false belief about what rationality requires. It takes a few more steps² to get to the full Fixed Point Thesis: Fixed Point Thesis: No situation rationally permits an a priori false belief about which overall states are rationally permitted in which situations. I spell out those argumentative details in “Right Reason.”³ But I hope here to have conveyed the essential dynamic of the argument. Rationality requires something of us. Attempts to exempt agents from putative rational requirements in the face of misleading evidence simply generate alternate rational requirements. The exemption maneuver must stop somewhere, on pain of contradiction. At that point we have the true rational requirements in hand. Violating those is a rational flaw; having false beliefs about them creates rational flaws as well. None of this surrenders the idea that rationality requires an agent’s attitudes to harmonize from her own point of view. Making good on that idea requires us not only to understand the agent’s point of view, but also to substantively fill out the notion of harmony in question. Rational requirements tell us what it is for a set of attitudes to harmonize—what it takes to avoid internal tension.

² Steps which, I should note, have been challenged by Skipper (forthcoming). It would take me too far afield here to respond to Skipper’s interesting critique of my argument. For what it’s worth, though, out of the two interpretations of the argument he proposes, I would favor the second one, and would suggest that on that interpretation the argument isn’t so benign as he thinks. ³ For instance, the restriction to a priori false beliefs in the Fixed Point Thesis is there to keep it from applying to cases in which an agent is wrong about the nature of her current situation, or her current overall state.

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Quine () wanted everything in the web of belief to be revisable. But regardless of what he wanted, it had to mean something for the web to cohere. Similarly, rationality requires some fixed content somewhere, if it is not to dissolve as a normative concept entirely.

. Types of rationality, and rational dilemmas The argument of section . is highly general.⁴ Take any normative category in the business of issuing requirements and permissions, or even just the business of approving and disapproving of overall states. If that normative category disapproves of states containing both an attitude and the belief that that attitude is disapproved, then the argument goes through, and the category will satisfy an analogue of the Fixed Point Thesis. I’ll be the first to admit that the Fixed Point Thesis has odd, counterintuitive consequences. When you have a false belief about an ordinary, garden-variety empirical fact, there is a sense in which your belief has gone wrong. But the notion of rationality is meant to capture a way in which your belief still might have gone right. Your belief may have been a perfect response to the evidence; it’s just that the evidence was misleading. Then your belief—and the overall state it’s a part of—could be rationally flawless even though the belief is false. The Fixed Point Thesis says that when it comes to beliefs about what’s rational, we can’t maintain this gap. In a non-akratic agent, a false belief about what’s rational may engender other rationally flawed attitudes, or lead you to do irrational things. So even possessed of certain kinds of misleading evidence, an agent cannot form false beliefs about the rational requirements while remaining rationally flawless. In some sense this shouldn’t be surprising, given the Akratic Principle. The Akratic Principle correlates the attitudes in a rational overall state with higher-order attitudes about the rationality of those attitudes. So rational constraints on the attitudes become constraints on rationally permissible higher-order contents. Beliefs about the requirements of rationality have a very special place in the theory of rationality. Nevertheless, it’s tempting to reinstall the gap. An inaccurate belief about rationality can’t be rational—but in the right circumstances, might it at least be reasonable (Schoenfield )? Or maybe there are two kinds of rationality: the Fixed Point Thesis applies to ideal rationality, but not to everyday rationality. Yet here the generality of the argument bites. Are some sets of attitudes reasonable, and others unreasonable? Is it reasonable to hold an attitude while also believing that attitude to be unreasonable? If not, then a Fixed Point Thesis can be derived for the normative category of reasonableness. And similarly for any normative category that is substantive and satisfies an analogue of the Akratic Principle. ⁴ Other highly general arguments towards the same conclusion can be found in Littlejohn (, §.), Field (, §), and Lasonen-Aarnio (forthcoming, §). Lasonen-Aarnio also makes the interesting point that one can argue not only from the Akratic Principle to the Fixed Point Thesis, but also in the opposite direction. Suppose Fixed Point is true, and an agent possesses both attitude A and the belief that A is forbidden in her situation. Either A is forbidden, in which case possessing A is rationally flawed, or A is not forbidden, in which case the belief generates a rational flaw by the Fixed Point Thesis.

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It’s worth working through the details of at least one proposal that tries to reintroduce the gap, and showing how the Fixed Point argument undermines that proposal. In her later work (; ), Schoenfield specifies more precisely what it takes to be “reasonable” in the sense of her (). She focuses on agents’ plans for handling various contingencies, and distinguishes two types of normative evaluations of such plans. On the one hand, we can evaluate which plan would turn out best were the agent to actually execute that plan. On the other hand, we can evaluate which plan would be best for the agent to make in advance—taking into account that an agent doesn’t always execute the plan that she makes. As far as I can tell, Schoenfield’s best plans to execute specify what’s rational in the sense I’ve been discussing, while what’s “reasonable” for an agent to do in Schoenfield’s earlier sense is specified by the best plan for an agent to make. Schoenfield then presents a case⁵ in which the best plan to execute requires the agent to adopt one attitude, yet the best plan to make requires a different one. While there’s some sense in which the former attitude would be rational, the latter attitude is the one that’s reasonable. Crucially, the two plans come apart because it’s been suggested to the agent by an authoritative source that she’s unable to discern what the best plan to execute requires. But a case like that isn’t the real crucible for Schoenfield’s normative category of reasonableness. In Schoenfield’s case, there’s a problem with the agent’s discerning the best plan to execute, but no potential problem discerning the best plan to make. Yet if we go back to the Fixed Point argument of section ., we’ll predict a problem for Schoenfield’s new normative category precisely when the best plan to make requires attitude A, but an authoritative source says the best plan to make forbids A. What’s the best plan to make for a case like that? By stipulation, it involves attitude A, but does it also involve believing the authoritative source? While Schoenfield acknowledges the possibility of cases like this (see Schoenfield , note  and the attached text), she sets them aside. But it’s exactly in cases like this that the problems for the old notion of rationality she hoped to avoid with her new notion of reasonableness come back with a vengeance.⁶ Let’s try another tack—perhaps the gap between rationality and accuracy is so central to the former concept that when it can’t be maintained, the possibility of being rational breaks down altogether. David Christensen () argues that cases in which an authoritative source provides misleading evidence about the requirements of rationality are rational dilemmas. This was one of the three available responses to the problem cases I noted in section .. And indeed, it’s consistent with the Fixed Point Thesis in a certain sense: on Christensen’s view, any overall state that includes a ⁵ Schoenfield (, §ff.), based on Adam Elga’s (ms) well-known “hypoxia” example. (I’m grateful to an anonymous referee for pointing out that a similar example appears in Williams (, p. ).) ⁶ What about a proposal on which reasonableness addresses the problem cases for old-style rationality, some other normative category addresses the problem cases for reasonableness, yet another category addresses the problem cases for that normative category, and so on up the line? A few philosophers have dabbled with infinite hierarchies of this sort, but Robinson (ms) helpfully points out why they won’t avoid the problem. With enough craftiness, we can build a case in which an authoritative source provides misleading information about what’s required by all of the normative categories past a particular point in the hierarchy.

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false belief about what rationality requires must be rationally flawed in these cases, because on his view every overall state available in such cases is rationally flawed. But to simply say that the problem cases are rational dilemmas leaves an important question unanswered. Consider a case in which the agent has a number of mutually exclusive attitudes available—perhaps the options are belief, disbelief, and suspension of judgment in some proposition, or intending to perform some act versus intending not to perform it. Call one of the available attitudes A, and let’s stipulate that in the agent’s situation A is rationally required. Now suppose also that an authoritative source tells the agent B, where B is the proposition that A is rationally forbidden while one of the other available attitudes is rationally permitted. Set aside for a moment what the agent should do about A, and focus on whether the agent is permitted to believe B. Even if one reads this case as a rational dilemma, B is a falsehood about what rationality requires.⁷ Anyone who accepts the Fixed Point Thesis (including rational dilemma theorists) must maintain that rationality forbids believing B. But to this point the only evidence bearing on B we’ve encountered in the agent’s situation is the authoritative source’s endorsement. Intuitively, that seems like evidence in favor of B. And we haven’t seen any evidence in the situation that either rebuts or undercuts it. So if the agent’s total evidence supports B, how can B be rationally impermissible to believe? One possible answer is that no rebutting or undercutting is required, because the authoritative source hasn’t actually provided any support for B. It might be that no evidence can ever rationally support (to any degree) a falsehood about the requirements of rationality. Thus because of its content in this case, the authority’s testimony is evidentially inert. In “Right Reason,” though, I granted the intuition that the authority provides at least pro tanto or prima facie evidence for B. So I needed to find some evidential factor that could defeat this support, yielding the result that belief in B is all-things-considered rationally forbidden. I settled on the suggestion that every agent always has available a body of a priori support for the truths about rationality, strong enough to defeat whatever empirical evidence tells against them. Importantly, this claim about the a priori was not a premise of my argument to the Fixed Point Thesis (contra Field ); it was a suggestion of one way the epistemological landscape might lie if the Thesis is true. Since then, some other approaches have been tried. One popular trend (e.g., Worsnip ) is to distinguish “structural” requirements of rationality, which place formal coherence constraints on combinations of attitudes, from “substantive” norms, which direct the agent to follow her evidence and respect the reasons it provides. When I wrote “Right Reason,” I assumed there was a single normative notion of rationality in play. I took that notion to both direct agents’ responses to evidence and be subject to the Akratic Principle. So I assumed, for instance, that if an agent’s total evidence all-things-considered supported a particular proposition, then ⁷ I’m grateful to an anonymous referee for prompting me to make explicit how this case reads on the rational dilemmas view. If the case is a rational dilemma in the sense that neither A nor any of its alternatives is rationally permitted, then we can say that A and all of the alternatives are vacuously both required and forbidden. (This is the standard approach to dilemmas in deontic logic.) Yet as none of them are permitted, B is still a falsehood about what rationality requires.

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it was rationally permissible for that agent to believe that proposition.⁸ Thus if I wanted to say that belief in B was rationally forbidden, I had to find some evidence to defeat the authoritative source’s support for B. (And any dilemmas theorist who believes all-things-considered evidentially supported beliefs are rationally permitted faces a similar problem.) But on the present view, the Akratic Principle is a structural requirement, while evidential requirements are substantive. So while the former may make it incoherent to be wrong about rationality, this doesn’t tell us anything about the agent’s total evidence. Yet even if we set aside structural requirements and the Akratic Principle, it might not do to be so sanguine about what “following the evidence” requires. Why think that the reasons evidence provides bear only on an agent’s beliefs about what she ought to do in her situation, and not directly on what that situation requires? If an authoritative source says you’re forbidden to adopt attitude A, doesn’t “following your evidence” require you not only to believe that A is forbidden, but also to avoid adopting A? Doesn’t evidence against expected utility theory also tell against maximizing utility? It may be objected that the reasons provided by evidence are theoretical reasons (not practical), and so may bear only on belief. Fine, then consider cases in which A is a belief. Suppose (to borrow an example from “Right Reason”) that Jane’s evidence contains the proposition ~(~X _ ~Y), and an authoritative (though sadly mistaken) logic teacher tells her that such evidence requires her to believe ~Y. If Jane follows her evidence and respects what her logic teacher has said, this seems to require not only believing that her evidence requires her to believe ~Y, but also believing ~Y simpliciter. This, after all, was the point of Feldman’s () “evidence of evidence is evidence” principle. If that’s right, and if the reasons provided by substantive norms apply to the first-order moves recommended just as much as to beliefs about those first-order moves,⁹ then the problem behind the Fixed Point argument recurs. We don’t need structural rationality or its ban on akrasia to generate our problem cases; they arise just from trying to understand what “following the evidence” requires. Again, the argument of section . is highly general; as long as a normative category contains some substance and parallel constraints at lowerand higher-orders, our argument is off and running.

. Internalism and the Fixed Point Thesis I often hear the complaint that it’s inappropriate to find rational fault in false beliefs about rationality, because an agent may not be “able to figure out” what rationality requires in a given situation. The requirements of rationality can be complex, and thoughtful people can get them wrong. (Consider, for example, folks on one side ⁸ This assumption of my argument has been noted by Littlejohn (, §) and Daoust (forthcoming, n. ). The first person to point it out to me was Kieran Setiya, in conversation. ⁹ Worsnip (, §IV) pushes against this suggestion; Daoust (forthcoming, §) pushes back. See also Feldman’s (), in which he argues in favor of “respecting the evidence,” described as follows: “a person respects the evidence about E and P by believing P when his or her evidence indicates that this evidence supports P or by not believing P when the evidence indicates that this evidence does not support P” (pp. –).

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of the Fixed Point debate.) This complaint can be linked to my earlier talk of rationality’s concerning what makes sense “from the agent’s own point of view.” If an agent can’t see the truths of rationality from where he cognitively stands, how can there be any rational error in his making mistakes about them? The Fixed Point Thesis can’t hold for the notion of rationality I’ve described; it’s plausible only on an externalist account of rationality, or an “objectivist” account, or a “highly idealized” account. All of this is wrong. The Fixed Point Thesis is perfectly compatible with internalist, subjectivist, everyday accounts of rationality. First off, such accounts typically require a version of the Akratic Principle, so we can argue as in section . that the Thesis applies to them. But second, the complaint just described fails even when it’s run using such accounts. Grant the complaint its assumption that on the notion of rationality I’ve identified, an attitude can be rationally required or forbidden in a situation only if the agent in that situation is able to figure out (whatever that means) that that attitude has the relevant rational status. Now suppose we have an agent who’s faced with a decision between two acts, one of which would maximize expected utility, while the other (given the agent’s attitudes toward risk) would maximize risk-weighted expected utility. In point of fact, maximizing traditional expected utility is rationally required. But the agent has been convinced by formidable arguments of Buchak () that the risk-weighted approach is correct, and it’s beyond his abilities to find the subtle flaws in those arguments. So the agent believes that maximizing risk-weighted utility is rationally permitted, when in fact it’s forbidden. It looks here like the Fixed Point Thesis will count that belief as rationally flawed, even though the agent is unable to see his mistake. Which contravenes the principle I granted at the start of this paragraph. This objection to the Fixed Point Thesis fails because under the objector’s own assumptions, the case described is impossible. The objection assumes that an attitude is rationally required/forbidden in a situation only if the agent in that situation can figure out that it’s required/forbidden. As the case has been described, the agent can’t see his way to the conclusion that maximizing risk-weighted utility is rationally forbidden. But then given the objection’s assumption, maximizing risk-weighted utility can’t be rationally forbidden for him. And so the belief that maximizing risk-weighted utility is permitted isn’t false, and the Fixed Point Thesis won’t fault him for maintaining it. The Fixed Point Thesis won’t forbid the agent to possess any belief he can’t figure out is forbidden. The point is that there’s a parallel between the constraints rationality places on an agent at the first order and the constraints Fixed Point places on his beliefs about those first-order constraints at higher orders. Go ahead and restrict rationality in any way you like to what an agent can figure out, or what’s “accessible” to him, or what’s “subjectively available.” This restriction will immediately limit the first-order attitudes required of that agent. And since the first-order rational requirements on the agent will be limited by this restriction, the truths about rational requirements that the Fixed Point Thesis concerns will also be limited. If being able to be figured out is necessary for something to be a rational requirement, and the Fixed Point Thesis only tracks rational requirements, then the Fixed Point Thesis won’t require an agent to believe anything he can’t figure out.

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We must keep this parallelism in mind as we consider the Fixed Point Thesis’s consequences for the peer disagreement debate. To repeat an example from “Right Reason,” suppose that at some initial time Greg and Ben, two mutually acknowledged epistemic peers, each possess the same total evidence E relevant to some proposition h. Suppose further that rationality requires Greg and Ben to each believe h in light of E. Greg reasons through the consequences of E correctly, and comes to believe h through that reasoning. Ben, on the other hand, reasons from E to ~h. (To keep the characters straight, remember that Greg does the good thing while Ben reasons badly.) Some time later, Greg and Ben interact, sharing their conclusions about h on the basis of E. I argued in “Right Reason” that if the Fixed Point Thesis is correct, then were Greg to change his opinion about what E supports after interacting with Ben, the resulting opinion would be rationally mistaken. This is a version of what’s sometimes called the “Right Reasons”—or “steadfast”—approach to peer disagreement. Abbreviated, my argument against the contrary “split the difference”—or “conciliatory”—approach was this: Any plausible view that required Greg to move his opinion towards Ben’s would also require Greg to believe ~h on the basis of E were he to encounter sufficiently many epistemic superiors who claimed that E supports ~h. But this would involve a false belief about the requirements of rationality, and so would contradict the Fixed Point Thesis. Thus the Fixed Point Thesis faults Greg’s overall state if he makes any change to his initial view about what E supports.¹⁰ Like the Fixed Point Thesis, the steadfast view on peer disagreement is disconcerting. When we investigate empirical matters, it’s often advisable to merge our observations and opinions with those of others, especially when those others have proven reliable in the past. I should be clear: Nothing about my steadfasting view tells against agents’ employing authoritative sources to determine what’s rational. It’s just that—as I’ve been emphasizing throughout this chapter—there’s an important rational asymmetry between empirical facts and facts about what’s rational. When we form an opinion about empirical matters by trusting reliable authorities, there’s a small chance those authorities are wrong. In that case we’ll have formed a false belief, but at least the belief will be rational. When we rely on authorities to form opinions about what’s rational, we take on additional risk: if the resulting belief is false, it will be rationally flawed as well. (Another casualty of collapsing the gap between inaccuracy and irrationality.) Suppose the earth is gradually warming, but  experts tell me otherwise. If I believe what they say, I will have made a factual error, but perhaps not a rational one (depending on the rest of my circumstances). Now suppose maximizing traditional expected utility really is what’s required by rationality. If  experts tell me

¹⁰ Allan Hazlett () argues that after interacting with Ben, Greg ought to continue to believe h, but ought to suspend judgment on the question of whether E supports h. Yet if I’m reading Hazlett correctly, his position that Greg should suspend on the higher-order question in light of peer disagreement would also endorse Greg’s adopting a false belief about what E supports should Greg receive enough (misleading) authoritative evidence. (“Misleading higher-order evidence undercuts good lower-order evidence when . . . it warrants belief that said lower-order evidence is not good.” §.) Thus Hazlett’s position seems to me also to run afoul of the Fixed Point Thesis.

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otherwise, and I believe what they say, then form the intention that maximizes risk-weighted utility, I will have made both a factual and a rational error. These are the stakes when we set out to determine what’s rational. But this shouldn’t be surprising given that description of the enterprise. And really, is the alternative view more plausible? The conciliationist holds that overwhelming expert testimony makes it perfectly rational to believe (and do) whatever the experts say is rationally required. For the conciliationist, consulting the experts about what’s rational is a wonderful idea, because it has a kind of magical quality. Presumably the experts rarely get these things wrong. But even when they do, the fact that you consulted them makes it rational to do what they said. In this sense, conciliationism makes experts infallible about what rationality requires of their audience.¹¹ People often ask me how, in the peer disagreement scenario, Greg is supposed to figure out that he’s the one who would be rationally mistaken if he changed opinions. Here’s where my earlier parallelism point kicks in. Suppose once more that agents can be rationally required to adopt an attitude only when they can figure out that that requirement holds. As we’ve constructed the case, before Greg interacts with Ben he’s rationally required to believe h on the basis of E. For this to be a rational requirement, it must be the case that Greg can figure out that E requires belief in h. And as we’ve told the story, Greg does figure that out: he reasons properly from E to h, and believes h on the basis of that reasoning. So after Greg talks to Ben, can he still figure out that E requires belief in h? Yes—all he has to do is repeat the reasoning that led to his initial position. Talking to Ben hasn’t somehow made it impossible for Greg to do that. Again, let me emphasize that I’m not saying there’s some objective, or external sense in which Greg would be correct to go on believing h, while there’s another (subjective/internal) sense that recommends conciliation. Whatever it means to “figure out” from one’s subjective point of view what rationality requires, Greg is capable of doing that after conversing with Ben. Why? Because he was able to do so before the conversation. And what about poor Ben? What should he do, and how can he tell? Again, let me be clear: I am not saying that rationality requires agents to adopt a policy on which, whenever they confront disagreement, they always stick to their initial opinions. My peer disagreement position is motivated by the Fixed Point Thesis, which says that false beliefs about rationality are rationally mistaken. Ben’s initial opinion about the rational relation between E and h was rationally mistaken, so were he to stick with it after interacting with Greg, he would simply perpetuate a rational mistake. Is Ben capable of figuring out that he’s made a rational mistake? Well, we started off the case by saying that both Greg and Ben are rationally required to believe h on the basis of E. Under the principle that being rationally constrained implies being able to figure out that you’re rationally constrained, Ben must initially have been capable of figuring out that E supports h. Interacting with Greg may have enhanced—and certainly can’t have degraded—that capability.

¹¹ Compare Feldman (, p. ).

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. Response to Smithies The best response I’m aware of to this parallelism point can be reconstructed from some arguments in Declan Smithies’s ().¹² Smithies begins by distinguishing propositional from doxastic justification. In order for a belief to be doxastically justified (and rational), it must not only be supported by the agent’s evidence; the agent must also base his belief on that evidence in the right way. We can grant that before interacting with Ben, Greg’s belief in h was both propositionally and doxastically justified—it was supported by Greg’s evidence, and correctly based on that evidence. Greg’s conversation with Ben provides evidence that his earlier reasoning may have been flawed. Smithies is willing to grant that even with this evidence, Greg is still propositionally justified to believe h after the interaction. Yet Smithies maintains that “evidence of one’s cognitive imperfection functions as a doxastic defeater, rather than a propositional defeater—that is to say, it does not defeat one’s propositional justification . . . but merely prevents one from converting this propositional justification into doxastic justification” (, p. , emphases in original). How does that work? Smithies holds that “What is needed for doxastic justification is . . . safety from the absence of propositional justification” (p. ). Beliefs are unsafe from the absence of propositional justification when “they are formed on the basis of doxastic dispositions that also tend to yield beliefs in the absence of propositional justification” (p. ). So even if, in a particular case, an agent reasons to a conclusion that correctly picks up on his propositional justification for that conclusion, he may not be doxastically justified. In order for him to be doxastically justified in believing the conclusion, the reasoning disposition that formed the belief must be such that it would not easily (in nearby similar cases, whether actual or counterfactual) yield beliefs lacking propositional justification. Let’s apply this to Greg’s case, while keeping the parallelism point in mind. Before Greg talked to Ben, Greg’s belief in h was doxastically justified. On Smithies’s view, this implies that the reasoning disposition that formed Greg’s initial belief in h was safe from the absence of propositional justification. So the disposition that formed it would not have yielded beliefs lacking propositional justification in nearby cases. In more concrete terms, whatever kind of reasoning Greg employed to determine h on the basis of E, had he deployed the same cognitive faculty on other, similar reasoning problems, the faculty would not have yielded beliefs unsupported by his evidence. Then Greg meets Ben, and has to decide whether to remain steadfast in his belief that h. Notice that if Greg simply relies on the reasoning he initially employed, he will maintain a belief that is propositionally justified for him. And since we’ve already seen that this reasoning is, in Greg, safe from the absence of propositional justification, Greg’s continued belief in h looks like it will be doxastically justified as well.

¹² In his (), Smithies focuses exclusively on beliefs in logical truths. His discussion involves credences as well as full beliefs. He draws a distinction between ideal and “ordinary” standards of rationality. And he isn’t directly talking about peer disagreements. (Though his footnote  suggests extending his position to that application.) That’s why the response I’ll present here has to be reconstructed from Smithies’s view, and why I’ve made some bracketed amendments in the quotations to come.

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Smithies acknowledges that the reasoning disposition behind Greg’s initial belief in h has not been impaired by the interaction with Ben: “Misleading evidence about your cognitive imperfection impacts neither the actual quality of our first-order reasoning nor the ideal ranking of options” (p. ). So why might Greg nevertheless lose doxastic justification? Because acquiring evidence about our cognitive imperfection brings new reasoning dispositions into play and so it does impact on the overall quality of our reasoning. We need to consider not just your first-order dispositions to reason . . . but also your second-order dispositions to respond to evidence of your own cognitive imperfection. Your first-order reasoning is reliable enough to make [belief in h] rational in ordinary contexts, but your second-order dispositions to respond to evidence of your cognitive imperfection are unreliable enough to make [belief in h] irrational in contexts where you have such evidence. (p. )

And why are those second-order dispositions unreliable? Because “exercising the same doxastic dispositions in response to the same empirical evidence could easily yield the false and unjustified belief that I correctly [reasoned] when in fact I didn’t” (p. ). The argument seems to be this: Admittedly, the first-order reasoning dispositions that formed Greg’s initial belief in h remain in reliable working order after the interaction with Ben. Yet the empirical evidence from Ben that Greg might have made a mistake brings into play a new reasoning disposition: Greg’s disposition to respond to empirical evidence of his own cognitive imperfection. If Greg remains steadfast, he deploys a disposition that would in many nearby cases yield a belief lacking propositional justification—when given evidence of cognitive imperfection, that disposition would blithely instruct him to keep his beliefs intact even when they weren’t propositionally justified. The unreliability of this second-order disposition defeats Greg’s doxastic justification to go on believing that h. And since Greg’s belief in h can’t be doxastically justified after he interacts with Ben, it can’t be rational either. My response to this line from Smithies comes in many stages. First, I’m honestly not sure how to differentiate reasoning dispositions, or rule on which dispositions were involved in basing a given belief. Suppose that in our case Greg hears the evidence from Ben, but goes on relying on his initial reasoning to govern his opinion about h. In that case has he employed only the initial reasoning disposition, the one we conceded was safe from the absence of propositional justification? Or has the recognition of Ben’s evidence forced him to deploy an additional reasoning disposition as well? How should we identify and assess that additional reasoning disposition? It may be question-begging to describe it as a disposition to set aside evidence of cognitive imperfection in all cases of peer disagreement. (As I said near the end of section ., the Fixed Point Thesis does not endorse a universal policy of sticking to one’s guns.) What if Greg’s higher-order disposition is to set aside such evidence just in cases in which his initial reasoning correctly latched on to what rationality requires? (And don’t tell me Greg “can’t figure out” which cases those are!) This brings us to the second stage of my response to Smithies: Which exactly are the nearby cases that determine the reliability of the higher-order disposition? If it’s just cases of reasoning about propositions very similar to E and h, then a higher-order

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disposition to remain steadfast might actually be safe for Greg across those cases. (Since his first-order reasoning about such cases was established to be safe initially.) But if it’s a wide variety of reasoning cases, including domains in which Greg’s first-order reasoning is not so reliable, then the higher-order disposition may have a problem. Suppose, though, that we set aside these detail concerns, and grant the entire Smithies-inspired argument. What does it give us in the end? Notice that after the interaction with Ben, there is no attitude towards h other than belief that would be doxastically justified for Greg.¹³ This is easily established, because Smithies grants that belief is still the attitude towards h propositionally justified for Greg after the interaction, and propositional justification is necessary for doxastic justification. So we don’t have an argument here that Greg would be doxastically justified(/rational) in conciliating with Ben, or that in cases with many experts the agent would be doxastically justified in believing their falsehoods about what rationality requires. At best these cases become rational and doxastic justification dilemmas. Perhaps I’ve made it too easy to reach this conclusion by focusing on Smithies, who grants that Greg’s propositional justification for h remains intact. Salas (forthcoming) takes a position similar to Smithies’s, which I haven’t focused on here because it isn’t as thoroughly worked out. But Salas thinks that the interaction with Ben defeats not only Greg’s doxastic justification but also the propositional. Perhaps Salas offers a view on which conciliating with Ben could be doxastically justified/rational for Greg? Or maybe all this talk of reliable first-order dispositions and unreliable second-order dispositions has put us back in mind of Schoenfield’s distinction between best plans to execute and best plans to make? But now the alarms from our earlier Schoenfield discussion should be ringing again. The Fixed Point argument from section . was highly general, and should alert us that there’s no stable landing point to be found in this direction. Any view that claims an agent is doxastically justified in believing misleading testimony from massed experts about rationality has to deal with the problem cases we identified in section .. And even if Greg were simply to suspend judgment about h after his interaction with Ben, he could be in serious rational trouble. Suppose h is the proposition that rationality requires maximizing traditional expected utility, which turns out to be not only supported by Greg’s initial evidence but also true. If Greg suspends judgment on h, then confronts a decision between two options (one of which maximizes traditional expected utility, the other risk-weighted), what is he going to do? So even if a line like Smithies’s goes through, it is not going to yield the verdict that conciliating in cases of peer disagreement is rationally permissible. At best, it will tell us that peer disagreement cases are rational dilemmas, in which no overall state is rationally permissible. This does not contradict the Right Reasons position I’ve defended, on which conciliating in peer disagreement cases generates a rational mistake. And it certainly doesn’t undermine the Fixed Point Thesis. ¹³ Van Wietmarschen () notes and concedes this point about his own doxastic/propositional peer disagreement stance. I haven’t discussed van Wietmarschen’s arguments for that stance here because they seem to me to rely on an independence principle that is question-begging against the steadfast position.

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. An amendment to Right Reason Nevertheless, I do need to amend and clarify the position on peer disagreement that follows from Fixed Point. While I didn’t recognize this when I wrote “Right Reason,” there are cases in which an agent who initially adopted the attitude towards a proposition required by her evidence nevertheless should change her attitude towards that proposition upon learning that an epistemic peer drew the opposite (and rationally incorrect) conclusion. To see why, let’s begin with a case somewhat different from the Greg and Ben setup; later we’ll adapt a lesson from this case to Greg and Ben’s. Sometimes a body of evidence mandates a single rational attitude towards a given proposition. But it’s possible that some bodies of evidence allow for multiple interpretations; multiple attitudes towards the same proposition are rationally permissible in light of that evidence. Following White (), call these “permissive cases.” There has been much recent philosophical debate about whether permissive cases exist. (Kopec and Titelbaum  provides a survey.) But for our purposes it will be helpful to imagine that such cases do exist, then consider how peer disagreement might act as evidence within them. One way to think about a permissive case is that in the case, two agents have different evidential standards—sets of principles that guide their interpretation of evidence and recommend attitudes. Given a particular body of total evidence E and proposition h, if the two evidential standards draw different lessons about h from E, yet both standards are permissible, then it might be rationally permissible for one agent to follow her standards and believe h on the basis of E, while the other follows her standards and believes ~h instead. In our (), Matthew Kopec and I examine a permissive case with one further wrinkle. Imagine that neither agent knows the details of how the other agent’s evidential standards work, or what those standards will say about any particular evidence-proposition pair. But each agent knows that both standards are reliable in the long term: for each standard, % of the propositions it recommends on the basis of evidence turn out to be true. Take one of the two agents in this situation—call her Anu. Anu initially evaluates E, applies her own evidential standards correctly, and comes to believe h on the basis of E. This is what rationality requires of Anu. But because Priya has different (though still rationally permissible) evidential standards, she is rationally required to believe ~h on the basis of E. This case is different from Greg and Ben’s, because the epistemic peers initially disagree about a given proposition based on the same body of evidence without either party’s making a rational mistake. Permissive cases allow for this possibility. Now suppose that Anu interacts with Priya. Anu learns that Priya’s evidential standards recommend belief in ~h on the basis of E. Kopec and I argue that in this case Anu should suspend judgment about h. This is not because Priya’s testimony is evidence that Anu made any sort of rational mistake. It’s because Priya’s testimony is evidence that h is false. As I said, a number of philosophers deny that permissive cases exist. But just by imagining that they do, we learn an important lesson. Schoenfield (, p. )

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helpfully distinguishes two worries an agent might have about one of her own beliefs: “My belief might not be rational!” versus “My belief might not be true!” Schoenfield winds up concluding that in peer disagreement cases, the worry about truth shouldn’t move one to change one’s opinions, but the worry about rationality might. She’s right about rationality, in the following sense: Sometimes disagreement may prompt you to realize that your previous opinion was rationally flawed, in which case it might be a good idea to amend it. (Perhaps this is what Ben should learn from his disagreement with Greg, and how he should respond.) But I think Schoenfield’s wrong about truth. In the permissive case, interacting with Priya need not indicate to Anu that her initial h belief was irrational. But the interaction should make her worry that h isn’t true, which rationally requires a change in attitude towards h. Interestingly, the same thing can happen in cases where permissivism is not an issue. In the Greg and Ben case, either Greg and Ben have the same evidential standard, or they both have evidential standards on which E requires belief in h. Up to this point, we’ve assumed that when Greg finds out about Ben’s belief in ~h, this disagreement poses a threat to Greg’s stance by worrying him that his initial belief in h was irrational. According to the Fixed Point Thesis, if this worry leads Greg to alter his previous opinion about what rationality requires, and if he changes his attitude towards h as a result, the resulting attitudes will be rationally mistaken. But now suppose we add some extra details to the story. Suppose Greg is epistemologically savvy, and incredibly epistemically careful. He knows that sometimes evidence is misleading. So upon receiving evidence E and initially concluding that it supports h, Greg enlists a confederate. This confederate is to go out into the world and determine whether h is actually true. If h is indeed true, the confederate will send Greg a peer to converse with who agrees with Greg that E supports h. But if h turns out to be false, the confederate will supply Greg an interlocutor who (falsely) believes that E supports ~h. Greg has full (and if you like, fully rational) certainty that the confederate will execute this task correctly; he dispatches the confederate and sets out to wait. Sometime later, Ben enters the room, and Greg discovers that Ben disagrees with him about whether E supports h. I submit that in this case rationality requires Greg to revise his initial attitude towards h. This is not a permissive case. We can suppose that both Greg’s and Ben’s evidential standards demand initial belief in h on the basis of E. It is also not a case in which Ben’s testimony should lead Greg to question whether E supports h. Nevertheless, Greg should change his opinion about h upon receiving Ben’s testimony. This is not because Greg receives evidence that his initial opinion about h wasn’t rational; it’s because he receives evidence that that opinion wasn’t true. I’ll admit this modified Greg/Ben case is somewhat baroque. Could the same effect occur in more realistic settings? In conversation, Kenny Easwaran suggested the following possibility: Suppose two detectives are partners—call them Brain and Brawn. When confronted with a case, Brain and Brawn collect the evidence together, and analyze it. Brain does a better analysis job; he almost always draws the rational conclusion from the evidence. Brawn is terrible at analyzing evidence. But Brawn has an interesting feature: he’s very intuitive. In the course of interviewing subjects, poking around crime scenes, etc., Brawn picks up on certain cues subconsciously. Brawn isn’t explicitly aware of this material, and he certainly wouldn’t list it as part of

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his evidence. But his subconscious processing leaks into his conscious thought; in particular, it introduces a bias when he goes to analyze the explicit evidence. So even though Brawn’s evidential interpretations are often terrible on the merits, they nevertheless are infected (not through explicit reasoning but instead through implicit processes) by factors that point them towards the truth. If that’s how it is with Brawn, and Brain knows that, then even after Brain has (rationally) interpreted the evidence, it seems rational for him to take Brawn’s analysis into account. If Brawn disagrees with Brain about a case, Brain should seriously consider the possibility that Brawn’s opinion—while not a rational response to the stated evidence—nevertheless may point towards the truth. Again, Brawn’s testimony shouldn’t change Brain’s view on whether his own analysis was rationally correct, but it may shift his opinion on whether the conclusion of that analysis is true. Philosophers have been discussing for millennia how an agent ought to respond to disagreement from his peers. In the recent epistemology literature, it’s been assumed that if peer disagreement changes an agent’s opinion, it must do so by providing higher-order evidence—evidence that changes his opinions about what’s rational.¹⁴ The debate between steadfasters and conciliationists is a debate about whether, in the unmodified version of the case, Ben’s testimony should affect Greg’s opinion that E supports h. Assuming the Akratic Principle, the expectation is that changes in Greg’s higher-order opinion will cause—or at least coincide with—a change in Greg’s firstorder attitude towards h. But the examples I’ve just given—the modified Ben/Greg case, and Brain v. Brawn—show that peer disagreement may rationally change an agent’s first-order opinions (Greg’s attitude towards h) without changing his attitudes about what’s rational. This means that my earlier, blanket steadfast view about peer disagreement was too general. The Fixed Point Thesis shows that an agent who draws the rationally required conclusion from her evidence makes a rational mistake if she allows testimony from others to change her belief that that conclusion was required. But the thesis does allow such testimony to change the agent’s attitude towards that conclusion—as long as this first-order change isn’t accompanied by a higher-order one. In fact, I now think it’s a mistake to classify a given piece of evidence as intrinsically higher-order or not. The evidential significance of a given fact will often vary according to context, and in particular according to the background information possessed by the agent receiving it. The very same fact may mean different things to different people; it may rationally alter their opinions in differing ways, or may produce the same doxastic effect through two different routes. Disagreeing testimony from an epistemic peer may lead you to question the rationality of your earlier reasoning, or it may leave your assessment of your earlier reasoning unmoved while nevertheless changing your attitude toward the result of that reasoning. This latter possibility was the one I missed in “Right Reason.”

¹⁴ See, e.g., Skipper (forthcoming, §), and the authors he cites there.

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Acknowledgments In addition to all of those who assisted me with the preparation of “Right Reason,” I am grateful to: my Philosophy Department colleagues at the University of Wisconsin-Madison for a lively discussion of an earlier draft, Clinton Castro, Tristram McPherson, Kenny Easwaran, Josh DiPaolo, the editors of this volume, and an extremely helpful anonymous referee. My work on this chapter was supported by a Romnes Faculty Fellowship and an Institute for Research in the Humanities Fellowship, both granted by the University of Wisconsin-Madison.

References Buchak, L. (). Risk and Rationality, Oxford University Press. Christensen, D. (). “Does Murphy’s Law Apply in Epistemology? Self-Doubt and Rational Ideals.” In: Oxford Studies in Epistemology , pp. –. Daoust, M.-K. (forthcoming). “Epistemic Akrasia and Epistemic Reasons.” In: Episteme. Elga, A. (ms). “Lucky to be Rational.” Unpublished manuscript. Feldman, R. (). “Respecting the Evidence.” In: Philosophical Perspectives , pp. –. Feldman, R. (). “Reasonable Religious Disagreements.” In: L. M. Antony (ed.), Philosophers without Gods: Meditations on Atheism and the Secular Life, Oxford University Press. Field, C. (). “It’s OK to Make Mistakes: Against the Fixed Point Thesis.” In: Episteme , pp. –. Greco, D. (). “A Puzzle about Epistemic Akrasia.” In: Philosophical Studies , pp. –. Hazlett, A. (). “Higher-Order Epistemic Attitudes and Intellectual Humility.” In: Episteme , pp. –. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs , pp. –. Kopec, M. and M. G. Titelbaum (). “The Uniqueness Thesis.” In: Philosophy Compass , pp. –. Lasonen-Aarnio, M. (forthcoming). “Enkrasia or Evidentialism? Learning to Love Mismatch.” In: Philosophical Studies. Littlejohn, C. (). “Stop Making Sense? On a Puzzle about Rationality.” In: Philosophy and Phenomenological Research , pp. –. Quine, W. (). “Two Dogmas of Empiricism.” In: The Philosophical Review , pp. –. Robinson, P. (forthcoming). “The Incompleteness Problem for Theories of Rationality.” Unpublished manuscript. Salas, J. (forthcoming). “Dispossessing Defeat.” In: Philosophy and Phenomenological Research. Schoenfield, M. (). “Chilling out on Epistemic Rationality: A Defense of Imprecise Credences (and Other Imprecise Doxastic Attitudes).” In: Philosophical Studies , pp. –. Schoenfield, M. (). “Permission to Believe: Why Permissivism is True and what it Tells us about Irrelevant Influences on Belief.” In: Noûs , pp. –. Schoenfield, M. (a). “Bridging Rationality and Accuracy.” In: Journal of Philosophy , pp. –. Schoenfield, M. (). “An Accuracy Based Approach to Higher Order Evidence.” In: Philosophy and Phenomenological Research (), pp. –. Skipper, M. (forthcoming). “Reconciling Enkrasia and Higher-Order Defeat.” In: Erkenntnis. Smithies, D. (). “Moore’s Paradox and the Accessibility of Justification.” In: Philosophy and Phenomenological Research , pp. –. Smithies, D. (). “Ideal Rationality and Logical Omniscience.” In: Synthese , pp. –.

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Titelbaum, M. G. (). “Rationality’s Fixed Point (or: In Defense of Right Reason).” In: T. S. Gendler and J. Hawthorne (eds), Oxford Studies in Epistemology , Oxford University Press, pp. –. Titelbaum, M. G. and M. Kopec (). “When Rational Reasoners Reason Differently.” In: M. Balcerak-Jackson and B. Balcerak-Jackson (eds), Reasoning: Essays on Theoretical and Practical Thinking, Oxford University Press, pp. –. Van Wietmarschen, H. (). “Peer Disagreement, Evidence, and Well-Groundedness.” In: The Philosophical Review , pp. –. Wedgwood, R. (). “The Aim of Belief.” In: Philosophical Perspectives , pp. –. White, R. (). “Epistemic Permissiveness.” In: Philosophical Perspectives , pp. –. Williams, B. (). Descartes: The Project of Pure Enquiry (nd edn), Routledge. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research , pp. –.

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12 Whither Higher-Order Evidence? Daniel Whiting

. Introduction Consider: MURDER (Part I) Maria, an experienced detective with an unbroken track record, is investigating a murder. She has gathered and considered carefully the various clues. The evidence they provide—concerning the motives of the suspects, their whereabouts at the time of the killing, their access to the murder weapon, and so on—suggests that the butler is innocent. In this case, one might think, it is rational for Maria to believe that the butler is innocent. Of course, the case is underdescribed, but it is highly plausible that there are ways of filling in the details which preserve this verdict. However, the story continues: MURDER (Part II) Maria is aware that the butler is her child, which suggests that she is not in a position to assess the evidence; more specifically, it suggests that her assessment of the evidence is prejudiced in favour of the butler’s innocence. In the full version of MURDER, one might think, it is not rational for Maria to believe (outright) that the butler is innocent, notwithstanding the fact that the clues suggest as much.¹ Indeed, one might think this even if the evidence is misleading and Maria does not in fact suffer from bias. Call evidence which bears on whether a proposition is true, that is, which indicates or makes it likely that a proposition is (or is not) true, first-order evidence. Call evidence which bears on whether one is able to assess or respond to one’s evidence concerning a proposition, higher-order evidence.² In MURDER, the clues provide first-order evidence while Maria’s relationship to the suspect provides higher-order evidence.

¹ Christensen (), DiPaolo (), Schechter (), Sliwa and Horowitz (), and Saul (), among many others, offer similar verdicts about similar examples. ² Some use the label instead, or in addition, for evidence about what evidence one has (or will have, or lacks, etc.), or for evidence about what one’s evidence supports. For reasons of space, I will not consider whether the points to follow generalize. Daniel Whiting, Whither Higher-Order Evidence? In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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Cases like MURDER might encourage us to think that higher-order evidence makes a difference to whether it is rational for a person to believe a proposition. In this chapter, I consider in what way it might do this.³ More specifically, I consider whether and how higher-order evidence plays a role in determining what it is rational to believe distinct from that which first-order evidence plays. To do this, I turn to a theory with considerable explanatory and expressive power, the theory of (normative) reasons, and try to situate higher-order evidence within it.⁴ Surprisingly perhaps, the only place I find for it there is as a reason for desire—for example, a reason for wanting to avoid certain beliefs—and as a reason for action—for example, as a reason to bring it about that one avoids those beliefs. There are two conclusions one might draw from this. First, one might think that the theory of reasons needs supplementation or revision so as properly to accommodate higher-order evidence. Second, one might instead take it to cast doubt on the idea that higher-order evidence makes a difference to whether it is rational to believe a proposition. I do not rule out the first but I do suggest that some of the points that emerge along the way support the second. Before proceeding to the main discussion, I will make some preliminary remarks about rationality and about the importance of the topic (over and above its selfstanding interest). I use ‘rational’ here as a label for the (or a) status that the beliefs of subjects in Gettier () scenarios possess. This is a positive epistemic status distinct from mere blamelessness.⁵ Some might prefer the label ‘justified’ or ‘reasonable’. The terminology is not important in what follows. I focus on what it is rational for a person to believe (ex ante rationality), not what she rationally believes (ex post rationality). In addition, I focus on what it is overall or all-things-considered rational for a person to believe, not what it is rational to some degree or in some respect for her to believe. I take no stand here on the dispute between internalists and externalists about rationality, or the corresponding dispute about evidence. As far as I can tell, the outcomes of those debates make little to no difference to the points I make.⁶ Why care about the role of higher-order evidence? For one thing, reflection on higher-order evidence might put pressure on a popular view: evidentialism.⁷ According to evidentialism, as I understand it here, it is rational for a person to believe a proposition if and only if her evidence suggests that it is true (cf. Conee and Feldman ). Consider again MURDER. Maria’s evidence suggests that the butler is

³ So as to keep things manageable, I focus on the rationality of full or outright belief, rather than partial belief or credence. How to extend the discussion to degrees of belief is an interesting question but not one I address. ⁴ Some have reservations about the importance of the notion of a reason for epistemology. For discussion, see Sosa and Sylvan (). For other examples of framing or approaching questions in epistemology using the framework the theory of reasons provides, see the contributions to Reisner and Steglich-Petersen (eds) (). ⁵ On the difficulties of specifying the conditions under which a belief is blameless, see Srinivasan (). ⁶ I also take no stand on the nature of the evidential relation. I assume only that it is objective in the sense that what a person’s evidence supports is independent of what she takes her evidence to support or whether she can tell what it supports. ⁷ DiPaolo () and Worsnip () make this point in different ways.

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innocent. But, given the likelihood of bias, it is not rational for Maria to believe that the butler is innocent. So, evidentialism is false. No doubt there are other ways to formulate evidentialism. Indeed, reflection on cases like MURDER might prompt one to explore alternatives. But this point only bolsters the claim about the significance of higher-order evidence. Another reason to care about higher-order evidence is that its effects might be surprising, even troubling. Some suggest, for example, that higher-order evidence generates rational dilemmas, situations in which a person is subject to conflicting requirements (see Christensen , , ).⁸ Consider again MURDER. If Maria fails to believe that the butler is innocent, she is ignoring the evidence the clues provide. But, if Maria believes that the butler is innocent, she is ignoring the threat of bias. More generally, rationality requires one to believe in accordance with one’s first-order evidence, and it requires one to believe in accordance with one’s higher-order evidence, but it is not always possible to satisfy both requirements. As Christensen puts it, situations involving higher-order evidence can be ‘rationally toxic’ (: ).⁹ Others suggest that reflection on higher-order evidence supports scepticism— either of a general sort or in particular domains.¹⁰ If the risk of bias in MURDER makes it irrational for Maria to believe that the butler is innocent, then many, perhaps most, of our beliefs are irrational. After all, there is plenty of evidence that ordinary thinkers are subject to biases and other irrational influences in the beliefs they form and revise, and are prone to make mistakes of various kinds in their reasoning and assessment of evidence. If many, or most, of our beliefs are irrational, then many, or most, of our beliefs fall short of knowledge. I am not endorsing these lines of thought; indeed, I query each of them below. The point for now is just that reflecting on higher-order evidence might lead to nontrivial conclusions.

. The motivational constraint In what follows, I often appeal to a motivational constraint on reasons: MC That p is a reason for a person to φ only if that person can φ for the reason that p.¹¹ ⁸ Some draw another surprising moral, namely, that higher-order evidence gives rise to cases in which it is rational to believe against one’s better judgement, that is, in which epistemic akrasia is rational. For discussion, see Coates (), Christensen (), Feldman (), Greco (), Horowitz (), Lasonen-Aarnio ( and forthcoming), Littlejohn (b), Sliwa and Horowitz (), Titelbaum (), and Worsnip (). I do not tackle that issue here. ⁹ In fact, I do not think that a person’s evidence ever rationally requires her to believe a proposition; it only ever rationally permits her to do so (see Nelson , Whiting ). I will not press this point, since I doubt for independent reasons that higher-order evidence generates rational dilemmas. ¹⁰ For relevant discussion, see Elga (), Feldman (), Feldman (), Saul (), Schechter (, §.), and Schoenfield (). ¹¹ Proponents of MC include Gibbons (), Kelly (), Kolodny (), Parfit (, p. ), Raz (, p. ), Shah (), and Williams (). MC entails but is not entailed by the principle reason implies can, according to which, if that p is a reason for a person to φ, she can φ (see Streumer ).

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To put the same point differently, a normative reason is a possible motivating reason.¹² As stated, MC is a condition on reasons in general, including, but not limited to, reasons for belief. MC offers a way to capture the intuitive thought that reasons provide guidance. As Korsgaard says, ‘A practical reason must function [ . . . ] as a guide’ (, p. ). Raz puts it in more general terms: ‘normative reasons can guide agents’ (, p. ). There is no point in providing guidance to a person, one might think, if she cannot be guided by it in what she does, thinks, or feels, that is, if she cannot act, think, or feel in light of it. To see that MC is plausible, suppose that it is good for Holly to grow taller. This improves her prospect of joining the basketball team. Holly can grow taller; indeed, she is growing taller. However, Holly cannot grow taller for the reason that it improves her prospect of joining the team, or any other reason for that matter. Her growth is not responsive to reasons. According to MC, that it improves her prospect of joining the team is not a reason for Holly to grow taller. In contrast, suppose that it is good for Holly to go to practice. This too improves her prospect of joining the team. Holly can go to practice; indeed, she is going to practice. Moreover, Holly can go to practice for the reason that it improves the prospect of joining the team. So, MC allows that the fact that it improves her prospect of joining the team is a reason for Holly to go to practice. These seem the right results. Accepting MC does not commit one to thinking that, if there is a reason for a person to do, think, or feel something, she can tell that she has that reason, or that it is a reason, or that she is doing, thinking, or feeling something for that reason. This follows only on the assumption that to respond to a reason one needs to be able to tell such things, which is questionable at best. Of course, there are challenges to MC. Since I tackle the main objections elsewhere (Way and Whiting ), I take it for granted here.

. Reasons for and against believing As mentioned above, I will consider how higher-order evidence fits into the theory of reasons, what place it might have within that framework. The starting-point is the notion of a reason. A reason is a consideration which counts in favour of or justifies an attitude or action. Reasons have weights and (so) one reason can be weightier than another. For example, that an interesting film is showing might be a reason of some weight for Miyuki to go the cinema, while that she promised to stay at home might be a weightier reason for her not to go. A plausible and widely held view is that a person’s reasons contribute to determining what it is rational for her to think, feel, or do.¹³ Another plausible and widely held view is that, if a consideration is evidence for or against the truth of a

¹² For discussion of the distinction between normative reasons and motivating reasons, see Alvarez (), Mantel (b). ¹³ For discussion of what it is to ‘have’ a reason, see Alvarez (), Comesaña and McGrath (), Schroeder (), and Lord ().

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proposition, it is a reason for or against believing that proposition.¹⁴ Does higherorder evidence play this role? Higher-order evidence might in this way provide a reason for a higher-order belief.¹⁵ In MURDER, the evidence of bias might suggest that it is not rational for Maria to believe that the butler is innocent, or that she cannot rationally believe this; it might thereby give Maria a reason for beliefs about the rational status of believing that the butler is innocent. However, the present concern is with the impact of higher-order evidence on first-order attitudes. Might higher-order evidence provide a reason for or against a first-order belief? It might. After all, any consideration can be evidence for the truth of any proposition, given a suitable background. Imagine that, in MURDER, Maria knows that, if the butler is her child, then the butler is guilty. Against this background, the evidence that her assessment of the clues will be prejudiced is evidence that the butler is guilty. In this way, it is a reason for Maria not to believe that the butler is innocent. In this way, in turn, it might make a difference to what it is rational for her to believe. In this role, higher-order evidence satisfies MC. Maria might not believe that the butler is innocent for the reason that the butler is her child, that is, in response to evidence that the butler is guilty. So, one way for higher-order evidence to affect the rationality of belief is for it to provide a reason for or against belief by providing evidence for or against the truth of a proposition. In this way, however, higher-order evidence does not play a distinctive part—the role it plays is simply that of first-order evidence. It is higher-order in name only or, to borrow Lasonen-Aarnio’s phrase (, p. ), ‘just more evidence’.¹⁶ I have some sympathy with this idea and return to it below. For now I will continue the search for some contribution higher-order evidence as such might make. To that end, suppose that, in MURDER, Maria has no background evidence relative to which the fact that the butler is her child is evidence for or against the proposition that the butler is innocent. Perhaps Maria and her child are estranged, and she knows nothing about the butler beyond the evidence the clues provide. In that version of the case, one might think, it remains irrational for Maria to believe that the butler is innocent, given the likelihood of bias. What role, then, is her higher-order evidence playing? At this point, one might point out that, even if every reason for believing a proposition is evidence for its truth, it is not the case that every reason against believing a proposition is evidence against its truth. By the same token, some reasons against believing are not reasons to disbelieve a proposition, or to believe its negation.¹⁷ For example, that Isabella’s evidence suggests neither that some proposition is true nor that ¹⁴ Kelly goes so far as to say, ‘ “reason to believe” and “evidence” are more or less synonymous’ (). That, I think, is going too far (cf. Whiting , §.). ¹⁵ Cf. Coates (), Kelly (), Lasonen-Aarnio (), and Worsnip (). Titelbaum () and Littlejohn (b) deny that higher-order evidence makes it rational for a person to believe such higherorder propositions, at least when it is misleading. But the claim here is the weaker one that it at least provides a reason for so believing. ¹⁶ This seems to be how Kelly (, though compare ) and Worsnip () view higher-order evidence. ¹⁷ Are they, then, reasons to suspend judgement with respect to that proposition? That depends on what it is to suspend. For discussion, see Friedman ().

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it is untrue is, plausibly, a reason for her not to believe it. But that reason is not itself evidence Isabella possesses for or against the relevant proposition; it is, rather, a fact about her evidence.¹⁸ By the same token, it is not a reason for Isabella to disbelieve the proposition. This opens up the prospect that higher-order evidence provides a reason against believing a proposition without providing evidence against it. In this way, it might have a distinctive role to play in determining whether it is rational for a person to believe a proposition. However, it is not enough simply to suggest that higher-order evidence is a reason against believing, especially given the puzzling or non-trivial consequences to which that suggestion might lead. We need a plausible account of how or in virtue of what higher-order evidence might provide a reason against believing, one which allows us to understand how that reason might interact with the reasons the first-order evidence provides so as to determine what it is rational to believe. In Isabella’s case, it is not hard to see why, given that her evidence suggests neither that the relevant proposition is true nor that it is false, it is not rational for her to believe it. After all, if her evidence does not support the proposition, she lacks a (sufficient) reason for believing it. Evidently, that is not what is going on in cases like MURDER. Moreover, in Isabella’s case, it is clear that her reason against believing is derivative. Once the (evidential) considerations that provide reasons for believing the proposition and reasons for disbelieving it are in place, the (non-evidential) reason for not believing it is in place. But, again, that is not what is going on in MURDER. In what follows, I will consider what might be going on instead.

. Modification It is widely recognized in the theory of reasons that, alongside reasons, there are modifiers. These come in two species: intensifiers and attenuators (cf. Dancy , pp. –). An intensifier is a consideration that increases the weight of a reason. An attenuator is a consideration that decreases the weight of a reason.¹⁹ For example, that Cora will be at the party might be a reason for Frank to go (since Cora is a fun person to be around). However, that Nick will be there makes the reason less weighty (since Cora is less fun when Nick is around). That Nick will be at the party is, then, an attenuator. To give another example, suppose that Juan testifies that the chauffer did it. This is evidence that the chauffer did it, and thereby a reason for Maria to believe that the chauffer did it. However, Juan and the chauffer are arch-enemies. This makes the reason less weighty—given the

¹⁸ For this point, see Schroeder (a), Littlejohn (a), and Lord (). ¹⁹ I focus in the remainder on attenuators. There are also disablers, considerations which make what would otherwise be a reason not one (cf. Dancy , pp. ff.). One might suggest that higher-order evidence disables, rather than attenuates, the reasons the first-order evidence would otherwise provide. This suggestion will face similar problems to those I discuss. In addition, it will struggle to account for cases in which the higher-order evidence seems to make it less rational, but not irrational, to believe the proposition the first-order evidence supports.

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animosity, Juan’s testimony is less likely to be true. That Juan and the chauffer are arch-enemies is, then, an attenuator.²⁰ If a consideration attenuates a reason for believing a proposition, it is thereby a reason against believing it, though it is not thereby a reason for disbelieving it (cf. Pollock , p. ).²¹ That Juan and the chauffer are arch-enemies is a reason for Maria not to believe that the chauffer did it. If Maria does not believe that the chauffer did it, she might (partially) justify the omission on the grounds that, though Juan said that the chauffer did it, Juan and the chauffer are arch-enemies. In view of this, one might ask whether higher-order evidence attenuates the reasons first-order evidence provides for a proposition and thereby provides a reason against believing it. This would explain why, in MURDER, it is not rational for Maria to believe that the butler is innocent. That the butler is her child reduces the weight of the reasons for believing that the butler is innocent which the clues provide (below whatever the threshold is for rational belief ).²² As others point out, cases involving higher-order evidence seem unlike typical cases of attenuation.²³ Juan’s relationship to the chauffer makes his testimony less reliable as a guide to whether the chauffer did it; their animosity makes it less probable that what Juan says is true. In contrast, Maria’s relationship to the butler does not make the clues less reliable as a guide to whether the butler is innocent. The risk of bias notwithstanding, the clues continue to suggest that the butler is innocent.²⁴ To bolster the view that higher-order evidence cannot play the role of an attenuator, I will appeal to MC. Higher-order evidence, I suggest, does not satisfy MC. One might think that higher-order evidence obviously satisfies MC. In MURDER, it is surely possible for Maria not to believe that the butler is innocent for the reason that she is not in a position to assess the evidence. At this point, further comment on MC is in order. The idea it is supposed to capture is that, if that p is a reason for a person to φ, it must be possible for her to φ for the reason that p in virtue of the fact that or because that p is a reason for her to φ (cf. Shah , pp. ff.; Way and Whiting , pp. ff.). This connects to the thought about guidance. A person is not guided by a consideration if she is not sensitive to the guidance it provides. It is possible for a person to φ for a reason for φing without doing so because it is a reason for φing. Suppose that Donald knows that he lives in the White House and, if he lives in the White House, he is President. On that basis, Donald concludes that he is President. However, when Donald forms his belief, he follows this rule of inference: ²⁰ Considerations that attenuate reasons for believing are, to use Pollock’s () terminology, (partial) undercutting defeaters. ²¹ This is a claim about attenuators on reasons for believing. I take no stand here on whether this holds for attenuators on reasons for other attitudes or for actions. ²² Feldman (, ) seems to view higher-order evidence in this way. ²³ Lasonen-Aarnio (, p. ) and Christensen () indicate ways in which defeat by higher-order evidence differs from undercutting defeat. ²⁴ I assume here that a consideration attenuates the weight of a reason for believing a proposition (only) by attenuating the weight of the evidence it provides for the truth of that proposition. In the practical domain, a consideration might attenuate the weight of a reason, not by attenuating the evidence it provides that some option will realize or promote some value, but by attenuating the value itself. Nothing analogous to this second dimension of attenuation occurs in the epistemic domain.

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 -  ?



from any consideration concerning Donald, infer that he is President. In that case, though Donald’s reasons for believing are in fact reasons which justify doing so, he is not responding to those reasons as such. He is not manifesting sensitivity to the reason-giving force of the relevant considerations. After all, Donald would have drawn the same conclusion had he believed that he does not live in the White House.²⁵ So, MC is to be understood as requiring that it be possible for subjects to respond to the relevant reason as such. My suggestion, then, is that higher-order evidence does not attenuate, since a person cannot respond to it as such.²⁶ Consider: Maria might revise her belief that the chauffer did it for the reason that Juan and the chauffer are enemies because their animosity makes it less likely that Juan’s testimony is true. In this way, she manifests sensitivity to the reason for not believing. But, in MURDER, Maria cannot revise her belief that the butler is innocent for the reason that her assessment of the evidence is likely prejudiced because the risk of bias means that the clues do not suggest as strongly that the butler is innocent. For one thing, to return to an earlier point, it is false that the risk of bias means that the clues do not suggest as strongly that the butler is innocent. For another, to respond to an attenuator as such or in its role as an attenuator a person must respond to its attenuating, that is, to its having an effect on the weight of the original or unmodified reason. To respond to that, to the difference the attenuator makes, a person must be responsive to the weight of the unmodified reason. But in cases involving higher-order evidence a person cannot simultaneously manifest sensitivity to the weight of the unmodified reason and manifest sensitivity to the supposed attenuator’s effect on it. In MURDER, if Maria treats the clues as having a certain weight apart from her higher-order evidence, she is not treating the risk of bias as genuine. Conversely, if she is sensitive to that risk, she will not treat those clues as having that weight apart from her higher-order evidence. After all, the higher-order evidence suggests that she is not in a position to assess the evidence the clues provide. To bolster this point, consider how Maria might manifest sensitivity to the weight of the original evidence, and the effect of her higher-order evidence on it, in her thought or talk: ‘Were there no risk of bias, the clues would suggest that the butler is innocent. But my assessment is probably biased. So, the clues do not suggest that.’ This is confused at best. It is helpful in this context to model sensitivity to an attenuator as a matter of applying a function to an unmodified reason with a certain weight to deliver

²⁵ For similar examples and discussion of what it is to a respond to a reason as such, see Arpaly and Schroeder (, ch. ), Lord and Sylvan (forthcoming), Mantel (a), and Way (). ²⁶ Coates () makes a similar point in different terms. He argues that a person cannot rationally regard higher-order evidence as defeating the rationality of her first-order belief. Coates then appeals to a Transparency Requirement, according to which a consideration is a defeater ‘only if it is possible for those who encounter the consideration rationally to regard it’ as such (, p. ). It follows that higher-order evidence is not a defeater. In what follows, I appeal, not to the Transparency Requirement, but to MC, which demands less cognitive sophistication. MC allows that there might be a reason for a person to respond in some way, or an attenuator on such a reason, even if she is unable to regard (think of, represent) it in those terms.

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

 

a modified reason with a different weight (which might then serve as input to deliberation).²⁷ In the testimony case, Maria enters the weight of the reason Juan’s testimony provides independent of context, applies the Juan-hates-the-chauffer function, and arrives at the modified weight of the reason. But we cannot model Maria’s sensitivity to her higher-order evidence in MURDER in this way. To what does she apply the I-am-probably-biased function? What does she input? Not the weight of the reason she takes the clues to provide. After all, if she is biased, that is the wrong thing to apply the function to. It seems, then, that is not possible for a person in a case involving higher-order evidence to respond to that evidence in the capacity of an attenuator on first-order evidence. By the same token, it is not possible for her not to believe the relevant proposition on the basis of the corresponding reason as such. Given MC, it follows that higher-order evidence does not attenuate.

. Second-order reasons So far, I have argued that higher-order evidence does not play the role of an attenuator. What other role might it play? Again, I turn to the theory of reasons. Raz (, pp. –, –) suggests that, in addition to reasons for and against acts and attitudes, there are second-order reasons. Second-order reasons are reasons for or against responding to certain reasons, that is, reasons for or against acting or having an attitude for or on the basis of other reasons.²⁸ To adapt Raz’s example, suppose that Kelly promises her partner to make decisions about their child’s schooling on the basis of education-related reasons alone, and not, say, in view of what might be best for their careers. Suppose further that Kelly is deliberating as to whether to send their child to a certain school. The promise, according to Raz, is not a reason for or against sending their child to that school—it does not reveal or indicate any positive or negative features of the school. Nor, for that matter, does it modify those reasons—the pros and cons remain as weighty. Instead, the promise is a second-order reason for Kelly not to decide to send their child to that school for the reason, say, that it will shorten their commute. In view of this, one might suggest that higher-order evidence plays the role of a second-order reason—a reason for or against basing one’s belief on certain first-order considerations. In MURDER, the risk of bias is a second-order reason for Maria not to believe that the butler is innocent on the basis of the evidence the clues provide.²⁹

²⁷ Cf. ‘The unmodified reason is always prior to the modified reason. The former is the input to the modification function, yielding the latter as its output’ (Bader , p. ). ²⁸ Second-order reasons are not merely reasons for considering certain reasons or directing one’s attention away from others. Such reasons are just first-order reasons—reasons to perform certain mental acts. ²⁹ This is how I understand Christensen’s ‘bracketing picture’ (, forthcoming; see also Elga ). Note that it is non-trivial to suggest that higher-order evidence provides second-order reasons. To talk of ‘second-order reasons’ in this context is not just to talk of considerations that operate at a higher order; it is to talk of considerations which play a distinctive role. It is a substantive question whether higher-order evidence plays that role.

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 -  ?



If one is going to appeal to this idea to explain why it is not rational for Maria to believe that the butler is innocent, one has to make the further assumption that only the reasons a person has which are not excluded by her second-order reasons contribute to determining what it is rational for a person to think. I will not spend long on this proposal as I have argued against it at length elsewhere (Whiting ). A key objection is that putative second-order reasons do not satisfy MC. It is possible to φ for a reason but it is not possible to φ-for-a-reason for a reason. Kelly can decide to send her child to a school. And she can decide to do so for educational reasons. But she cannot decide to send her child to a school for educational reasons for the reason that she promised to do so. Since second-order reasons do not satisfy MC, they are not really reasons at all, at least, not reasons for or against acting or having an attitude for a reason. If there are no second-order reasons, then higher-order evidence cannot provide such reasons. That is not to say that the relevant considerations provide no reasons at all. In Whiting (), I suggest that Kelly’s promise, for example, is a reason for wanting to make schooling decisions on basis of educational reasons alone, or for bringing it about that she does so. More generally, putative second-order reasons are really firstorder reasons for desires and actions. Might higher-order evidence play a similar role? I return to this later.

. State-given reasons Is there another place for higher-order evidence in the theory of reasons? A familiar and widely discussed distinction is between object-given and state-given reasons for attitudes.³⁰ Object-given reasons indicate or reveal something about the object of the attitude, while state-given reasons indicate or reveal something about the attitude itself. Suppose that Maria believes that the butler is innocent. Evidence that the butler is innocent—for example, that the butler lacked a motive—indicates that what she believes, the object of her belief, is true. It is, then, an object-given reason for believing. In contrast, suppose that believing that he is President makes Donald happy. This concerns, not what he believes, but his believing it. So, if it is a reason for believing, it is a state-given reason. The issue at hand is what role higher-order evidence plays in determining what it is rational to believe. First-order evidence, like the clues in MURDER, provides object-given reasons for and against believing. Perhaps higher-order evidence, like Maria’s relationship to the butler in MURDER, provides state-given reasons. This is a suggestion DiPaolo () makes. According to DiPaolo, state-given reasons for or against an attitude are valuebased; they concern respects in which having that attitude is good or bad. For example, that it makes Donald happy to believe that he is President is a respect in which so believing is good (for him). DiPaolo’s proposal is that in the same way higher-order evidence can indicate that believing a proposition is bad in a respect ³⁰ For this terminology, see Parfit (). For some important discussions of the distinction, though not always in these terms, see D’Arms and Jacobson (), Hieronymi (), Olson (), Rabinowicz and Rønnow-Rasmussen (), Schroeder (b), Sharadin (), and Way ().

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(, p. ). In MURDER, Maria’s relationship to the butler indicates that her believing that the butler is innocent is irrational, which is a bad thing. In this way, higher-order evidence can provide state-given reasons against believing a proposition (though not reasons for believing its negation). If the state-given reason not to believe a proposition the higher-order evidence provides outweighs the object-given reasons to believe it which the first-order evidence provides, it can make it irrational to believe that proposition. This is an interesting proposal. One advantage of it, to return to an early theme, is that in addition to suggesting that there are non-evidential reasons against believing it tells us what grounds those reasons, namely, expected values (or, rather, disvalues). One challenge facing it is to explain how the object-given reasons weigh against or interact with the state-given reasons so as to determine what it is rational for a person to believe. It is not clear that they are commensurable.³¹ Here is a different worry. State-given reasons are thought by many not to make a difference to whether it is rational to have an attitude (see Schroeder b). That so believing makes him happy does not make it rational for Donald to believe that he is President. That might suggest that the idea that higher-order evidence gives stategiven reasons cannot explain how such evidence bears on what it is rational to believe. The point about rationality connects to a further point. Many take a characteristic feature of state-given reasons to be that it is not possible to respond to them, in the sense that it is not possible to hold an attitude for such reasons (see Kelly , Parfit , Raz , Schroeder b, and Shah ). Donald cannot believe that he is President for the reason that so believing makes him happy. This might explain why state-given reasons do not contribute to determining whether it is rational to have an attitude. If it is not possible to respond to them, it is not a failure of rationality not to do so. This point in turn connects to MC. Given MC, if subjects cannot have attitudes for state-given reasons, they are not really reasons (in anything but name) for the relevant attitudes (see Kelly , Shah ).³² Proponents of this line of thought typically add that, what might seem to be state-given reasons for an attitude are in fact object-given reasons for wanting to have that attitude (or for causing that attitude). For example, that it makes him happy to do so is a reason, not for Donald to believe that he is President, but for him to want to believe this. Building on these points, I will suggest that, if higher-order evidence provides stategiven reasons, it is not possible for a person to believe or to refrain from believing for those reasons. Given MC, it follows that they are not really reasons at all.³³ ³¹ DiPaolo acknowledges this challenge but does not try to address it (, fn. ). For a story about how practical considerations might weigh against or alongside evidential considerations, see Reisner (). It is a nice question whether Reisner’s proposal combines with DiPaolo’s. ³² In Whiting (), I develop and defend a theory of reasons which, together with the plausible assumption that truth is the correctness-condition for belief, entails that state-given reasons for belief are not (really) reasons. But I do not rely on that theory here. ³³ Schroeder (a) argues that there are state-given reasons against believing of a certain sort, namely, reasons provided by what is at stake in holding a belief. According to Schroeder, those reasons bear on whether it is rational to believe a proposition and (because) it is possible to respond to them, hence, they satisfy MC. In both respects, Schroeder takes the stakes-based reasons to differ from more familiar

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 -  ?



Again, it might seem that it is possible for Maria not to believe that the butler is innocent for the reason that the butler is her child. Recall, however, that the issue is whether it is possible for her to respond to that consideration as a state-given reason for believing. To support the suggestion that this is not really possible, I will proceed in an indirect fashion. According to DiPaolo, higher-order evidence provides a state-given reason against believing by indicating a way in which so believing is bad, namely, that it is irrational. In view of this, suppose that Hilary knows that Donald is President.³⁴ In this case, it seems rational for her to believe that Donald is President or the moon is made of cheese. After all, her evidence a priori entails this. However, a malicious and allpowerful demon tells Hilary that, if she believes the disjunction, it will ensure that her belief is irrational, say, by making her forget that Donald is President. The demon’s testimony indicates a way in which believing the disjunction would be bad, namely, that it would be irrational. So, parity of reasoning suggests that it is a state-given reason against so believing. It is, of course, possible for Hilary not to believe the disjunction—she might not have bothered to draw out this consequence. Plausibly, however, it is not possible for her not to believe the disjunction for the reason that the demon will make her forget her evidence if she does so. The demon’s threat cannot be Hilary’s reason for not believing what is entailed by her evidence. She cannot refrain from believing the proposition which her evidence supports in light of the fact that, were she to do so, the demon would interfere with her mental states. Given MC, the demon’s threat is not a reason against believing. Setting aside MC, it seems clear that, in this case, it is rational for Hilary to believe the disjunction, the demon’s threat notwithstanding.³⁵ Although the demon will make that belief irrational, if Hilary forms it, it is presently the rational thing for her to believe. So, even if the demon’s threat provides a state-given reason against believing, it is not one which bears on what it is rational for her to believe. Here, then, is the argument. If the evidence that Maria’s assessment of the clues will be prejudiced in MURDER is a state-given reason against believing, one which bears on what it is rational to believe, then so is the demon’s threat. The demon’s threat is not a state-given reason against believing, at least, not one which bears on what it is rational to believe. So, the threat of bias is not a state-given reason against believing, at least, not one which bears on what it is rational to believe. More generally, higher-order evidence does not provide state-given reasons against believing, at least, not reasons which bear on what it is rational to believe. A proponent of the view under consideration might insist that cases like MURDER and the case involving the demon are disanalogous. Alternatively, she might insist that, in the demon case, Hilary is able to respond to its threat in the relevant fashion. Rather than explore such responses, I will argue that the view is independently implausible. examples of state-given reasons, such as Donald’s happiness-based reason. I do not here take a stand on Schroeder’s view. For critical discussion, see Mueller (). ³⁴ One can add that she knows this for certain, or knows that she knows this, or that she has mastered the rule for disjunction-introduction. ³⁵ The issue here is what it is ex ante (not ex post) rational for Hilary to believe.

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

 

The proposal that higher-order evidence provides state-given reasons against believing faces what one might call a problem of containment. Consider this transmission principle (cf. Way ): If that p is a state-given reason for φing, and ψing is a means to φing, then that p is a state-given reason for ψing. For example, that it makes Donald happy to believe that he is President is a stategiven reason for him so to believe. If believing that he lives in the White House is a means to believing that he is President, then that it makes Donald happy to believe that he is President is a state-given reason for him to believe that he lives in the White House. The transmission principle does not hold for object-given reasons (cf. Way ). That Donald is President is a reason for Hilary to believe that Donald is President or the moon is made of cheese. Suppose that believing that the moon is made of dairy would facilitate believing the disjunction. Nevertheless, that Donald is President is not an object-given reason for believing that the moon is made of dairy; it is not evidence that this is true. Nor is it a state-given reason for so believing; that Donald is President does not indicate or reveal some respect in which it is good for Hilary to believe that the moon is made of dairy. In view of this, consider again MURDER. Suppose that Maria’s relationship to the butler is a state-given reason for her not to believe that the butler is innocent, insofar as it provides evidence that so believing is irrational. Suppose also that disbelieving the (first-order) evidence—say, that the butler was out of the country at the time—or believing without evidence that some defeating consideration obtains—say, that the butler’s prints are on the gun—would allow Maria not to believe that the butler is innocent. Given the transmission principle, it follows that Maria’s relationship to the butler gives her a state-given reason against believing that the butler was out of the country, or for believing (without evidence) that the butler’s prints are on the gun. In turn, Maria’s relationship to the butler might make it irrational for her to believe that the butler was out of the country, or rational for her to believe that the butler’s prints are on the gun. While it might be plausible to think that, given her relationship to the child, it is not rational for Maria to believe that the butler is innocent, it is not plausible to think that, given her relationship to the child, it is rational for her to disbelieve her first-order evidence or to form beliefs without evidence. The point here is that, while the suggestion that higher-order evidence provides state-given reasons against believing might deliver the desired verdict when it comes to Maria’s belief that the butler is innocent, it does so at the cost of delivering highly problematic verdicts concerning any number of other beliefs. So, quite apart from MC, there is reason to reject the suggestion. While I do not accept DiPaolo’s proposal, I think that it contains an important insight. Higher-order evidence indicates or suggests that believing a proposition is bad in a respect, say, that it is irrational. This is an object-given reason for wanting not to believe that proposition (perhaps also for causing oneself not to believe it). So, just as state-given reasons are (really) reasons for desire, higher-order evidence provides reasons for desire. I explore this further in section ..

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 -  ?



. Reasons for and against wanting and acting Reflection on cases like MURDER might encourage the thought that higher-order evidence can make a difference to what it is rational to believe. The question I have asked is: how so? What role might higher-order play such that it bears on whether it is rational to believe a proposition? Higher-order evidence, I allow, might provide (‘just more’) first-order evidence. In that case, it plays the familiar role of a reason for or against belief. I suspect that, for many cases in which higher-order evidence seems to affect what it is rational to believe, it does so, or we imagine that it does so, in this capacity. But does higher-order evidence play any other role? One might think that it makes a difference to what it is rational to believe, even when it does not provide firstorder evidence. How so? Drawing on the theory of reasons, I have considered various possibilities. Specifically, I have considered whether higher-order evidence plays the role of an attenuator on reasons for belief, or a second-order reason against believing for certain first-order reasons, or a state-given reason against believing. I have argued that it cannot play these roles—in each case, the case rests in part on an independently plausible motivational constraint on reasons. As noted at the outset, there are two lessons one might take from this. One is that the theory of reasons as it stands is incomplete or inadequate. Another is that higherorder evidence does not really make a distinctive difference to what it is rational to believe. I cannot here rule out the former option. But there is something to be said for the latter. First, when surveying the candidate roles higher-order evidence might play, it turned out in each case that higher-order evidence is unable to guide our thinking about first-order matters. In this way, a pattern has emerged, one which might be projectable. At this point one might object. Surely Maria might say or think to herself, ‘I’m probably biased, so I shall withhold belief as to whether the butler is innocent.’ This suggests that it is possible for Maria not to believe that the butler is innocent in light of the risk of bias (or the evidence which suggests it). By way of response, note that, in general, ‘I shall’ expresses a practical attitude, such as a decision. Accordingly, I suggest that ‘I shall withhold belief ’ expresses, not a doxastic attitude, but a practical attitude, for example, a decision. It is commonplace that one cannot decide (not) to believe. But one can decide to bring it about that one does (not) believe. This brings me to a second point. The discussion of state-given reasons highlights that higher-order evidence in cases like MURDER indicates a respect in which having a certain belief is bad. A consideration that suggests or indicates that φing is bad in some respect is, in general, a reason to want not to φ. So, higher-order evidence in such cases is a reason to want not to believe certain propositions.³⁶ By the same token, it might serve as a reason to perform (mental or non-mental) actions that result in one’s not believing those propositions. In a similar fashion, higher-order evidence might provide a

³⁶ Or, if one prefers, a consideration that rationalizes so desiring.

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

 

reason to want to counter the risk of bias, for example, by securing expert advice, or as a reason to do this (when possible). Note that the reason for desire that higher-order evidence provides satisfies MC. It is possible for Maria to want not to believe that the butler is innocent, or to want a second opinion, for the reason that the butler is her son (which suggests her assessment is biased). Here we have found a role for higher-order evidence to play, that of a reason for desire. This might seem a surprising conclusion; it suggests that the reasons higherorder evidence provides are non-epistemic or practical. On this view, if Maria believes that the butler is innocent but does not, say, doublecheck her reasoning, or at least want to do so, she might exhibit irrationality of a sort, but the irrationality in question is practical, not epistemic. She is failing to respond to evidence of disvalue, rather than evidence of truth. So, what is irrational is not Maria’s belief, but her lack of concern.³⁷ This proposal, if correct, goes some way toward domesticating higher-order evidence. It shows that higher-order evidence is no threat to evidentialism. In MURDER, assuming that the evidence that Maria’s assessment of the clues is prejudiced is not also evidence of the butler’s guilt, Maria’s evidence suggests that the butler is innocent. So, it is rational for her to believe this. The higher-order evidence might give Maria a reason to want not to believe this but it does not affect what it is rational for her to believe (at the first order). By the same token, if the proposal is correct, it shows that higher-order evidence does not generate a distinctive sceptical threat. If it remains rational for Maria to believe that the butler is innocent, her belief remains a candidate for knowledge. One might, however, suspect that on the view I am exploring higher-order evidence continues to give rise to rational dilemmas. Consider again MURDER. It is rational for Maria to believe that the butler is innocent, given the clues, but rational for her to want not to believe this, given her relationship to the child. If Maria forms the relevant attitudes, is she not conflicted or at odds with herself? Perhaps. But the conflict is of an unproblematic and familiar sort. For one thing, Maria can accord with both the reasons for believing and the reasons for desiring—by having a belief she wants not to have. The situation is not one in which she is unable to satisfy competing considerations. For another, there are many cases not involving anything like higher-order evidence in which it is rational to want not to have rational attitudes. It is rational for Hilary to believe that Donald is President but, since that thought upsets her, it is rational for her to want not to believe this. So, if higher-order evidence gives reasons for desire, rather than reasons against belief, it does not generate rational dilemmas of a troubling sort. Not so fast! Above I suggested that higher-order evidence might provide reasons, not only to want to lack a belief, but also to act so as to bring it about that one lacks a belief (when possible). Doesn’t this allow for problematic dilemmas? In MURDER, Maria is rationally required to believe that the butler is innocent, given the first-order

³⁷ This way of presenting things avoids Schechter’s (, p. ) objection to the idea that a failure to respond to higher-order evidence is a ‘moral or pragmatic’ failure rather than an epistemic one.

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

evidence, and rationally required to bring it about that she does not believe this, given the higher-order evidence. If she satisfies the first requirement, she violates the second, and vice versa.³⁸ However, this is not a case in which there is a requirement (or corresponding reasons) to respond in a certain way and a requirement (or corresponding reasons) not to respond in that way. In MURDER, according to the view under consideration, the evidence the clues provide gives Maria reasons for believing but it does not give her reasons against acting, and the evidence of bias gives Maria reasons for acting but it does not give her reasons against believing. To use Parfit’s terminology (, p. ), the epistemic reasons and practical reasons in this case and others like it compete, in the sense that it is not possible to accord with both, but they do not conflict, in the sense that they support different answers to the same question, namely, ‘What to believe?’ (or, for that matter, ‘What to do?’). In contrast, the suggestion I mentioned in the introduction was that higher-order evidence generates cases of conflict, not just competition, that in cases like MURDER the first-order evidence and the higher-order evidence support different answers to the same question, namely, ‘What to believe?’ So, if the reasons higher-order evidence provides are practical rather than epistemic, the (so-called) dilemmas higher-order evidence gives rise to are of a more benign sort than feared at the outset. Moreover, the view under consideration is not the only one to countenance competitions of the above kind. Any view according to which, for example, the fact that it would make him happy to do so is not a reason for Donald to believe that he is President but, instead, a reason for him to cause this belief will have to tolerate such situations. In closing I will consider a final objection. One might complain that it is simply counterintuitive to think that higher-order evidence (as such) has no bearing on what it is rational to believe at the first order. Note, however, that the competing views which allow for rational conflicts or scepticism are also counterintuitive. So, no position here seems wholly in accord with untutored intuition. Moreover, the positive proposal might help to explain away opposing intuitions—they are tracking, not reasons not to believe, but reasons to want not to believe. While I have shown that higher-order evidence plays the role of a reason for desire, I have not shown that that is the only role it plays (distinct from that of first-order evidence). Nonetheless, if this turn to be the case, it helps to bring higher-order evidence down to earth.

Acknowledgements Thanks to Alex Gregory, Ellie Gwynne, Sophie Keeling, Conor McHugh, Brian McElwee, Genia Schönbaumsfeld, Nils-Hennes Stear, Andrew Stephenson, Kurt Sylvan, Jonathan Way, and anonymous reviewers for helpful feedback on earlier versions of this material.

³⁸ As mentioned in fn. , I do not think that subjects are ever rationally required to believe propositions but I grant here that they are for the sake of argument.

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Titelbaum, M. (). “Rationality’s Fixed Point (or, In Defense of Right Reason).” In: T. Gendler and J. Hawthorne (eds), Oxford Studies in Epistemology , Oxford University Press. Way, J. (). “Transmission and the Wrong Kind of Reason.” In: Ethics , pp. –. Way, J. (). “Creditworthiness and Matching Principles.” In: M. Timmons (ed.), Oxford Studies in Normative Ethics , Oxford University Press. Way, J. and D. Whiting (). “Reasons and Guidance (or, Surprise Parties and Ice Cream).” In: Analytic Philosophy , pp. –. Whiting, D. (). “Truth: The Aim and Norm of Belief.” In: Teorema , pp. –. Whiting, D. (). “Against Second-Order Reasons.” In: Noûs , pp. –. Whiting, D. (). Right in Some Respects: Reasons as Evidence.” In: Philosophical Studies , pp. –. Williams, B. (). “Internal and External Reasons.” In: Moral Luck, Cambridge University Press. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research , pp. –.

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13 Evidence of Evidence in Epistemic Logic Timothy Williamson

The slogan ‘Evidence of evidence is evidence’ may sound plausible, but what it means is far from clear. It has often been applied to connect evidence in the current situation to evidence in another situation. The relevant link between situations may be diachronic (White , p. ): is present evidence of past or future evidence of something present evidence of that thing? Alternatively, the link may be interpersonal (Feldman : ): is evidence for me of evidence for you of something evidence for me of that thing? Such inter-perspectival links have been discussed because they can destabilize inter-perspectival disagreements. In their own right they have become the topic of a lively recent debate (Fitelson , Feldman , Roche , Tal and Comesaña ). This chapter concerns putative intra-perspectival evidential links. Roughly, is present evidence for me of present evidence for me of something present evidence for me of that thing? Unless such a connection holds between a perspective and itself, it is unlikely to hold generally between distinct perspectives. Formally, the singleperspective case is also much simpler to study. Moreover, it concerns issues about the relation between first-order and higher-order evidence, the topic of this volume. The formulations in this chapter have not been tailored for optimal fit with previous discussions. Rather, they are selected because they make an appropriate starting-point, simple, significant, and not too ad hoc. The issue is not what people meant all along by ‘Evidence of evidence is evidence’. Rather, the issue is what there is worth meaning by it in the vicinity. In particular, I will not discuss existential generalizations to the effect that a given hypothesis has some evidential support, in the sense that some part of the evidence supports it. Such principles are usually too weak to be of interest, since from unequivocally negative evidence one can typically carve out a gerrymandered fragment that in isolation points the opposite way—for instance, by selecting a biased sample of data points. Instead, the focus will be on the total evidence. For the sake of rigour and clarity, evidence will be understood in probabilistic terms. On some probabilistic readings of the slogan ‘Evidence of evidence is evidence’, it can be straightforwardly refuted by standard calculations of probabilities for playing cards, dice, and so on. For example, it might be interpreted as saying that Timothy Williamson, Evidence of Evidence in Epistemic Logic In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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 

if p is evidence for q, and q is evidence for r, then p is evidence for r. Counterexamples to such transitivity principles are easy to construct, whether ‘p is evidence for q’ is understood as ‘p raises the probability of q’ or as ‘the conditional probability of q on p’ is high (see also Fitelson ). This chapter is not concerned with principles refuted by standard first-order probabilistic calculations. The principles it discusses all involve readings of the phrase ‘evidence of evidence’ in terms of second-order evidence, evidence for propositions about evidence: more specifically, secondorder probabilities, probabilities of propositions about probabilities. Formal models will be used throughout, within the framework of epistemic logic, since it provides a natural way of integrating first-level epistemic conditions (such as evidence of a coming storm) and second-level epistemic conditions (such as evidence of evidence of a coming storm). An integrated framework is needed to give a fair chance to the idea that evidence of evidence is evidence. We will be asking questions like this: if the probability on the evidence that the probability on the evidence of a hypothesis H is at least % is itself at least %, under what conditions does it follow that the probability on the evidence of H is indeed at least %, or at least more than %? Such principles may remind one of synchronic analogues of more familiar probabilistic reflection principles, and turn out to be sensitive to similar structural features of the underlying epistemic relations (compare Weisberg  and Briggs ). Bridge principles between first-level and higher-level epistemic conditions often turn out to imply versions of highly controversial principles in standard epistemic logic, most notably the contentious principle of so-called positive introspection, that if one knows something, one knows that one knows it, and the more obviously implausible principle of negative introspection, that if one doesn’t know something, one knows that one doesn’t know it (Williamson , pp. –; ). To anticipate, various natural formalizations of the intra-perspectival principle that evidence of evidence is evidence also turn out to have such connections, although more complicated ones than usual. Since the overall direction of this chapter is against those principles, it is dialectically fair to use a formal framework that presents them with no unnecessary obstacles.

. The formal framework For clarity, the formal framework will first be explained, though some readers will already be familiar with it. The underlying models come from modal logic, as adapted to single-agent epistemic logic (Hintikka ), to which we can add the required probabilistic structure (Williamson ). For present purposes, a frame is an ordered pair , where W is a nonempty set and R is a dyadic relation over W, a set of ordered pairs of members of W. Informally, we conceive the members of W as worlds, or as relevantly but nonmaximally specific states of affairs, mutually exclusive and jointly exhaustive. We model (coarse-grained) propositions as subsets of W; thus the subset relation corresponds to entailment, set-theoretic intersection to conjunction, union to disjunction, complementation in W to negation, and so on. If w 2 X, the proposition X is true in the world w; otherwise, X is false in w. We use the relation R to model evidence

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     



for a given agent (expressed by ‘one’) at a given time (expressed by the present tense). More precisely, if w and x are worlds (w, x 2 W) then Rwx ( 2 R) if and only if it is consistent with one’s total evidence in w that one is in x. We define the proposition R(w) as {x: Rwx}, which is the strongest proposition to follow from one’s evidence in w; in effect, R(w) is one’s total evidence in w. Since our interest here is in what follows from one’s evidence, and what follows is automatically closed under multi-premise entailment, concerns about the logical omniscience imposed by such models are less pertinent than elsewhere in epistemic logic. At the very least, the total evidence should be consistent, otherwise it both entails everything and excludes everything. In a probabilistic setting, we want to conditionalize on the evidence, which makes no obvious sense when the evidence is the empty set of worlds. Thus we require that R(w) 6¼ {}; in other words, the relation R is serial in the sense that each world has it to some world. The epistemic interpretation motivates the formal development but plays no further role in it. Formally, we will generalize over every (finite) nonempty set W of entities of any kind and every serial relation R over them. We need to add probabilities to the frames. We assume W to be finite in order to avoid the complications inherent in infinite probability distributions. There is enough complexity and variety in finite probability distributions for most epistemological modelling purposes. A probabilistic frame is an ordered triple where is a frame, W is finite, R is serial, and Pr is a probability distribution over W. Thus Pr maps each subset of W to a real number between  and , where Pr(W) =  and Pr(X [ Y) = Pr(X) + Pr(Y) whenever X and Y are disjoint. We impose one further constraint on Pr: it is regular, in the sense that Pr(X) =  only if X = {} (the converse follows from the other axioms). The reason for that constraint will emerge shortly. Informally, we regard Pr as the prior probability distribution. For present purposes, it need not be absolutely prior; it may embody one’s previously acquired background information. Posterior probabilities in a world w are defined by conditioning Pr on one’s total evidence in w: thus the posterior probability of X in w, the probability Prw(X) of X on the evidence in w, is the prior conditional probability of X on R(w). These conditional probabilities are themselves defined in the usual way as ratios of unconditional probabilities, giving this equation: EVPROB

Prw ðXÞ ¼ PrðX j RðwÞÞ ¼ PrðX \ RðwÞÞ=PrðRðwÞÞ

Of course, the ratio in EVPROB is well defined only if Pr(R(w)) > . Since R is serial, R(w) is nonempty, but we still need regularity to conclude, from that, that Pr(R(w)) > . That is why the constraint was imposed. Informally, the probability of a proposition X on one’s evidence in w is the weighted proportion of worlds consistent with one’s evidence in w in which X is true. A stronger constraint on Pr than regularity is uniformity, which says that for any two worlds w and x, Pr({w}) = Pr({x}): all worlds have equal weight. In a finite probability space, uniformity entails regularity; uniformity equates Pr(X) with the unweighted proportion of members of X that are members of W. We do not impose uniformity, despite the simplicity it brings. For the members of W may represent less than maximally specific possibilities, which may themselves vary in their level of

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

 

specificity: in that case, uniformity in the model would require non-uniformity at a level of resolution finer than that represented in the model. Permitting nonuniformity makes for more robust results. Mathematically, this probabilistic framework is just like the one in Williamson  and . However, there R(w) was informally explained as what the agent knows in w, rather than as the evidence in w. This chapter does not impose the equation E = K of total evidence with total knowledge. The reason is not any loss of confidence in E = K, but simply a preference for addressing a wider range of views of evidence. Obviously, E = K remains compatible with the present framework. A formal correlate of this difference is that since knowledge entails truth, if R(w) is what the agent knows then w 2 R(w), so R is reflexive. All reflexive relations are serial but not vice versa. Even without E = K, the assumption that evidence consists of true propositions is attractive (Williamson : –). Nevertheless, for the sake of generality, we do not impose it. Thus the framework allows one’s total evidence to be what one reasonably believes (in some sense that does not entail truth), rather than knows, so long as one’s reasonable beliefs are jointly consistent. The framework does in effect assume that evidence consists of propositions, true or false. After all, the standard probabilistic operation of conditionalizing on evidence treats evidence as propositional (for more support for the assumption see Williamson : –). The framework automatically includes within the models non-trivial propositions about probabilities on the evidence. For instance, for any proposition X and real number c, we may define Pc[X] as {w: Prw(X)  c}, the proposition that the probability on one’s evidence of X is at least c, which may be true in some worlds and false in others. Thus Pc[X] itself receives a probability Prw(Pc[X]) on one’s evidence in a world w, so Pc[Pc[X]] is in turn well defined: it is the proposition that the probability on one’s evidence that the probability on one’s evidence of X is at least c is itself at least c. Other propositions about probabilities can be defined similarly; for instance, P>c[X] is {w: Prw(X) > c}, the proposition that the probability on one’s evidence of X is greater than c. When interpreting English renderings in which epistemic terms such as ‘one’s evidence’ and ‘the probability of X on one’s evidence’ occur within the scope of further epistemic vocabulary, it is crucial to remember that the embedded occurrences are here to be read de dicto rather than de re. Thus even if the probability of X on one’s evidence is %, one cannot substitute ‘%’ for ‘the probability of X on one’s evidence’ without loss in the scope of another probability operator, for doing so would in effect presuppose that it is certain on one’s evidence that the probability of X on one’s evidence is %. To read the terms for probability de re would preclude by fiat the non-rigid behaviour that represents uncertainty in an epistemic modal setting, and so prevent us from addressing the very epistemic issues we want to discuss. De dicto readings will therefore be understood throughout. They are unambiguously written into the definitions of the formal notation, but one must bear that in mind when paraphrasing formulas into natural language. Of course, the setup just described is not the only formal framework conceivable for theorizing about probabilities on evidence of probabilities on evidence. For instance, one could assign each world its own probability distribution, with no

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     



requirement that they all be derived by conditionalization on one’s evidence in the world from a single prior probability distribution. However, that liberalization would have less chance of vindicating any interesting version of the idea that evidence of evidence is evidence. The present framework is simple, perspicuous, and tightly integrated. In particular, it avoids the ad hoc move of postulating a second probability distribution to handle second-order probabilities, which would also tend to undermine the prospects for inferences from evidence of evidence to evidence, by leaving too many degrees of freedom. In such respects the framework provides a comparatively hospitable environment for those inferences. If they do not thrive here, they are not robust. The framework is also mathematically tractable, permitting us to prove general results and through them to understand what the successes and failures of ‘evidence of evidence’ principles depend on. We can use the framework to formulate principles of forms such as this: Pa ½Pb ½X  Pc ½X

ðiÞ

This says that whenever the probability on the evidence that the probability on the evidence of X is at least b is itself at least a, the probability on the evidence of X is at least c. Such an inclusion is valid on a finite serial frame if and only if it holds for every proposition X  W and every regular probability distribution Pr over W. Given real numbers a, b, c between  and , under what conditions on is (i) valid on that frame? We shall not be much concerned with validity on a probabilistic frame , where the probability distribution is held fixed rather than generalized over. This is partly for reasons of mathematical tractability: validity on a probabilistic frame tends to be ultra-sensitive to complicated combinatorial effects. Generalizing over probability distributions helps smooth out such complications. But it is anyway unclear what general constraints on the probability distribution might be appropriate, particularly once the worlds are allowed to represent possibilities of varying degrees of specificity, motivating different probabilistic weights. Of course, if the particular set W happens to consist of genuine epistemic situations, which we interpret at face value, both the appropriate accessibility relation R and the appropriate probability distribution Pr may be fixed, but that is to go outside the structural realm in which a formal inquiry proceeds. If two probabilistic frames and are isomorphic, in the sense that some bijection maps W onto W* and induces mappings of R onto R* and Pr onto Pr*, then whatever formal constraint one frame satisfies, so does the other. Features of that depend on the intrinsic natures of the members of W are not formal in that sense. What is hard to motivate is a purely formal constraint passed by but failed by , where Pr and Pr are two regular probability distributions over W. A reasonable default hypothesis is that all regular probability distributions over the given frame are equally legitimate, from a formal perspective. But this chapter is intended to open up more questions than it closes down: obviously, readers are welcome to try their hand at formulating well-motivated formal constraints that privilege one probability distribution over another for a given frame .

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

 

Section . starts to explore the formal prospects for principles of the rough shape of (i), and to assess the epistemological issues arising when they are interpreted in terms of evidence.

. Positive and negative introspection for evidence Here is a simple example of what might go wrong with evidence of evidence principles. Consider a frame with just three worlds, where W = {, , }, R() = {}, R() = R() = {} (so R is serial, cf. Figure .). Informally, if one is in world , it is certain on one’s evidence that one is in world ; if one is in world  or world , it is certain on one’s evidence that one is in world . Let the proposition X be {}, true in world  and false in the other two worlds. Then P[X] is true in world , so P[P[X]] is true in world . But in world , one is certain on one’s evidence not to be in world , so P>[X] is false. In this frame, even the weakest non-trivial evidence of evidence principles fail, because X is certain on one’s evidence to be certain on one’s evidence to be true, but also certain on one’s evidence to be false. A fortiori, any principle of the form Pa[Pb[X]]  P>c[X] (for fixed a, b, c between  and ) fails in this frame. A relevant feature of that frame is that R is not reflexive; neither world  nor world  has R to itself. In both those worlds, one’s total evidence is a falsehood. That is why probability  on one’s evidence fails to entail truth. In a reflexive frame, if a proposition X is false at a world w, one’s evidence R(w) contains w, so the probability of {w} conditional on R(w) is Pr({w})/Pr(R(w)), which is nonzero because Pr({w}) is nonzero, since Pr is regular, so the probability of X conditional on R(w) is less than  because X excludes {w}. Thus in a reflexive frame, both these principles hold: P1 ½Pb ½X  Pb ½X

ðiiÞ

Pa ½P1 ½X  Pa ½X

ðiiiÞ

For, in regular frames, P[Y] entails Y for any proposition Y; what is certain is true. If a proposition is certain on one’s evidence to be likely to some degree on one’s evidence, then it is likely to at least that degree on one’s evidence; similarly, if it is likely to some degree on one’s evidence to be certain on one’s evidence, then it is likely to at least that degree on one’s evidence. By contrast, in any non-reflexive frame, P is not a truth-entailing operator, for if a world w fails to have R to itself, then the proposition W{w} (true everywhere except w) is true throughout R(w), and so certain on one’s evidence in w, even though it is false in w. In effect, for regular probability distributions, P is simply the familiar necessity operator ☐, and reflexivity corresponds to the T axiom ☐p ⊃ p. Similarly, P> is simply the familiar possibility operator ◊, and another form of the T axiom is p ⊃ ◊p (every substitution instance of one form is equivalent to a substitution instance of the other form).

0

1

Figure .

2

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     



The principles (ii) and (iii) demonstrate that one should resist any temptation to regard a requirement for evidence to be true as automatically ‘externalist’, and so in conflict with ‘internalist’ evidence of evidence principles. For the truth axiom entails the evidence of evidence principles (ii) and (iii). Another relevant feature of the above frame is that R is non-transitive: world  has R to world , and world  has R to world , but world  does not have R to world . In modal logic, transitivity corresponds to the  axiom ☐p ⊃ ☐☐p, which in epistemic logic is interpreted as the positive introspection principle that if one knows, one knows that one knows. In the present setting, the modal operator ☐ is interpreted as ‘one’s evidence entails that . . .’, which is equivalent to ‘it is certain on one’s evidence that . . .’ (P), because the regularity of the prior probability distribution guarantees that the conditional probability of a proposition X on one’s evidence R(w) is  if and only if R(w) entails (is a subset of ) X. Thus transitivity corresponds to the principle that if one’s evidence entails a proposition, then one’s evidence entails that one’s evidence entails that proposition, or equivalently that if the proposition is certain on one’s evidence, then it is certain on one’s evidence to be certain on one’s evidence. So stated, the principle is not of the form we have been considering, since the second-level condition is in the consequent rather than the antecedent. However, the validity of the  axiom is equivalent in any frame to the validity of its contraposed form ◊◊p ⊃ ◊p, where the modal operator ◊ is interpreted in the present setting as ‘it is consistent with one’s evidence that . . .’, which is equivalent to ‘there is a nonzero probability on one’s evidence that . . .’ (P>), by regularity again. For any frame and real number a between  and , the probability operators P>a and Pa are dual to each other in a sense analogous to that in which ◊ and ☐ are: just as ¬☐¬p is equivalent to ◊p and ¬◊¬p is equivalent to ☐X, so WPa[WX] = P>a[X] and WP>a[WX] = Pa[X] (since Prw(WX) = Prw(X)). Since, for regular probability distributions, the operator P> is the possibility operator ◊, the transitivity of R is necessary and sufficient for the validity on the frame of this principle: Positive introspection

P>0 ½P>0 ½X  P>0 ½X

If it is consistent with one’s evidence that X is consistent with one’s evidence, then X is consistent with one’s evidence; nonzero evidence of nonzero evidence is nonzero evidence—though of course in a sense in which nonzero evidence for X is compatible with much stronger evidence against X. Perhaps surprisingly, the validity of Positive Introspection on a frame is equivalent to the validity of much weaker principles on that frame, of this form (for  < a < ): Weak a Positive Introspection

Pa ½Pa ½X  P>0 ½X

For example, Weak% Positive Introspection says that if it is at least % probable on one’s evidence that a given proposition is at least % probable on one’s evidence, then that proposition is more than % probable on one’s evidence. Even that very mild-looking principle, though valid on all transitive frames, is valid on no others, and so requires the full power of Positive Introspection (proposition .; all such references are to the appendix). The same goes for the principle that Pa[Pb[X]]  P>[X], since Pc[Pc[X]]  Pa[Pb[X]]  Pd[Pd[X]], where  < min{a, b} = d  c = max{a, b} < .

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

 

Weaka Positive Introspection is weaker than Positive Introspection in the sense that the former may hold on a given non-transitive frame for every proposition X  W while the latter does not for a given regular probability distribution over W. But for Weaka Positive Introspection to be valid on is for it to hold for every proposition X  W for every regular probability distribution over W. Generalizing over probability distributions erases the sensitivity to the quantitative threshold a in Weaka Positive Introspection because a sufficiently non-uniform distribution will exceed the threshold in any case of non-transitivity. We will encounter other examples below where the condition for a principle to be valid is strikingly insensitive to the specific numerical value of a probabilistic threshold, because one can concentrate almost all the weight of prior probability on a few ‘rogue’ worlds. Even requiring the prior probability distribution to be uniform would make less difference than one might expect, because one can often simulate the effect of a highly non-uniform distribution with a uniform distribution, by replacing individual worlds with clusters of worlds mutually indiscernible with respect to the accessibility relation R, where the comparative sizes of the clusters approximate the comparative non-uniform probabilities of the original worlds. If one reads the positive introspection principle for evidence carelessly, it may sound more or less trivial. But it is not, for the truth of one’s total evidence proposition X may not entail that X is at least part of one’s total evidence. Given the equation E = K of one’s total evidence with one’s total evidence, positive introspection for evidence reduces to positive introspection for knowledge.¹ Elsewhere, I have argued against the latter principle (Williamson , pp. –; ). Thus I am committed to rejecting positive introspection for evidence, though I will not rehearse the details of the argument here. Of course, positive introspection for knowledge still has defenders (Greco , Das and Salow , Dorst ; see also the exchange between Hawthorne and Magidor ;  and Stalnaker ). Further reason to doubt positive introspection for evidence will emerge at the end of this section. However, the principle requires no extensive critique here, for salient evidence of evidence principles turn out to imply even more obviously problematic principles, such as negative introspection. Negative introspection is equivalent to a principle formulated in the required way, with the second-level condition in the antecedent: if for all one knows one knows something, then one does know that thing. In the language of modal logic, that is the  axiom ◊☐p ⊃ ☐p. It is valid on all and only frames where R is euclidean, in the sense that any points seen from the same point see each other (if Rxy and Rxz then Ryz). Since our evidential interpretation of the frame makes ◊ equivalent to P> and ☐ to P, negative introspection amounts to this principle: ¹ There is a subtlety here, for if E = K is true, why should it follow that the agent knows E = K? If one is an agent who doesn’t know E = K, couldn’t one know that one knows X without knowing that X is part of one’s evidence, or vice versa? In practice, however, the putative counterexamples to positive introspection for knowledge do not depend on whether the agent knows E = K, so one can stipulate that the agent does know it. Thus positive introspection principles for knowledge and for evidence stand or fall together, given E = K, even though, if they fall together, the nonempty class of counterexamples to one differs slightly from the nonempty class of counterexamples to the other. Epistemic logic in any case tends not to treat such subtle differences in mode of presentation as differences in what is known, since it treats knowledge as closed under truth-functional entailment.

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      Negative Introspection



P>0 ½P1 ½X  P1 ½X

If it is consistent with one’s evidence that one’s evidence entails X, then one’s evidence does entail X. Negative introspection can be shown to entail positive introspection, given the T axiom. Negative introspection for knowledge has been recognized by most philosophically sophisticated formal epistemologists since Hintikka () as implausible. For example, consider a good case in which you know by sight that there is an apple on the table, and a corresponding bad case in which you appear to yourself to be in the good case, and still believe that there is an apple on the table, but your belief is false, because what looks like an apple is just a wax replica. By hypothesis, the bad case is indistinguishable from the inside from the good case. In the bad case, for all you know you are in the good case, so for all you know you know that there is an apple on the table; but you do not know that there is an apple on the table, for there is no apple on the table. Thus the bad case is a counterexample to negative introspection for knowledge. Given the limitations of our cognitive powers, the possibility of such mild sceptical scenarios follows almost inevitably from an antisceptical view of human knowledge. To idealize away such possibilities is to turn one’s back on one of the main phenomena that epistemology is tasked with coming to understand. Nevertheless, much mainstream epistemic logic outside philosophy, notably in computer science and theoretical economics, has continued to treat negative introspection for knowledge as axiomatic. This has done less harm than one might have expected, because the focus of such work has been on multi-agent epistemic logic, which is mainly concerned with what agents know about what other agents know, and iterations thereof, in particular with common knowledge. When modelling multi-agent epistemic phenomena, it is legitimate to idealize away single-agent epistemic complications, because they constitute noise with respect to the intended object of study. Similarly, in modelling dynamic epistemic phenomena, it is legitimate to idealize away synchronic epistemic complications, because they too constitute noise with respect to the intended object of study. But an idealization may be legitimate at one level of magnification and not at another: the astronomer can sometimes treat planets as point masses; the geologist cannot. Mainstream epistemology turns up the magnification on individual epistemic processes to a level at which negative introspection for knowledge is no longer a legitimate idealization. What is the relation between negative introspection for knowledge and for evidence? Given E = K, the two interpretations of the principle stand or fall together. In the bad case, for all one knows one knows X, but one doesn’t know X; similarly, it is consistent with one’s evidence that one’s evidence entails X, but one’s evidence doesn’t entail X. Even if one’s evidence is restricted to the contents of one’s noninferential observational knowledge, mild sceptical scenarios can still arise for it, at least on the sensible assumption that the contents of observation can concern one’s physical environment. If all evidence must be true, then negative introspection requires all evidence to be in principle immune to sceptical scenarios, for example by concerning only mental states that are essentially just as they appear to be to the agent. Such radical foundationalism depends on an antiquated view of the mind which there is no need to argue against here.

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

 

The position is more complex for views that allow false evidence. If my evidence may entail that there is an apple on the table, even though there is no apple on the table, then one’s evidence in the bad case may include the proposition that there is an apple on the table after all. For instance, suppose that one’s evidence is just what one rationally takes for granted, and that one can rationally take for granted something false. Then, in effect, negative introspection says that if it is consistent with what one rationally takes for granted that what one rationally takes for granted entails X, then what one rationally takes for granted does entail X. In contraposed form: if what one rationally takes for granted does not entail X, then what one rationally takes for granted entails that what one rationally takes for granted does not entail X. How to assess that principle may not be immediately obvious. Although negative introspection does not imply that one’s evidence is true, it does imply that one’s evidence entails that one’s evidence is true. More precisely, the consequence is that one’s evidence entails that if one’s evidence entails X, then X is true. Some such conditionals will be amongst one’s evidence’s false entailments, if it has any. Negative introspection implies the principle because the former corresponds to the euclidean frame condition that if Rxy and Rxz then Ryz; putting z = y gives the condition that if Rxy then Ryy, which we may restate by calling R quasi-reflexive: every seen world sees itself. In modal logic, quasi-reflexivity corresponds to the quasitruth principle ☐(☐p ⊃ p). It follows from negative introspection in any normal modal logic.²,³ Whether it is rational to take for granted that something is rationally taken for granted only if it is true may again not be obvious. The quasi-reflexivity of the accessibility relation is also equivalent to the validity of the evidence of evidence principle (iii) above on the frame, for any given real number a strictly between  and  (proposition .): if the probability on one’s evidence that X is certain on one’s evidence is at least a, then the probability of X on one’s evidence is at least a. Again, the exact value of the numerical probability parameter a makes no difference, as long as it is not extremal. We can consider negative introspection for evidence from a different angle. Imagine various pieces of evidence coming in from various sources. Normally, no one of these evidence propositions entails about itself that it exhausts one’s total evidence. Suppose, for instance, that your evidence includes both the proposition FLASH that there is a flash and the proposition SQUEAK that there is a squeak. FLASH is simply neutral as to whether your evidence also includes SQUEAK, and vice versa. Now suppose that no other evidence comes in. Then the conjunction FLASH & SQUEAK exhausts your evidence, but it does not itself entail that it exhausts your total evidence. For all the conjunction says, your evidence might also ² In this context a normal modal logic, identified with the set of its theorems, is a set of formulas of a standard language for monomodal propositional logic (with ☐ as the only primitive modal operator) containing all truth-functional tautologies and the K axiom ☐(p ⊃ q) ⊃ (☐p ⊃ ☐q) and closed under uniform substitution, modus ponens, and the rule of necessitation. The correspondence result can be proved by the standard method of canonical models (see, e.g., Hughes and Cresswell , pp. –). ³ To derive quasi-truth from negative introspection, consider two cases: (a) p truth-functionally entails ☐p ⊃ p, so by normality ☐p entails ☐(☐p ⊃ p); (b) ¬☐p truth-functionally entails ☐p ⊃ p, so by normality ☐¬☐p entails ☐(☐ p ⊃ p); but by negative introspection and normality, ¬☐p entails ☐¬☐p, so ¬☐p entails ☐(☐p ⊃ p). Either way, we have ☐(☐p ⊃ p).

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     



include the proposition STINK that there is a stink. Although some special evidence might somehow manage to entail of itself that it exhausts your evidence, there is no reason to expect evidence to do that in general. On almost any view, one’s total evidence is usually much richer than FLASH & SQUEAK, but that does not mean that it entails its own totality. Nor is there any reason to postulate a meta-device, guaranteed to be in perfect working order, for surveying all one’s evidence, including the evidence generated by the meta-device itself. Thus, for the sake of simplicity, we can work with the case where your total evidence is just FLASH & SQUEAK, since it is not structurally misleading. Although it is consistent with your total evidence that your total evidence entails (by including) STINK, your total evidence does not entail STINK, since FLASH & SQUEAK may be true while STINK is false. Hence the case is a counterexample to negative introspection for evidence. Moreover, as a template for counterexamples it works on a wide range of theories of evidence. It does not assume E = K; it does not even assume that all evidence is true. Thus, except under extreme idealizations, negative introspection for evidence is not a reasonable hypothesis. That simple case also casts doubt on positive introspection for evidence. On the face of it, the flash and the squeak could occur without being part of your evidence, even implicitly. Thus FLASH & SQUEAK does not entail that your evidence entails FLASH & SQUEAK. So if FLASH & SQUEAK just is your evidence, your evidence may entail FLASH & SQUEAK even though your evidence does not entail that your evidence entails FLASH & SQUEAK. Someone might try to avoid that result by positing that one’s evidence consists wholly of propositions that can only be true by being part of one’s evidence, but that idea too threatens to degenerate into radically naive foundationalism. So far, we have mainly assessed evidence of evidence principles that in effect transcribe principles from epistemic logic into the probabilistic idiom. Sections . and . discuss a wider range of evidence of evidence principles, and consider whether they imply problematic forms of introspection, or other problematic epistemic principles.

. Threshold Transfer The simplest interesting frames for single-agent epistemic logic are partitional: the accessibility relation R is an equivalence relation—it is reflexive, symmetric, and transitive—and so partitions W into mutually exclusive, jointly exhaustive subsets of the form R(w). Since any symmetric transitive relation is euclidean, and any reflexive euclidean relation is symmetric, R is an equivalence relation if and only if it is reflexive, transitive, and euclidean. Thus what the class of such frames validate about knowledge is that it entails truth and satisfies positive and negative introspection. We saw in section . how problematic positive and negative introspection are for both knowledge and evidence. However, simple cases make good starting-points, so we begin by reinterpreting partitional frames in terms of evidence, so that the evidence forms a partition. We start with principles which identify there being evidence for X with the probability of X on the evidence reaching a fixed threshold. Suppose that accessibility for evidence is an equivalence relation. Consequently, if one’s evidence in a world w is

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

 

consistent with one’s being in a world x, so Rwx, then one’s evidence in w is the same as one’s evidence in x, so any proposition has the same probability on the evidence in w as in x. This validates a strong principle about posterior probabilities: Transfera

P>0 ½Pa ½X  Pa ½X

In other words, if the probability on one’s evidence that the probability on one’s evidence of X is at least a is itself nonzero, then the probability on the evidence of X is at least a. Transfera holds for every proposition X and real number a in any partitional frame. For if P>[Pa[X]] is true in x, then Pa[X] is true at some world y in R(x); if the frame is partitional, R(y) = R(x), so Pry(X) = Prx(X), so Pa[X] is true in x too. In such frames, evidence of evidence is always perfect evidence of evidence, so no wonder evidence of evidence is evidence. Many principles reminiscent of Transfera hold only in partitional frames (see, e.g., Williamson , pp. –). However, what was needed to validate Transfera was only for R to be quasi-partitional, in the sense that whenever y 2 R(x), R(x) = R(y), in other words, if Rxy then for all z, Rxz if and only if Ryz. As is easily checked, a relation is quasi-partitional if and only if it is both transitive and euclidean. Transfera does not require reflexivity; it can hold even if some evidence is false. An example of a quasi-partitional but not partitional frame is one with just two worlds,  and , where R() = R() = {} (see Figure .). It is not partitional because the world  is not in any set of the form R(w). The relation R is non-reflexive, because  does not have R to itself. R is also non-symmetric, because R but not R. Nevertheless, Transfera holds in such a frame. Conversely, for  < a < , Transfera is valid only on quasi-partitional frames (Appendix, proposition .). Thus Transfera is equivalent to quasi-partitionality. Again, the numerical value of the parameter a does not matter, as long as it is not extremal. Since Transfer requires quasi-partitionality, it implies positive and negative introspection. Given the implausibility of that combination, we must seek weaker evidence of evidence principles, ones that do not make the outer probability operator redundant. Here is a natural candidate: Thresholda Transfer

Pa ½Pa ½X  Pa ½X

In other words, whenever the probability on the evidence that the probability on the evidence of X is at least a is itself at least a, the probability on the evidence of X is at least a. Taking a as the threshold for something to be probable on the evidence (with a > ½), we can read Thresholda Transfer as saying that if it is probable on one’s evidence that a hypothesis is probable on one’s evidence, then that hypothesis is probable on one’s evidence. Transfer entails Thresholda Transfer for all a. For when a = , Thresholda Transfer is trivial, and when a > , Pa[Pa[X]]  P>[Pa[X]]  Pa[X] by Transfer. But not even Thresholda Transfer for all values of a together entails Transfer.

0

1

Figure .

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      0



1

Figure .

Here is an example of a frame on which Thresholda Transfer is valid while Transfera is not. As before there are just two worlds,  and , where R but not R, and R, but this time R too (cf. Figure .). Thus R is reflexive as well as transitive, though not symmetric. R is also not euclidean (since R and R but not R). Transfera fails on this frame for any non-extremal value of a and any regular probability distribution. For P[{}] is true at  but false at , even though P>[P[{}]] is true at . Since R() = {, } while R() = {}, the total evidence propositions are not even mutually exclusive. Nevertheless, Thresholda Transfer is valid on this frame, whatever the probability distribution and the value of a. This follows from the general result that for all a between  and , Thresholda Transfer is valid on every frame that is near-partitional in this sense: whenever y 2 R(x), either R(y) = R(x) or R(y) = {y}. The new two-world frame is clearly near-partitional. The converse also holds for all a strictly between  and : every frame on which Thresholda Transfer is valid is near-partitional (proposition .). This is another instance of the phenomenon already noted in relation to Weaka Positive Introspection: the validity on a frame of ‘evidence of evidence’ principles with a probabilistic threshold is often insensitive to the value of that parameter. In this case, whenever a and b are both strictly between  and , Thresholda Transfer and Thresholdb Transfer are valid on exactly the same frames. For R(y) to be {y} is for one’s evidence in y to entail all and only truths in the world y: one’s evidence tells the whole truth and nothing but the truth about y. That is a wildly idealized scenario. Thus, in practice, near-partitionality is very close to partitionality, and Thresholda Transfer very close to Transfer, too close for it to be a useful weakening. Another way to assess the strength of Thresholda Transfer is by noting that the propositional modal logic of near-partitional (and serial) frames can be axiomatized by this set of axioms: D Q-T

☐p ⊃ ◊p ☐ð☐p ⊃ pÞ

4

☐p ⊃ ☐☐p

Q-5

◊p ⊃ ☐ð◊p ∨ ðq ⊃ ☐qÞÞ

More precisely, the smallest normal modal logic with D, Q-T, , and Q- as theorems is sound and complete for the class of near-partitional serial frames. D corresponds to seriality, Q-T to quasi-reflexivity,  to transitivity, and Q- to a slight weakening of the euclidean property. The two disjuncts in the consequent of Q- correspond to the two disjuncts in the definition of near-partitionality (in the same order). On the relevant interpretation of the modal operators in terms of evidence, D says that one’s evidence is compatible with what it entails, Q-T that one’s evidence entails that one’s evidence is true, and  that one’s evidence obeys positive introspection. Q- weakens

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

 

negative introspection (corresponding to the  principle ◊p ⊃ ☐◊p, equivalent to the contraposed principle ◊☐p ⊃ ☐p) by saying that if one’s evidence is consistent with a proposition, then one’s evidence entails that either one’s evidence is consistent with the proposition or any given truth is entailed by one’s evidence. Positive introspection for evidence was already discussed in section .. As for axiom Q-, it is scarcely more plausible than negative introspection, even though logically it is slightly weaker. Under E = K, Q- implies claims like this about the bad case (at least if one knows E = K): if for all one knows there is no apple on the table, then one knows that either for all one knows there is no apple on the table or if there is life on other planets then one knows that there is life on other planets. But the antecedent is true in the bad case: since there is no apple on the table, for all one knows there is no apple on the table. Thus Q-, read epistemically, generates the claim that, in the bad case, one knows that either for all one knows there is no apple on the table or if there is life on other planets then one knows that there is life on other planets. But since there is no useful connection between the disjuncts, one’s only way of knowing the disjunction is by knowing one of the disjuncts (sometimes one knows a disjunction without knowing any disjunct, for instance when the disjunction is an instance of the law of excluded middle or the content of some disjunctive testimony, but in such cases there is a useful epistemic connection between the disjuncts). Hence either one knows that for all one knows there is no apple on the table, or one knows that if there is life on other planets then one knows that there is life on other planets. But one does not know that for all one knows there is no apple on the table, because for all one knows one knows that there is an apple on the table. One also does not know that if there is life on other planets then one knows that there is life on other planets, for one has no special access to whether there is life on other planets. Even if we bracket E = K, and do not assume that all evidence is true, the case in section . where one’s total evidence is just the conjunction FLASH & SQUEAK raises as severe a problem for Q-, read in terms of evidence, as it does for negative introspection. Thus weakening negative introspection by the second disjunct in the consequent makes no significant difference to the plausibility of Q-. In brief, Thresholda Transfer principles weaken negative introspection too slightly to regain plausibility. We can also consider principles intermediate between Thresholda Transfer and Weaka Positive Introspection, in other words, principles of the form Pa[Pa[X]]  Pb[X] for a > b > . For example, we might set b = a². Such principles are only valid on transitive frames, since they all entail Positive Introspection; the question is where the corresponding frame conditions come between transitivity and near-partitionality. In general, such intermediate principles require far more than transitivity. In this setting, an important frame condition is quasi-nestedness: is quasi-nested if and only if whenever Rwx, Rwy, Rxz, and Ryz, then either Rxy or Ryx. Very roughly, if two points visible from a given point are invisible from each other, then their fields of vision are disjoint. Quasi-nestedness has a long history in epistemic logic, under varying terminology, going back to Geanakoplos () (see Dorst  for much more discussion relevant to connections between quasi-nesting and various probabilistic conditions). One can show that

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     



for all a,b 2 [,], the natural-looking Product Rule is valid on every finite serial transitive quasi-nested frame: Product Rule

Pra ½Prb ½X  Prab ½X

Conversely, any finite serial frame on which the Product Rule is valid for all a,b 2 [,] is transitive and quasi-nested (proposition .). The difference in validity conditions between this principle and Thresholda Transfer is one way in which the numerical values of the thresholds do make a difference. Quasi-nesting does not correspond to the validity of a formula in the language of propositional modal logic, for the modal system S is sound and complete both for the class of all reflexive transitive frames, many of which are not quasi-nested, and for the class of reflexive transitive tree frames, all of which are quasi-nested (see Blackburn, de Rijke, and Venema , p. ). Thus no formula of the language is valid in all and only quasi-nested frames. The distinctive consequences of quasi-nesting appear only in a more expressive language, such as one with probability operators. How plausible is quasi-nesting as a condition on evidence? To see why it is problematic, note that the contrast in sceptical setups between good and bad cases is not all-or-nothing. A case may be good in one respect, bad in another. For example, one may suffer a minor illusion about the size, shape, or distance of a particular building, or the direction from which a particular sound is coming, while continuing to gain large amounts of perceptual evidence about one’s environment in many other respects. Indeed, such mixed cases may be usual in everyday life. For simplicity, consider worlds which differ from each other only in goodness or badness in two independent minor respects,  and . Let each subset of {, } label the world (or scenario) which is good in the respects it contains and bad in the other respects. Thus world {} is bad in both respects, world {} is good in respect  but bad in respect , world {} is good in respect  but bad in respect , and world {, } is good in both respects. In line with that interpretation, we naturally identify accessibility with the subset relation, for S  S* ( W) just in case there is no respect in which S is good and S* is not (which is what would block accessibility). This accessibility relation is reflexive and transitive, but not symmetric.⁴ Thus {} has R to all worlds, each of {} and {} has R just to itself and {, }, and {, } has R just to itself (see Figure .). Hence quasi-nesting fails, for {} has the accessibility relation R to both {} and {}, and they both have R to {, }, but neither of {} and {} has R to the other. Quasinesting fails similarly in frames corresponding to more than two respects of good case/bad case contrast. Moreover, quasi-nesting is problematic for reasons independent of the KK principle, since the frame in question is transitive. Although quasi-nesting holds in the two-world non-symmetric frame corresponding to a sceptical scenario in only one respect (since that frame is connected), it is

⁴ More generally, we can interpret any set X as comprising the respects in which things can be epistemically good or bad, yielding a frame in which the worlds are the subsets of X and the subset relation is accessibility. All such frames are reflexive, transitive and convergent in the sense that if Rwx and Rwy then for some z both Rxz and Ryz, which corresponds to the modal axiom ◊☐p ⊃ ☐◊p. Thus they all validate the modal system proposed as a logic of knowledge in Stalnaker (). They are used for a related purpose in the appendix to Salow ().

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

 

{1, 2} {1}

{2} {}

Figure .

y1

y2

…

x

z

yn2

Figure .

hard to think of any principled reason for permitting the one-respect setup but rejecting all the multiple-respects setups. Thus a principled defence of quasi-nesting would be forced to reject the natural modelling of sceptical scenarios. On its epistemic reading, the Product Rule is correspondingly unreasonable, for many values of a and b (it is trivial when a =  or b = ). For a variant of the example, consider frames where W = {x} [ Y [ {z}; {x}, Y, and {z} are pairwise disjoint; R(x) = W, R(y) = {y, z} for y 2 Y, and R(z) = {z} for z 2 Z; |Y|= n² (see Figure .). Informally, think of z as the good case, the members of Y as n² moderately bad cases, and x as a very bad case. Such frames are reflexive and transitive but neither symmetric nor euclidean. Let Pr be the probability distribution such that Pr({z}) = n/(n² + n + ) and Pr({w}) = /(n² + n + ) for every world w 6¼ z; thus the good case is given more weight than any one bad case, but less than all the bad cases put together. Then Pry({z}) = n/(n + ) for y 2 Y, and Prz({z}) = . Thus, for any given b < , Y [ {z}  Pb[{z}] for sufficiently large n, in which case Prx(Pb[{z}])  (n² + n)/(n² + n + ), so for any given a < , x 2 Pa[Pb[{z}]] for sufficiently large n. But Prx({z}) = n/(n² + n + ), so for any given c > , x 2 = Pc[{z}] for sufficiently large n. Thus, for any given non-extremal a, b, c, for sufficiently large n the frame invalidates the principle Pa[Pb[X]]  Pc[X], although it accepts Positive Introspection. Is there any good reason to deny that such frames do a reasonable job of modelling coherent epistemic situations? Given E = K, or many other views on which evidence

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     



consists of true propositions, which may be about the external world, the likely price of denying that such situations can arise is scepticism. Some philosophers propose that evidence can be false. For example, they may suggest that one’s evidence in both bad and good cases is just what one knows in the good case: in effect, one’s evidence is what, for all one knows, one knows. That implies a very different structure for evidential accessibility.⁵ However, the epistemological cost is high. On that picture, when one is in the bad case, it is certain on one’s evidence that one is not in the bad case. Moreover, in the scenario described at the end of section ., for all one knows, one knows that there is a stink, simply because one’s capacity to survey the totality of one’s knowledge is so limited. Thus, on the envisaged view, one’s evidence entails the falsehood that there is a stink, even though there does not even appear to one to be a stink. That is not an attractive consequence. Of course, some epistemologists will find the positive consequences of quasinesting (combined with transitivity) very attractive. From that perspective, there is good reason to seek a solution to the epistemological problems raised by quasinesting. But the positive consequences themselves are no solution.

. Comparative Transfer Probabilistic confirmation can be understood in two ways, absolute and comparative. In the absolute sense, evidence confirms a hypothesis if the probability of the hypothesis on the evidence reaches some fixed threshold (such as %). In the comparative sense, evidence confirms a hypothesis if the probability of the hypothesis on the evidence exceeds its prior probability. The absolute sense concerns high probability, the comparative sense higher probability, or probability raising. Various hybrids of the two standards are also conceivable, but it is better to start with the simple contrast. Clearly, evidence of evidence principles can also be read in either the threshold way or the comparative way. Again, various hybrid readings are also conceivable, and again it is better to start with the simpler ones. In section ., threshold readings of ‘evidence of evidence’ principles turned out rather unpromising. This section considers the comparative reading. Probability-raising is easy to formalize in the present framework, since it provides both the prior probability distribution Pr and, for each world w, the posterior probability distribution Prw which conditionalizes Pr on one’s (new) evidence in w, R(w). For present purposes, it does not matter whether Pr embodies earlier background evidence or not. Thus the proposition that the (new) evidence has raised the probability of a proposition X is simply {w: Prw(X) > Pr(X)}, which we notate as P>[X]. It is just the set of worlds in which the posterior probability of X is greater than its prior probability. When the ‘evidence of evidence’ slogan is so understood in terms of probability-raising, it becomes this: Comparative Transfer

P> ½P> ½X  P> ½X

⁵ This is one way of interpreting evidential relations over the framework of Stalnaker ().

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

 

In other words, if the evidence raises the probability that the evidence raises the probability of X, then the evidence does raise the probability of X. What are the prospects for Comparative Transfer? In which frames is it valid? An initial observation is encouraging: Comparative Transfer is valid on some frames on which Negative and Positive Introspection both fail. In that way it is less demanding than the Threshold Transfer principles. In fact, the validity of the Threshold Transfer principles on a frame is neither necessary nor sufficient for the validity of Comparative Transfer (proposition .). On closer inspection, however, the picture is more complicated and less rosy. We can start with the case of Positive Introspection, or in modal terms the  axiom. Although its validity does not follow from that of Comparative Transfer, the validity of this weakening of it does (corollary .): ☐☐ p ⊃ ☐☐☐ p

ðivÞ

In other words, although the evidence may entail a proposition without entailing that it entails it, if the evidence entails that it entails a proposition, then it entails that it entails that it entails that proposition. In terms of the accessibility relation between worlds, the  axiom corresponds to transitivity, in other words, if you can get from x to z in two steps of accessibility, you can get there in one step. For comparison, (iv) corresponds to the feature that if you can get from x to z in three steps of accessibility, you can get there in two. It is hard to see what independent theoretical reason there might be for accepting the weaker principle that would not also be a reason for accepting the stronger. Relevant objections to Positive Introspection for knowledge generalize to objections to the weaker principle (Williamson , pp. –).⁶ If the proposition that one’s evidence entails SQUEAK is added to one’s evidence, it does not follow that the proposition that it has been added to one’s evidence has itself been added to one’s evidence. As with previous ‘evidence of evidence’ principles, the problems for Comparative Transfer do not end with (slight weakenings of ) the contested principle of Positive Introspection. They extend to (slight weakenings of) the far more generally rejected principle of Negative Introspection. For example, on any frame on which Comparative Transfer is valid, so is (v): ð☐ p ⊃ pÞ ∨ ð◊q ⊃ ☐◊qÞ

ðvÞ

Since p and q are independent variables, (v) says in effect that each world is either reflexive or euclidean: if the former, all instances of the T axiom hold at it, if the latter, all instances of the  axiom do. For those (like the author) who hold that all evidence ⁶ One objection to Positive Introspection for knowledge is that a simple creature with no grasp of the distinction between knowing and not knowing may still know truths about its environment, but without being able to so much as entertain the truth that it knows them. This objection does not generalize against the principle that if you know that you know, then you know that you know that you know, for if you know that you know then in the required sense you do grasp the distinction between knowing and not knowing. However, such objections are too fine-grained for the present setting, where logical omniscience is assumed: if you know one truth but cannot grasp another you presumably cannot grasp their disjunction either, which follows from the truth you do know. For present purposes we are accepting the principle ☐p ⊃ ☐(p ∨ q), which is valid in all standard epistemic models.

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     



is true, (v) is unproblematic, because it is true thanks to the first disjunct. But for those who hold that evidence may be false, (v) has the effect that Negative Introspection must hold of all propositions whatever whenever one has some false evidence. Epistemologically, it is quite unclear why the T and  axioms should play such complementary roles. To turn the screw, (v) requires that if one has false evidence about one topic, then one’s evidence about some completely different topic conforms perfectly to Negative Introspection. But if the T axiom for evidence has false instances in some situations, and Negative Introspection has false instances, it is almost inevitable that in some combined situations both principles will have false instances, perhaps about unrelated topics, which is enough to violate (v). For example, if one’s evidence is consistent with the proposition that one’s evidence entails STINK, even though one’s evidence does not in fact entail STINK, why should that prevent one from getting false evidence about something else, if false evidence is in general an option? However, even for those who accept the T axiom, Comparative Transfer has problematic consequences. They include weakenings of the B axiom p ⊃ ☐◊p, which corresponds to the symmetry of the accessibility relation. B is implausible in epistemic logic, for reasons closely connected to the implausibility of Negative Introspection. Read in terms of knowledge, B says that if something obtains, one knows that for all one knows it obtains. But consider any pair of a good case, in which all goes well, things are as they seem and one has plenty of knowledge, and a bad case, a sceptical scenario in which one seems to oneself to be in the good case but things are not as they seem, much of what one seems to oneself to know is false, and one knows very little. In the good case, one knows things incompatible with being in the bad case. In the bad case, for all one knows one is in the good case, so for all one knows one knows things incompatible with being in the bad case. Thus, if one is in the bad case, one does not know that for all one knows one is in the bad case. Hence the B axiom fails for knowledge. Consequently, on the equation E = K, the B axiom also fails for evidence. Even if evidence is not equated with knowledge, the B axiom faces similar counterexamples to those for knowledge. There are two cases to consider: either false evidence is disallowed (R must be reflexive), or false evidence is allowed (R can be non-reflexive). First, suppose that false evidence is disallowed. Start with a good case where E is true and part of one’s evidence. Typically, there will be a bad case where one’s evidence is consistent with being in the good case, but E is false and so not part of one’s evidence. The bad case may be one which seems from the inside just as the good case does, or it may simply be that in both cases one’s capacity to survey one’s evidence as a whole is imperfect, and in the bad case one cannot ascertain that one’s evidence lacks E and add that fact to one’s evidence. Either way, in the bad case, ¬E is true, but one’s evidence does not entail that one’s evidence is consistent with ¬E, for one’s evidence in the bad case is consistent with being in the good case, where one’s evidence is inconsistent with ¬E. Thus, on its evidential reading, the B axiom fails in the bad case. Now suppose instead that false evidence is allowed. Typically, the motivation for allowing it is to let one’s evidence in both good and bad cases be the content of

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi



 

appearances common to those cases. Thus one’s evidence is the same in the two cases, but true in the good case and false in the bad case. Since one’s evidence is false in the bad case, one’s evidence in both cases entails that one is not in the bad case. Therefore, since one’s evidence is true in the good case, and entails that one is not in the bad case, one’s evidence in both cases does not entail that one’s evidence is consistent with one’s being in the bad case. Thus, on its evidential reading, the B axiom fails again in the bad case. In brief, whether false evidence is allowed or not, the B axiom fails on reasonable views of evidence. As already noted, the validity of Comparative Transfer does not require the validity of the B axiom for evidence. Comparative Transfer is valid on some reflexive but non-symmetric frames, such as the two-world frame where W = {good, bad}, R(good) = {good} and R(bad) = {good, bad} (by proposition .). Since that is exactly the problematic structure under discussion, Comparative Transfer may look to be out of the wood. But it is not. For although its validity on a frame does not require the accessibility relation to be symmetric everywhere, it does limit how widespread the failures can be. More specifically, if Comparative Transfer is valid on a finite serial frame , then for all worlds w, x, y, z in W, if Rwx, Rxy, and Ryz, then either Rxw, or Ryx, or Rzy. In other words, if there is a chain of three successive links of accessibility, at least one of those links is bidirectional, an instance of symmetry. In the language of modal logic, that frame condition corresponds to this axiom (proposition .): p ⊃ ☐ðq ⊃ ð◊p ∨ ☐ðr ⊃ ð◊q ∨ ☐◊rÞÞÞ

ðviÞ

The three variables p, q, and r are mutually independent. To see why (vi) is problematic, consider a three-respect version of the frames used at the end of section .to show how quasi-nesting can fail. Each subset of {, , } labels the world which is good in the respects it contains and bad in the other respects. Thus world {} is bad in all three respects, while world {, , } is good in all three respects. As before, we identify accessibility, R, with the subset relation, for S  S* ( W) just in case there is no respect in which S is good and S* is not, which is what would block accessibility. Thus R is reflexive and transitive, but not symmetric. In this frame, {} has R to {}, {} has R to {, }, and {, } has R to {, , }, and none of these R links is reversible: {} lacks R to {}, {, } lacks R to {}, and {, , } lacks R to {, } (see Figure .). Thus the frame violates the condition, so Comparative Transfer is violated. In terms of (vi) itself, there is a counterexample where p is the proposition that one is in a bad case in respect , q is the proposition that one is in the bad case in respect , and r is the proposition that one is in the bad case in respect .⁷ For reflexive frames, where evidence is always true, the counterexamples can be slightly simplified, for we only require a chain of two steps of R for one of the steps to be bidirectional. More specifically, if Comparative Transfer is valid on a finite reflexive frame , then for all worlds w, x, y in W, if Rwx and Rxy, then either

⁷ See fn.  for such frames.

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

     



{1, 2, 3}

{1, 2}

{1, 3}

{2, 3}

{1}

{2}

{3}

{}

Figure .

Rxw or Ryx (proposition .). In the language of modal logic, that frame condition corresponds to this axiom (proposition .): p ⊃ ☐ðq ⊃ ð◊p ∨ ☐◊qÞÞ

ðviiÞ

In this case, counterexamples can make do with two respects rather than three. Since the frame just described is already reflexive, there is no need to elaborate. In brief, although Comparative Transfer is on balance weaker than the Thresholda Transfer principles, its epistemological consequences are still implausibly strong. From a technical point of view, Comparative Transfer behaves rather differently from the other principles considered in this chapter. Unlike them, it essentially involves the prior probability distribution Pr, in ways that cannot be reduced to the posterior distributions Prw. Whether it holds at a given world depends on global features of the frame, and can depend on what happens at worlds to which the starting world does not even bear the ancestral of the accessibility relation, for those worlds contribute to the priors with which the posteriors are being compared. One manifestation of this global aspect is that in some cases Comparative Validity is valid on each of two mutually disjoint frames but invalid on their union, even when the two original frames are isomorphic to each other (proposition .). That cannot happen in the standard model theory of ordinary modal logic or the other principles considered in this chapter, because at every stage the generalizations over worlds are restricted by the accessibility relation, and so never cross the boundary between one of the disjoint subframes and the other. As a corollary, by contrast with the Thresholda Transfer principles, there is no set of formulas in the standard language of propositional modal logic whose validity on a frame is equivalent to the validity of Comparative Transfer, for the class of frames on which such formulas are valid is closed under disjoint unions. Obviously, once we add the operator P> to the objectlanguage, we can express Comparative Transfer for that language. In part for the reason just discussed, the model theory is more intricate for Comparative Transfer than for the principles discussed earlier. The partial results in the appendix give some indication of its complexity. They do not include a

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi



 

necessary and sufficient condition, in non-probabilistic terms, for Comparative Transfer to be valid on a frame . That is left as an open problem for the interested reader. The epistemological implausibility of Comparative Transfer makes it non-urgent for present purposes.

. Conclusion This chapter has considered by no means all imaginable renderings of the slogan ‘evidence of evidence is evidence’ in the framework of single-agent synchronic epistemic logic with probabilities. However, it has assessed the most natural candidates, with unpromising results: when combined with reasonable views of evidence, they all have epistemologically implausible consequences, which fail even in very mild sceptical scenarios. To overturn this provisional negative verdict, it will not be enough to formulate another candidate rendering that has not been considered here. Such moves come far too cheap. Rather, what would be needed are serious results to the effect that the candidate principle is valid in a good variety of frames invalidating the implausible

Table . Frame conditions Condition

Label

Relational formula

Modal formula

Serial

D

8w9xRwx

☐p ⊃ ◊p

Reflexive

T

8wRww

☐p ⊃ p

Quasi-reflexive

Q-T

8w8x(Rwx ⊃ Rxx)

☐(☐p ⊃ p)

Symmetric

B

8w8x(Rwx ⊃ Rxw)

p ⊃ ☐◊p

Transitive



8w8x8y((Rwx ∧ Rxy) ⊃ Rwy)

☐p ⊃ ☐☐p

-

(iv)

8w8x8y8z((Rwx ∧ Rxy ∧ Ryz) ⊃ 9v(Rwv ∧ Rvz))

☐☐p ⊃ ☐☐☐p

Euclidean



8w8x8y((Rwx ∧ Rwy) ⊃ Rxy)

◊p ⊃ ☐◊p

8w8x(Rwx ⊃ 8y(Rxy = Rwy))

∧

8w8x(Rwx ⊃ (8y(Rwy ⊃ Rxy) ∨ 8y(Rxy ⊃ x = y))

◊p ⊃ ☐(◊p ∨ (q ⊃ ☐q))

Near-partitional

8w8x(Rwx ⊃ (8y(Rxy $ Rwy) ∨ 8y(Rxy = x = y))

Q-T ∧  ∧ Q-

Quasi-nested

8w8x8y8z((Rwx ∧ Rwy ∧ Rxz ∧ Ryz) ⊃(Rxy ∨ Ryx))

None

Quasi-partitional Quasi-Euclidean

Q-

-

(v)

8w(Rww ∨ 8x8y((Rwx ∧ Rwy) ⊃ Rxy)

T∨

-

(vi)

8w8x8y8z((Rwx ∧ Rxy ∧ Ryz) ⊃(Rxw ∨ Ryx ∨ Rzy))

p ⊃ ☐(q ⊃ (◊p ∨ ☐(r ⊃ (◊q ∨ ☐◊r)))

-

(vii)

8w8x8y((Rwx ∧ Rxy) ⊃(Rxw ∨ Ryx))

p ⊃ ☐(q ⊃ (◊p ∨ ☐◊q))

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

     



introspection principles, or else a properly developed epistemology that somehow makes those introspection principles less implausible. If that challenge is unmet, we may tentatively conclude that, in any reasonable sense, evidence of evidence is not always evidence—not always in the single-agent synchronic case, and a fortiori not always in the multi-agent and diachronic cases either. Of course, none of this means that evidence of evidence is not typically evidence. It would also be useful to have positive results giving less demanding sufficient conditions in the present framework for most evidence of evidence to be evidence, in senses close to those in this chapter.

Appendix For convenience, definitions in the main text are repeated here as they arise. Weaka Positive Introspection

Pa ½Pa ½X  P>0 ½X

Proposition .: For a 2 (, ), Weaka Positive Introspection is valid on a finite serial frame iff it is transitive. Proof: Suppose that Pa[Pa[X]]  P>[X] is valid on a finite serial frame . Suppose further that Rxy, Ryz, but not Rxz. Thus y 6¼ z. If W = {y, z}, define a probability distribution Pr over W thus: PrðfygÞ ¼ 1  a PrðfzgÞ ¼ a If W 6¼ {y, z}, so |W| = n > , instead define Pr thus: PrðfygÞ ¼ að1  aÞ PrðfzgÞ ¼ a PrðfugÞ ¼ ð1  aÞ2 =ðn  2Þ for u 2 = fy; zg: The following argument works on both definitions. Since z 2 R(y), Pry({z}) = Pr({z} \ R(y))/Pr(R(y)) = Pr({z})/Pr(R(y))  Pr({z}) = a, so y 2 Pa[{z}]. Moreover, since y 2 R(x) and z 2 = R(x): Prx({y}) = Pr({y})/Pr(R(x))  Pr({y})/Pr (W–{z}) = Pr({y})/(–a)  a(–a)/(–a) = a. Hence x 2 Pa[{y}], so Prx({y})  a. But {y}  Pa[{z}], so Prx(Pa[{z}]  a, so x 2 Pa[Pa[{z}]]. Thus, by hypothesis, x 2 P>[{z}], so Px({z}) > 0 since R is regular, so z 2 R(x), contrary to hypothesis. Thus R is transitive after all. Conversely, suppose that R is transitive and Pr is a regular probability distribution over W. Suppose that x 2 Pa[Pa[X]]. Thus Prx(Pa[X])  a > , so there is a y 2 Pa[X] \ R(x). Thus Pry(X)  a > , so there is a z 2 X \ R(y). But then z 2 R(x) because y 2 R(x), z 2 R(y), and R is transitive. Hence z 2 X \ R(x), so Prx(X) >  because Pr is regular, so x 2 P>[X]. This shows that Pa[Pa[X]]  P>[X], as required. Corollary .: For any a, b 2 (, ) and finite serial frame , Weaka Positive Introspection is valid on iff Weakb Positive Introspection is. Transfera

P>0 ½Pa ½X  Pa ½X

for a 2 ½0; 1:

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi



 

A frame is quasi-partitional: for all x 2 W, if y 2 R(x) then R(x) = R(y) (equivalently, R is transitive and euclidean). Proposition .: For a 2 (,), Transfera is valid on a finite serial frame iff it is quasipartitional. Proof: Suppose that Transfera is valid on a finite serial frame . If |W| =  the result is trivial, so we may assume that |W| = n  . Let x, z 2 W, and y 2 R(x). Choose b so that max{a,  – a} < b < . Define a probability distribution Pr over W by setting: PrðfzgÞ ¼ b PrðfugÞ ¼ ð1  bÞ=ðn  1Þ

for u 6¼ z:

If z 2 R(y), Pry({z}) = Pr({z})/Pr(R(y)  Pr({z}) = b > a, so y 2 Pa[{z}]. Since y 2 R(x), x 2 P>[Pa[{z}]], so, by Transfera, y 2 Pa[{z}], so z 2 R(x). On the other hand, if z 2 = R(y), then Pry(W{z}) = , so y 2 Pa[W{z}]. Since y 2 R(x), x 2 P>[Pa[W{z}]], so x 2 Pa[W{z}] by Transfera, so  – Prx({z}) = Prx(W–{z})  a. Hence Prx({z})   – a. But, as before, if z 2 R(x) then Prx({z})  b >  – a. Thus z 2 = R(x). So for all z 2 W, z 2 R(y) iff z 2 R(x). Thus R(y) = R(x). Hence R is quasi-partitional. The converse is routine. Corollary .: For any a, b 2 (, ), Transfera is valid on a finite serial frame iff Transferb is. Proposition .: For a 2 (, ), the principle Pa[P[X]]  Pa[X] is valid on a finite serial frame iff it is quasi-reflexive. Proof: Suppose that Pa[P[X]]  Pa[X] is valid on a finite serial frame . If |W| =  the result is trivial, so we may assume that |W| = n  . Let x, z 2 W. Define a probability distribution Pr over W just as in the proof of .. Suppose that z 2 = R(z). Thus: R(z)  W{z}, so {z}  P[W{z}], so Prx(P[W{z}])  Prx({z}) = Pr({z})/Pr(R(x))  Pr({z}) = b > a. Hence x 2 Pa[P[W{z}]]. By hypothesis, Pa[P[W{z}]]  Pa[W{z}]. Thus x 2 Pa[W{z}]. By an argument as in the proof of ., we can show that z 2 = R(x). By contraposition, if z 2 R(x) then z 2 R(z). Thus R is quasi-reflexive. Conversely, suppose that R is quasi-reflexive and Pr is a regular probability distribution over W. Suppose that z 2 Pa[P[X]]. So Prz(P[X])  a. Let u 2 P[X] \ R(z). Since u 2 R(z), by quasi-reflexivity u 2 R(u). Hence if u 2 = X, Pru(X) <  because Pr is regular. So, since u 2 P[X], u 2 X. Thus P[X]] \ R(z)  X, so a  Prz(P[X])  Prz(X), so z 2 Pa[X]. Thus Pa[P[X]]  Pa[X], as required. Corollary .: For any a, b 2 (, ), one of these principles is valid on a finite serial frame iff the other is: Pa[P[X]]  Pa[X] and Pb[P[X]]  Pb[X]. Thresholda Transfer

Pa ½Pa ½X  Pa ½X

for a 2 ½0; 1:

A frame is near-partitional: for all x 2 W, if y 2 R(x) either R(y) = R(x) or R(y) = {y}. Proposition .: For a 2 (, ), Thresholda Transfer is valid on a finite serial frame iff it is nearpartitional. Proof: Suppose that Thresholda Transfer is valid on a finite serial frame , but is not near-partitional. Thus for some w and x 2 R(w), R(x) 6¼ R(w) and R(x) 6¼ {x}. Since Thresholda Transfer is valid on , so is the principle Pa[P[X]]  Pa[X], for: P[X]  Pa[X], so Pa[P[X]]  Pa[Pa[X]]  Pa[X]. Thus, by ., R is quasi-reflexive, so {x}  R(x), for x 2 R(w). Since R(x) 6¼ {x}, there is a y 2 R(x) with y 6¼ x. Furthermore, by Thresholda Transfer, the principle Pa[Pa[X]]  P>[X] is also valid on , because

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

     



Pa[X]  P>[X]. Thus, by ., R is transitive, so R(x)  R(w); since R(x) 6¼ R(w), there is a z 2 R(w)R(x), so z 6¼ y. Moreover, since x 2 R(x), z 6¼ x. Thus x, y, and z are mutually distinct. There are two cases: Case (i): W = {x, y, z}. Define a probability distribution Pr over W thus: PrðfxgÞ ¼ að1  aÞ PrðfygÞ ¼ a2 PrðfzgÞ ¼ 1  a Since R(x)  W{z}, Pr(R(x))  Pr(W{z}) =  – Pr({z}) = a. Thus, since y 2 R(x): Prx ðfygÞ ¼ PrðfygÞ=PrðRðxÞÞ  a2 =a ¼ a Since y 2 R(x), R(y)  R(x) because R is transitive, so Pr(R(y))  Pr(R(x))  a. Also y 2 R(y), because R is quasi-reflexive. Thus: Pry ðfygÞ ¼ PrðfygÞ=PrðRfygÞ  PrðfygÞ=PrðRfxgÞ  a Thus {x, y}  Pa[{y}]. Moreover, {x, y}  R(w) because R is transitive. Since z 2 R(w), R(w) = W. Hence Prw(Pa[{y}])  Prw({x, y}) = Pr({x, y})/Pr(R(w)) = Pr({x, y}) = a( – a) + a² = a. So w 2 Pa[Pa[{y}]]. Hence w 2 Pa[{y}] because Thresholda Transfer is valid on by hypothesis. But R(w) = W, so Prw({y}) = Pr({y}) = a² < a because a < . Hence w 2 = Pa[{y}], which is a contradiction. Thus is near-partitional after all. Case (ii): |W| = n > . Define a probability distribution Pr over W thus: PrðfxgÞ ¼ að1  aÞ=ð1 þ aÞ PrðfygÞ ¼ 2a2 =ð1 þ aÞ PrðfzgÞ ¼ ð1  aÞ=ð1 þ aÞ PrðfugÞ ¼ að1  aÞ=ðn  3Þð1 þ aÞ

for u 2 W  fx; y; zg:

The argument resembles that of case (i); some overlapping parts are omitted. Pr(R(x))   – Pr ({z}) = a/( + a). Thus: Prx({y}) = Pr({y})/Pr(R(x))  (a²/( + a))/(a/( + a) = a. Similarly, Pry({y})  Pr({y})/Pr(R(x))  a. Thus {x, y}  Pa[{y}]. But Prw(Pa[{y}])  Prw({x, y}) = Pr ({x, y})/Pr(R(w))  Pr({x, y}) = (a( – a) + a²)/( + a) = a. So w 2 Pa[Pa[{y}]]. Hence w 2 Pa[{y}] by Thresholda Transfer. But {x, y, z}  R(w). Thus: Prw{x, y, z}  Prw(R(w)), so: Prw({y}) = Pr({y})/Pr(R(w))  Pr(({y})/Pr({x, y, z}) = a²/(a( – a) + a² +  – a) = a²/( + a²). But ( – a)² > , so  > a/( + a²), so a > a²/( + a²), so Prw({y}) < a. Hence w 2 = Pa[{y}], again a contradiction. Thus is near-partitional after all. Conversely, suppose that is semi-partitional. Let w 2 W and X  W. Suppose that Prw(X) < a. We first note (): fx: Prx ðXÞ  ag \ RðwÞ  X \ RðwÞ For if Prx(X)  a > Prw(X) then R(x) 6¼ R(w), so by near-partitionality R(x) = {x}; since Prx(X) > , some member of R(x) belongs to X, so x 2 X. But () entails (): Prw ðfx: Prx ðXÞ  agÞ  Prw ðXÞ < a By contraposition, if Prw({x: Prx(X)  a})  a then Prw(X)  a. In other words, Pa[Pa[X]]  Pa[X], so Thresholda Transfer at a is valid on , as required.

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi



 

Corollary .: For a, b 2 (, ), Thresholda Transfer is valid on a finite serial frame iff Thresholdb Transfer is. Lemma .: Let be a finite serial frame , w 2 W, and X  W. Then Pr(X) < Prw(X) for some regular probability distribution Pr on iff R(w) \ X ¼ 6 {} and R(w) [ X 6¼ W. Proof: Suppose that R(w) \ X 6¼ {} and R(w) [ X 6¼ W. Let x 2 R(w) \ X and y 2 W(R(w) [ X)), so x 6¼ y. Thus |W| = n  . Define a probability distribution Pr over W thus: If n ¼ 2; PrðfxgÞ ¼ 1=2 and PrðfygÞ ¼ 1=2: If n > 2; PrðfxgÞ ¼ 1=3; PrðfygÞ ¼ 1=2; and PrðfzgÞ ¼ 1=6ðn  2Þ for z 2 W  fx; yg: Either way, since X  W{y}, Pr(X)  Pr(W{y}) = ½. Since R(w)  W{y}, Pr(R(w))  Pr (W{y}) = ½. But Prw(X) = Pr(X \ R(w))/Pr(R(w))  Pr({x})/Pr(R(w))  (/)/(/) = / > ½  Pr(X). For the converse, let Pr be any regular probability distribution on . First, suppose that R(w) \ X = {}. Then Prw(X) =  so Pr(X)  Prw(X). Second, suppose that R(w) [ X = W. Then WR(w)  X so Pr(WX|WR(w)) = . Hence, by total probability: Pr(WX) = Pr(WX|R(w))Pr(R(w)) + Pr(WX|WR(w))Pr(WR(w)) = Pr(WX|R(w))Pr(R(w))  Pr(WX|R(w)) = Prw(WX) so Pr(X) =  – Pr(WX)   – Prw(WX) = Prw(X), as required. Notation : R1 ðxÞ ¼ fw: Rwxg and NR ¼ fw: RðwÞ 6¼ Wg: Comparative Transfer : P> ½P> ½X  P> ½X Proposition .: Let be a finite serial frame and x 2 W. Then: P>[P>[{x}]]  P>[{x}] for every regular probability distribution Pr on iff: for all w, x 2 W: if R(w) \ (R1(x) \ NR) 6¼ {} and R(w) [ (R1(x) \ NR) 6¼ W then x 2 R(w). Proof: First note that for any regular probability distribution Pr and x 2 W: P> ½fxg ¼ R1 ðxÞ \ NR For if x 2 R(w) and R(w) 6¼ W then w 2 P>[{x}] since Pr(R(w)) <  so Pr({x}) < Pr({x})/Pr(R(w)) = Prw({x}). If R(w) = W then w 2 = P>[{x}] because Pr({x}) = Prw({x}). If x 2 = R(w) then w 2 = P>[{x}] because Prw({x}) =  and Pr({x}) > . Thus it suffices to prove: P>[P>[{x}]]  P>[{x}] for every regular probability distribution Pr on iff: for all w 2 W if R(w) \ P>[{x}] 6¼ {} and R(w) [ P>[{x}] 6¼ W then x 2 R(w). Suppose that P>[P>[{x}]]  P>[{x}] for every regular probability distribution Pr on , and that R(w) \ P>[{x}] 6¼ {} and R(w) [ P>[{x}] 6¼ W. By lemma .0, Pr(P>[{x}]) < Prw(P>[{x}]) for some regular probability distribution Pr on , so w 2 P>[P>[{x}]], so w 2 P>[{x}] by hypothesis, so w 2 R1(x) \ NR, so x 2 R(w). Conversely, suppose: whenever R(w) \ P>[{x}] 6¼ {} and R(w) [ P>[{x}] 6¼ W, x 2 R(w). Let Pr be a regular probability distribution on . Suppose that w 2 P>[P>[{x}]]. Thus Pr (P>[{x}]) < Prw(P>[{x}]). Then, by ., R(w) \ P>[{x}] 6¼ {} and R(w) [ P>[{x}] 6¼ W, so by hypothesis x 2 R(w). Therefore w 2 R1(x), and R(w) 6¼ W, so w 2 NR, so w 2 P>[{x}]. Thus P>[P>[{x}]]  P>[{x}], as required. Corollary .: If Comparative Transfer is valid on a finite serial frame , then for all v, w, x, y, z, if Rwy, Ryx, not Ryv, not Rwz, and either not Rzx or for all u Rzu, then Rwx. Corollary .: If Comparative Transfer is valid on a finite serial frame , then for all x, y, z, v, if neither Rxx nor Ryv, then if Rxy and Ryz then Rxz.

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

     



Proof: Substitute x for w and z, and z for x in .. Proposition .: Let be a finite serial frame and x 2 W. Then: P>[P>[W{x}]]  P>[W{x}] for every regular probability distribution Pr on iff: for all w, x 2 W, if x 2 R(w) then either R(w)  R1(x) or R1(x)  R(w). Proof: Note that P>[W{x}] = WR1(x). For if w 2 R1(x) and R(w) 6¼ W then w 2 = P>[W{x}] because Pr({x}) < Prw({x}) (as in .) so Pr(W{x}) > Prw(W{x}). If R(w) = W then, again as in ., w 2 = P>[{x}]. If w 2 = R1(x) then w 2 P>[W{x}] because Prw(W{x}) =  and Pr(W{x}) < . Thus it suffices to prove: P>[P>[W{x}]]  P>[W{x}] for every regular probability distribution Pr on iff: for all w 2 W if R(w) \ P>[{x}] 6¼ {} and R(w) [ P>[{x}] 6¼ W then w 2 WR1(x). The rest of the proof is similar to that of .. Corollary .: If Comparative Transfer is valid on a finite serial frame , then for all w, x, if Rwx then either for all z if Rwz then Rzx or for all z if Rzx then Rwz. Corollary .: If Comparative Transfer is valid on a finite serial frame , then whenever Rwx, either Rww or Rxx. Proof: By ., substituting x for z in the first disjunct and w for z in the second. Corollary .: On any finite serial frame on which Comparative Transfer is valid, so is (☐p ⊃ p) ∨ ☐(☐q ⊃ q). Proof: Suppose that ☐p ⊃ p is false at w 2 W in a model based on such a frame . Thus not Rww. Hence, by ., Rxx for all x such that Rwx, so ☐(☐q ⊃ q) is true at w in the model. Corollary .: On any finite serial frame on which Comparative Transfer is valid, for all w, x, if Rwx then there is a u such that Rwu and Rux. Proof: From .. Corollary .: On any finite serial frame on which Comparative Transfer is valid, so is ☐☐p ⊃ ☐p. Proof: From .. Corollary .: On any finite serial frame on which Comparative Transfer is valid, for all w, x, y, if Rwx and Rwy but not Rww then Rxy. Proof: By ., since Rwy, either for all z if Rwz then Rzy or for all z if Rzy then Rwz. The first disjunct implies that Rxy because Rwx. The second disjunct implies that Rww because Rwy. Corollary .: On any finite serial frame on which Comparative Transfer is valid, so is (☐p ⊃ p) ∨ (◊☐q ⊃ ☐q). Proof: Suppose that ☐p ⊃ p is false at some w in a model based on such a frame . Then not Rww. Suppose that ◊☐q is true at w in the model, so ☐q is true at some x such that Rwx. Then whenever Rwy, by . Rxy, so q is true at y. Thus ☐q is true at w. Corollary .: On any finite serial frame on which Comparative Transfer is valid, for all w, x, y, z, if Rwx, Rxy, and Ryz, then either Rxw, or Ryx, or Rzy. Proof: Suppose that Rwx, Rxy, and Ryz. By ., since Rxy, either Rxx or Ryy. Suppose that Rxx. So by . either for all u if Rxu then Rux or for all u if Rux then Rxu. Hence either if Rxy then Ryx or if Rwx then Rxw; thus either Rxw or Ryx. Similarly, if Ryy then either Ryx or Rzy. Hence either Rxw, or Ryx, or Rzy.

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi



 

Corollary .: On any finite serial frame on which Comparative Transfer is valid, so is p ⊃ ☐(q ⊃ (◊p ∨ ☐(r ⊃ (◊q ∨ ☐◊r))). Proof: Suppose that the formula is false at some w in some model based on the frame. Then p is true and ☐(q ⊃ (◊p ∨ ☐(r ⊃ (◊q ∨ ☐◊r))) at w. Hence q ⊃ (◊p ∨ ☐(r ⊃ (◊q ∨ ☐◊r)) is false at some x such that Rwx. Hence q is true and both ◊p and ☐(r ⊃ (◊q ∨ ☐◊r)) false at x. The former implies that not Rxw. The latter implies that r ⊃ (◊q ∨ ☐◊r) is false at some y such that Rxy, so r is true and both ◊q and ☐◊r are false at y. The former implies that not Ryx. The latter implies that ◊r is false at some z such that Ryz. Hence not Rzy. That is impossible by .. Corollary .: On any finite reflexive frame on which Comparative Transfer is valid, for all w, x, y, if Rwx and Rxy, then either Rxw or Ryx. Proof: A simplification of the proof of .. Corollary .: On any finite reflexive frame on which Comparative Transfer is valid, so is p ⊃ ☐(q ⊃ (◊p ∨ ☐◊q)). Proof: A simplification of the proof of .. Corollary .: Suppose that Comparative Transfer is valid on a finite serial frame . Then if Rwx but not Rww, either R(x) = W or R(x) = R(w). Proof: From . and .. Corollary .: If a serial frame with |W|   satisfies the necessary conditions in . and . for Comparative Transfer to be valid, then Comparative Transfer is valid on . Proof: Suppose that X  W. If |X| =  then by . P>[P>[X]]  P>[X]; if |X| =  then by . P>[P>[X]]  P>[X]. If |X| =  or |X| =  then P>[P>[X]] = {} so the case is trivial. Proposition .: If Comparative Transfer is valid on a finite serial frame , then for all w, x, y, z, if Rwx, Rxy, and Ryz, then for some u, Rwu and Ruz. Proof: It suffices to derive a contradiction from the hypothesis that Rwx, Rxy, Ryz, and there is no u such that Rwu and Ruz. First note that there are no shortcuts in the chain from w to z, in the sense that not Rwy, not Rxz, and not Rwz. The first two are easy because we can put u = y and u = x respectively; the third follows from .. But by an instance of .: if Rwx, Rxy, not Rxz, not Rwz and not Rzy then Rwy. Thus, since Rwx, Rxy, not Rwz, and not Rwy, it follows that Rzy. By another instance of .: for all v, if Rxy, Ryz, not Ryv, not Rxw, and not Rwz, then Rxz. Thus, since Rxy, Ryz, not Rwz, and not Rxz, this follows:

(*)

Either Rxw or Ryv for all v.

But by an instance of .: if Rxy then either for all u if Rxu then Ruy or for all u if Ruy then Rxu. Thus, since Rxy, either if Rxw then Rwy or if Rzy then Rxz. But on the initial hypothesis we already have that Rzy, not Rxz, and not Rwy. Thus not Rxw. Hence, by (*), Ryv for all v. But by yet another instance of .: if Rwx, Rxy, not Rxz, not Rwy, and for all v Ryv, then Rwy. This is a final contradiction, since on the initial hypothesis we already have that Rwx, Rxy, not Rxz, not Rwy, and for all v Ryv. Corollary .: If Comparative Transfer is valid on a finite serial frame, then so too is ☐☐p  ☐☐☐p. Proof: From . and . by standard model theory of modal logic. Proposition .: If is a finite serial frame such that whenever w 2 W, x 2 R(w), either R(w) = W or R(x) = W or R(x) = R(w), then Comparative Transfer is valid on .

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

     



Proof: If R(w) = W then for all Y  W Prw(Y) = Pr(Y) so w 2 = P>[Y]. Suppose that w 2 = P>[X] and w 2 P>[P>[X]]. Thus R(w) 6¼ W (let Y = P>[X]). Suppose that for all x 2 R(w) either R(x) = W or R(x) = R(w). If R(x) = W, then x 2 = P>[X]. If R(x) = R(w) then Prx(X) = Prw(X) so x 2 = P>[X] because w 2 = P>[X]. Thus for all x 2 R(w), x2 = P>[X]. Hence Prw(P>[X]) = , so w 2 = P>[P>[X]], contrary to hypothesis. Corollary .: On some finite serial frames, Comparative Transfer is valid but none of ☐p ⊃ ☐☐p, ◊p ⊃ ☐◊p, p ⊃ ☐◊p, and ☐p ⊃ p is valid. Proof: Consider the three-world frame where W = {a, b, c}, R(a) = {b}, R(b) = W, and R(c) = {c}. By ., Comparative Transfer is valid on . R is non-transitive, because Rab and Rbc but not Rac, so ☐p ⊃ ☐☐p is not valid. R is non-euclidean because Rba and Rbc but not Rac, so ◊p ⊃ ☐◊p is not valid. R is non-symmetric because Rbc but not Rcb, so p ⊃ ☐◊p is not valid. R is non-reflexive because not Raa, so ☐p ⊃ p is not valid. Proposition .: Let be a finite serial frame such that for some x, z 2 W, x 6¼ z, x 2 R(x) and for all y 2 W, if y 6¼ x then R(y) = {z}. Then Comparative Transfer is valid on . Proof: Let , x, z be as specified. Suppose that y 2 = P>[X] for some y 2 W and X  W. If y 6¼ x then R(y) = {z} = R(z), so y 2 = P>[P>[X]] (as in the proof of .). Thus it suffices to show that if x 2 = P>[X] then x 2 = P>[P>[X]]. There are two cases. Case (i): z 2 X. We may assume that X 6¼ W, otherwise P>[X] = {} so P>[P>[X]] = {} and we are done. Hence Pr(X) < . Now if y 6¼ x, R(y) = {z}  X so Pry(X) = , so y 2 P>[X]. Since x 2 R(x), WR(x)  P>[X], so Pr(P>[X]|WR(x)) = . Hence, by the principle of total probability: PrðP> ½XÞ ¼ PrðP> ½XjRðxÞÞPrðRðxÞÞ þ PrðP> ½XjW  RðxÞÞPrðW  RðxÞÞ ¼ Prx ðP> ½XÞPrðRðxÞÞ þ PrðW  RðxÞÞ  Prx ðP> ½XÞPrðRðxÞÞ þ Prx ðP> ½XÞPrðW  RðxÞÞ ¼ Prx ðP> ½XÞ Hence x 2 = P>[P>[X]], as required. Case (ii): z 2 = X. Hence if y 6¼ x, Pry[X] = , so y 2 = P>[X]. Thus if x 2 = P>[X], then Prx(P>[X]) = , hence x 2 = P>[P>[X]] and we are done. Proposition .: Let be a finite serial frame, with some z 2 W such that for all x 2 W, R(x)  {x, z}. Then Comparative Transfer is valid on . = P>[P>[X]]. Proof: Let , z be as specified, and x 2 = P>[X]. It suffices to show that x 2 There are two cases. Case (i): z 2 X. As in the proof of ., we may assume that X 6¼ W, so Pr(X) < . Observe that X  P>[X], for if y 2 X then R(y)  {y, z}  X, so Pry(X) =  > Pr(X), so y 2 P>[X]. Thus: Pr(X)  Pr(P>[X]). Since x 2 = P>[X] and z 2 X, P>[X] \ R(x)  X \ R(x), so: Prx(P>[X])  Prx(X). Since x 2 = P>[X]: Prx(X)  Pr(X). Thus Prx(P>[X])  Prx(X)  Pr(X)  Pr(P>[X]), so x2 = P>[P>[X]], as required. Case (ii): z 2 = X. Since R(z) = {z}, Prz[X] = , so z 2 = P>[X]. Hence if if x 2 = P>[X] then P>[X] \ R(x) = {}, so Prx(P>[X]) = , so x 2 = P>[P>[X]], and we are done. Proposition .: The condition on R for Comparative Transfer to be valid on a finite serial frame is not expressible in first-order logic without identity. Proof: Let W = {, , }, R() = {, }, R() = {}, R() = {}, W* = {, , , }, f be the function from W* onto W such that f = , f = , f = f = , and for all j, k in W*, R*jk iff Rfjfk. Thus  and  are indiscernible in terms of R*. By standard non-modal model theory, the models and satisfy the same formulas in the language of first-order logic with just

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi



 

one predicate letter, which is binary, and without identity. Since for all j in W, R(j)  {j, }, by . Comparative Validity is valid on . But Comparative Transfer is not valid on . For let Pr be the uniform probability distribution over W*, and X = {, }, so Pr(X) = ½. Now Pr₀(X) = Pr₂(X) = Pr₃(X) = ½, Pr₁(X) = , so Pr>[X] = {}, so Pr(Pr>[X]) = ¼. Consequently, Pr₀ (Pr>[X]) = ½, so  2 Pr>[Pr>[X]], but  2 = Pr>[X]. Since differs from with respect to the validity of Comparative Transfer but not with respect to which formulas of firstorder logic without identity are satisfied, the former cannot be expressed in terms of the latter. Corollary .: The necessary conditions in . and . for Comparative Transfer to be valid on a finite serial frame are not jointly sufficient. Proof: By . and ., those necessary conditions are expressible in first-order logic without identity, so by . they are not jointly sufficient. Proposition .: There is a finite serial frames on which Comparative Transfer is valid such that Comparative Transfer is not valid on the union of two disjoint copies of that frame. Proof: Let W = {, }, R() = {, }, R() = {} W* = {, }, R*() = {, }, R*() = {}; thus and are isomorphic. By ., Comparative Transfer is valid on and . But it is not valid on . For let Pr be the uniform probability distribution over W and X = {, }, so Pr(X) = ½. Thus Pr₀(X) = Pr₂(X) = ½, Pr₁(X) = , and Pr₃(X) = , so Pr>[X] = {}, so Pr(Pr>[X]) = ¼. Thus Pr₀(Pr>[X]) = ½, so  2 Pr>[Pr>[X]], but  2 = Pr>[X]. Proposition .: For a 2 (, ), the validity of Thresholda Transfer and the validity of Comparative Transfer are independent conditions on a frame. Proof: The union frame in the proof of ., on which Comparative Transfer is invalid, meets the condition for the validity of Thresholda Transfer. Conversely, the latter condition entails transitivity, so Thresholda Transfer is invalid on those non-transitive frames on which Comparative Transfer is valid (see .). We make the following definitions for a frame ( expresses proper subsethood): is quasi-nested iff for all w, x, y, z 2 W, if Rwx, Rwy, Rxz, and Ryz, then either Rxy or Ryx. R0 ðwÞ ¼ RðwÞ \ fu: RðuÞ ¼ RðwÞg R1 ðwÞ ¼ RðwÞ \ fu: RðuÞ  RðwÞ and for no v 2 RðwÞ: RðuÞ  RðvÞ  RðwÞg Observation .: A quasi-nested serial frame is quasi-reflexive. Proof: Put x = y in the definition of ‘quasi-nested’. Proposition .: Let be a finite serial transitive quasi-nested frame. If x, y 2 R₁(w) then either R(x) = R(y) or R(x) \ R(y) = {}. Proof: Suppose that R(x) \ R(y) 6¼ {}. Hence, for some z, Rxz and Ryz. Since R₁(w)  R(w), Rwx and Rwy. Thus, since R is quasi-nested, either Rxy or Ryx. Without loss of generality, suppose that Rxy. Thus, since R is transitive, R(y)  R(x). Suppose that R(x) 6¼ R(y). Hence R(y)  R(x). But x 2 R₁(w), so R(x)  R(w). Thus R(y)  R(x)  R(w), contradicting the hypothesis that y 2 R₁(w). Hence R(x) = R(y). Proposition .: Let be a finite serial transitive quasi-nested frame, w 2 W, and J consist of one member from each equivalence class of the equivalence relation defined by R(x) = R(y) over R₁(w). Then R₀(w) and R(j) for j 2 J partition R(w). Moreover, if j and k are distinct members of J then R(j) \ R(k) = R₀(w) \ R(j) = {}.

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

     



Proof: By definition, R₀(w)  R(w) and R(j)  R(w) for j 2 J  R₁(w), so R₀(w) [ ([ j 2 J R(j))  R(w). For the converse, let x 2 R(w). Since R is transitive, R(x)  R(w). If R(x) = R(w) then x 2 R₀(w) and we are done, so suppose that R(x)  R(w). Since W is finite, there is a sequence x₀, x₁, . . ., xn all in R(w) of maximal length such that x₀ = x and for   i < n, R(xi)  R(xi+₁)  R(w). So there is no v 2 R(w) such that R(xn)  R(v)  R(w), otherwise we could put v = xn+₁. Thus xn 2 R₁(w), so for some k 2 J: R(k) = R(xn). Since x 2 R(w) and by . R is quasi-reflexive, x 2 R(x) = R (x₀)  R(xn) = R(k), so x 2 [ j 2 J R(j). Thus R(w) = R₀(w) [ ([ j 2 J R(j)). Moreover, if j and k are distinct members of J then R(j) 6¼ R(k), so R(j) \ R(k) = {} by ., and if x 2 R(j) then by transitivity R(x)  R(j)  R(w), so x 2 = R₀(w). Proposition .: Let be a finite serial transitive quasi-nested frame, b 2 [, ], X  W, w 2 W, and Pr a regular probability distribution on . Then bPrw(Pb[X])  Prw(X). Proof: We use induction on |R(w)|. The base case is vacuous because R is serial. Suppose that for all x 2 W such that |R(x)| . Second, we show that is quasi-reflexive. For suppose that Rwx but not Rxx, so w 6¼ x. Let |W| = n  . Define a probability distribution Pr over W thus:

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi



  PrðfxgÞ ¼ 2=3 PrðfygÞ ¼ 1=3ðn  1Þ for y 6¼ x

Then Prx(W – {x}) = , so x 2 Pr[W – {x}], so Prw(Pr[W – {x}])  Prw({x})  /, so w 2 Pr/[Pr[W – {x}]], but Prw(W – {x}) =  – Prw({x})    Pr({x}) = /, so w 2 = Pr/ (W – {x}). Finally, we show that is quasi-nested. For suppose otherwise. Then for some w, x, y, z 2 W, if Rwx, Rwy, Rxz, Ryz, but neither Rxy nor Ryx; thus w 6¼ x, w 6¼ y, x 6¼ z, and y 6¼ z. Since R is quasi-reflexive, Rxx, Ryy, and Rzz; thus x 6¼ y. Since R is transitive, Rwz but neither Rzx (otherwise Ryx) nor Rzy (otherwise Rxy); thus w 6¼ z. Hence |W| = n  . Define a probability distribution Pr over W thus: PrðfxgÞ ¼ PrðfygÞ ¼ 6=25 PrðfzgÞ ¼ 12=25 PrðfugÞ ¼ 1=25ðn  3Þ for u 2 = fx; y; zg Since {z}  R(z)  W–{x, y}), Prz({z})/ > /. Since {x, z}  R(x)  W – {y}: Prx({z}) = Pr({z})/Pr(R(x))  Pr({z})/Pr(W – {y}) = /. Similarly, Pry({z})  /. Hence {x, y, z}  P/[{z}], so Prw(P/[{z}])  Prw({x, y, z})  Pr({x, y, z}) = /. Hence w 2 P/ [P/[{z}]]. But {x, y, z}  R(w), so Prw({z}) = Pr({z})/Pr(R(w)  Pr({z})/Pr({x, y, z}) = ½ < (/)(/) = /, so w 2 = Pr/({z}). Thus the principle Pr/[Pr/[X]]  Pr/[X]] is not valid on the frame, contrary to hypothesis.

Acknowledgements Earlier versions of this material were presented at Oxford and a British Academy-funded conference on epistemology in Cambridge; I thank participants for useful suggestions, and Bernhard Salow, Kevin Dorst, and John Hawthorne for valuable discussion and correspondence. Many thanks are also due to Mattias Skipper in his capacity as editor and Kevin Dorst in his capacity as referee for providing detailed constructive comments; space and time do not permit full responses to all of them. More thanks are also due to Skipper for the diagrams.

References Blackburn, P., M. de Rijke, and Y. Venema (). Modal Logic, Cambridge University Press. Briggs, R. (). “Distorted Reflection.” In: The Philosophical Review , pp. –. Das, N. and B. Salow (). “Transparency and the KK Principle.” In: Noûs , pp. –. Dorst, K. (). “Evidence: A Guide for the Uncertain.” In: Philosophy and Phenomenological Research. Online first. Feldman, R. (). “Reasonable Religious Disagreements.” In: L. Antony (ed.), Philosophers Without God: Meditations on Atheism and the Secular Life, Oxford University Press, pp. –. Feldman, R. (). “Evidence of Evidence is Evidence.” In: J. Matheson and R. Vitz (eds), The Ethics of Belief, Oxford University Press, pp. –. Fitelson, B. (). “Evidence of Evidence is not (Necessarily) Evidence.” In: Analysis , pp. –. Geanakoplos, J. (). “Game Theory without Partitions, and Applications to Speculation and Consensus.” Cowles Foundation Discussion Paper, , Yale University. Greco, D. (). “Could KK be OK?” In: Journal of Philosophy , pp. –.

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

Hawthorne, J. and O. Magidor (). “Assertion, Context, and Epistemic Accessibility.” In: Mind , pp. –. Hawthorne, J. and O. Magidor (). “Assertion and Epistemic Opacity.” In: Mind : –. Hintikka, J. (). Knowledge and Belief, Cornell University Press. Hughes, G. and M. Cresswell (). A Companion to Modal Logic, Methuen. Roche, W. (). “Evidence of Evidence is Evidence under Screening-off.” In: Episteme , pp. –. Salow, B. (). “Elusive Externalism.” In: Mind , pp. –. Stalnaker, R. (). “On Logics of Knowledge and Belief.” In: Philosophical Studies , pp. –. Stalnaker, R. (). “On Hawthorne and Magidor on Assertion, Context, and Epistemic Accessibility.” In: Mind , pp. –. Tal, E. and J. Comesaña (). “Is Evidence of Evidence Evidence?” In: Noûs (), pp. –. Weisberg, J. (). “Conditionalization, Reflection, and Self-Knowledge.” In: Philosophical Studies , pp. –. White, R. (). “Problems for Dogmatism.” In: Philosophical Studies , pp. –. Williamson, T. (). Knowledge and its Limits, Oxford University Press. Williamson, T. (). “Very Improbable Knowing.” In: Erkenntnis , pp. –.

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14 Can Your Total Evidence Mislead About Itself? Alex Worsnip

. Introduction: what would it be for total evidence to mislead about itself? It’s fairly uncontroversial that you can get misleading evidence about your evidence. Suppose the clues at the crime scene do not support believing p, but then some expert tells you (in a rare lapse of judgment) that the clues do support believing p. Then you have misleading “higher-order evidence” about your “first-order evidence” (and what it supports believing). More generally, we can define misleading higher-order evidence as follows: Misleading Higher-Order Evidence. A case of misleading higher-order evidence is one where either: (a) (i) Your first-order evidence supports some doxastic attitude D toward p, but (ii) You have higher-order evidence that your first-order evidence does not support D toward p; or (b) (i) Your first-order evidence does not support some doxastic attitude D toward p, but (ii) You have higher-order evidence that your first-order evidence supports D toward p.¹ What is more controversial is whether such a phenomenon can ever result in a situation whereby one has misleading total evidence about one’s total evidence. That is: Misleading Total Evidence (about Total Evidence). A case of misleading total evidence (about total evidence) is one where either: (a) (i) Your total evidence supports some doxastic attitude D toward p, but

¹ You may notice that, under both (a) and (b), clause (i) refers to evidential support for attitudes, whereas clause (ii) refers to (higher-order) evidence for propositions. (Moreover, in the definition of misleading total evidence below, both clauses (i) and (ii) refer to evidential support for attitudes.) This is intentional; I refer you to section . below to understand my terminology. Alex Worsnip, Can Your Total Evidence Mislead About Itself? In: Higher-Order Evidence: New Essays. Edited by: Mattias Skipper and Asbjørn Steglich-Petersen, Oxford University Press (). © the several contributors. DOI: ./oso/..

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       ?



(ii) Your total evidence supports believing that your total evidence does not support D toward p. (b) (i) Your total evidence does not support some doxastic attitude D toward p, but (ii) Your total evidence supports believing that your total evidence supports D toward p. Such cases, if there are such, are ones where one’s total evidence misleads about what it itself supports. These are general characterizations of misleading higher-order evidence and misleading total evidence, but I am going to focus on the particular kind of misleading higher-order evidence that I mentioned in the opening paragraph: that is, cases where your first-order evidence does not support believing p, but your higher-order evidence supports believing that your first-order evidence does not support believing p. The question is: can such a case amount to a case of misleading total evidence, where your total evidence does not support believing p, but your total evidence supports believing that your total evidence supports believing p? If we can show that it does, then we have shown that at least one kind of case of misleading total evidence, defined in the more general above way, is possible.

. Straightening out some terminology In trying to make progress on this question, we need to be clear about our terminology. One source of confusion in the literature is that not all participants use the term “support” in the same way. First, some talk of the evidence supporting propositions, while others talks of the evidence supporting attitudes toward those propositions. Secondly, to say that the evidence “supports” something might mean either that it supports it to some degree, or that it supports it to some relevantly sufficient degree. I think there is probably more than one permissible way of talking here, but there are some considerations that affect choice of terminology. First, we want to allow ourselves the capacity to say that in some cases, the attitude toward a proposition p that is best supported by one’s total body of evidence is that of suspending judgment about whether p. Secondly, though, within such cases, we want to be able to distinguish between those in which one’s total body of evidence, on balance, supports p to some extent (just not enough to support believing it), and those in which it doesn’t. We don’t want to mistakenly suggest that whenever the evidence on balance favors p over not-p, it warrants believing p. In light of that, I will talk (and have already been talking, in section .) in the following way. Evidential support for propositions fundamentally comes in degrees. We can talk about how strongly a body of evidence supports a proposition p. When I say that a body of evidence supports p, without a qualifier as to how strongly or weakly it supports it, I mean only that it supports p, on balance, at least somewhat better than it supports not-p. This is still compatible with its supporting p (over not-p) rather weakly—not strongly enough to warrant believing p, rather than suspending judgment. In addition to this, however, I will also talk of evidential support for attitudes. When I talk of a body of evidence supporting an attitude, I mean that given that body of evidence (taken on its own), the attitude is warranted. This is an all-or-nothing

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matter, and is not something that comes in degrees. But plausibly, it is systematically related to degreed evidential support for propositions. Plausibly, for a body of evidence to support believing p just for it to support p to some (perhaps situationally variable) sufficient degree (to warrant belief ). It’s also natural to say that what it is for a body of evidence to support suspending judgment about whether p is for it neither to support p to a sufficient degree (to warrant belief ), nor to support not-p to a sufficient degree. But this latter claim will have to be complicated if there are permissive cases where the evidence permits either believing p or suspending judgment about whether p.² For the sake of simplicity, I will ignore permissive cases in what follows, though the core of my argument does not turn on assuming that there are no such cases. In a sense, then, “support” means something different when I talk about support for propositions than it does when I talk about support for attitudes. Perhaps it would be good to have two different words instead of using “support” in these two different ways. But my meaning can always be discerned by whether I am talking of support for propositions or for attitudes. I also need to clarify what does the supporting. On the way I’m talking, a “body of evidence” can be one’s total evidence, or it can be a subset of one’s total evidence (at the limit, it could be a single item, e.g., a single proposition). So, we can ask both whether one’s total evidence supports p, and whether some particular subset of one’s evidence supports p. Slightly less obviously, I think that we can also talk either about whether one’s total evidence supports an attitude toward p, or about whether a subset of one’s total evidence supports an attitude toward p.³ Indeed, notice that, as I defined “misleading higher-order evidence” in section ., it refers explicitly in clauses (a.i) and (b.i) to the support provided for attitudes by subsets of one’s evidence. It might seem a little unclear what this talk comes to, given my stipulations about what it means to talk of evidential support for attitudes. But I propose to understand it as follows: to say that a subset of one’s evidence supports a doxastic attitude toward p is to say that, given that subset of one’s evidence, and bracketing all one’s other evidence, the attitude is warranted. (This is what the “taken on its own” clause in my definition of support for attitudes was supposed to indicate.) Put counterfactually, if that helps: were that subset of one’s evidence to exhaust one’s total evidence, that doxastic attitude toward p would be warranted.⁴ Note that this does not require that the

² If this is so, we might try saying that for the evidence to support suspending judgment about whether p is for it not to support p so strongly that only believing p is warranted, and likewise for not-p. Or we might want to introduce notions of “decisive” and “sufficient” support, where decisive support for an attitude entails that only that attitude is warranted. ³ In light of these points, I’ll avoid using the language of “pro tanto support,” because it’s not clear whether it refers to support to a degree that may not be sufficient to warrant belief, or whether it refers to support by a part, rather than the whole, of one’s evidence. These two statuses are orthogonal and should not be confused. ⁴ This counterfactual is tricky to manage when we are talking about higher-order evidence, though. Since higher-order evidence is (at least often) evidence about one’s first-order evidence, it’s hard to envisage a situation where a particular body of higher-order evidence exhausts one’s evidence. Still, I think there’s still a good sense in which we can evaluate whether the support provided by the higherorder evidence for a proposition would be enough to warrant belief if we at least bracket the force of the first-order evidence.

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

subset of one’s evidence makes that doxastic attitude warranted even given the presence of one’s other evidence. However, I do want a word to refer to the stronger feature, of a subset of one’s evidence, of making it the case that one’s total evidence supports a doxastic attitude. I will say that a subset of one’s evidence is “dispositive” with respect to a particular doxastic attitude when, given the other evidence one has, its presence makes it the case that one’s total evidence supports this attitude. Given my terminology, then, a subset of your evidence can support a doxastic attitude without being dispositive with respect to that attitude: this is so when, were it to exhaust your total evidence, it would warrant belief, but the other evidence you have is such that overall, your total evidence does not (sufficiently) support believing this proposition. I will characterize higher-order evidence, somewhat vaguely, as evidence about one’s evidence, or what one’s evidence supports, or how one has responded to one’s evidence. I won’t try to characterize exactly what this aboutness relation comes to. I assume that we have a good enough grip on it to classify some evidence as first-order and some as higher-order: for example, the clues at the crime scene are first-order evidence, whereas testimony about what the clues support is higher-order evidence. Even if it turned out that no principled or completely sharp distinction between first-order evidence and higher-order evidence could be drawn, though, this would not be a problem for my argument. In principle, I could expunge all reference to “first-order” and “higher-order” evidence and just talk about the clues and the testimony, arguing that ultimately this package of evidence can lead to a situation where one’s total evidence, consisting of both the clues and the testimony, misleads about itself. With that point made, I will continue to talk of “first-order” and “higher-order” evidence in what follows. One point about the meaning of “higher-order evidence” bears stressing, however. It is yet another frequent source of confusion in the literature that the adjectives “higher-order” and “first-order” can attach either to different bodies of evidence, or to different propositions (and attitudes that have those propositions as their objects). A higher-order proposition would be a proposition like the evidence supports believing p, where the corresponding first-order proposition is p. It might be tempting to characterize the higher-order evidence as the evidence that evidentially bears on the higher-order proposition, and the first-order evidence as the evidence that evidentially bears on the first-order proposition. But this would be a mistake. As we’ll see shortly, most philosophers (on both sides of the debate about misleading total evidence) admit that higher-order evidence, such as testimony about whether the evidence supports believing p, can be at least some evidence for p itself. (And, slightly more arguably, first-order evidence, such as the clues, may provide some evidence concerning higher-order propositions about what they themselves support.) So the higher-order evidence cannot be distinguished from the first-order evidence just by what propositions it bears on.⁵ Maybe there is a sense in which higher-order evidence bears more directly on the relevant higher-order proposition than it does on the ⁵ Given this, the “aboutness” relation referred to in the previous paragraph had better not be that of evidential bearing. I assume it is something stronger. One stronger way of characterizing it, assuming that evidence is propositional, would be to say that the higher-order evidence is evidence about one’s evidence in that its propositional content ineliminably refers to one’s evidence. But I leave it open whether this is exactly right.

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relevant first-order proposition, but it can still bear on the latter in a perfectly good sense—or at least, this should not be ruled out by terminological fiat.

. (Re)framing the debate With this terminology in hand, we can reframe the debate. In particular, we can state two different ways in which one might have misleading higher-order evidence without having misleading total evidence (about total evidence). First, one’s higher-order evidence might be dispositive both with respect to believing the (higher-order) proposition that one’s total evidence supports believing p, and with respect to believing the (first-order) proposition p itself. Many philosophers have thought that, at least usually, “evidence of evidence is evidence”—that is, slightly more precisely, evidence that one has evidence for p is itself evidence for p. If this is so, then even when one lacks first-order evidence that supports believing p, the misleading higherorder evidence that one has might itself be dispositive with respect to believing p. In such a case, one’s total evidence would support believing that one’s total evidence supports believing p, and one’s total evidence would support believing p. Thus, though we are in a case of misleading higher-order evidence, we would not be in a situation of misleading total evidence (about total evidence). Call such cases doubly-dispositive cases—since they are cases where the misleading higher-order evidence is dispositive both with respect to the higher-order belief and the first-order belief. Secondly, one’s higher-order evidence might be dispositive neither with respect to believing the proposition that one’s total evidence supports believing p, nor with respect to believing the proposition p itself. In such a situation, one’s higher-order evidence would provide some support for the proposition that one’s total evidence supports believing p—perhaps even enough such that, taken on its own, it supports believing this proposition—but, given the presence of the first-order evidence, not enough to be dispositive with respect to believing this proposition. Remember that in the case of misleading higher-order evidence that I’m focusing on, one’s first-order evidence does not support believing p. Maybe this fact carries with it some kind of rational self-evidence that is not defeated by higher-order evidence such as testimony. In such a case, one’s total evidence would not support believing p, nor would one’s total evidence support believing that one’s total evidence supports believing p. Thus, again, though we are in a case of misleading higher-order evidence, we are not in a position of misleading total evidence (about total evidence). Call such cases nondispositive cases—since they are cases where the misleading higher-order evidence is dispositive neither with respect to the higher-order belief nor the first-order belief. Faced with cases of misleading higher-order evidence, some philosophers try to block the possibility of misleading total evidence (about total evidence) by claiming that all cases of misleading higher-order evidence are doubly-dispositive cases;⁶ others by claiming that all cases of misleading higher-order evidence are non-dispositive cases.⁷ But, in my view, it is more promising for the denier of misleading total evidence (about total evidence) to make the weaker, and more ⁶ Cf., e.g., Feldman (), Bergmann (), and (more qualifiedly) Horowitz (). ⁷ Cf., e.g., Titelbaum ().

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

plausible, claim that all cases of misleading higher-order evidence are either doubly-dispositive cases or non-dispositive cases. That is: the misleading higherorder evidence may be dispositive with respect to the higher-order belief, or may not be; the crucial claim is simply that if it is dispositive with respect to the higher-order belief, it is also dispositive with respect to the first-order belief. The challenge for those who think that there can be misleading total evidence (about total evidence), then, is to show that this is mistaken. In other words, they must show that there can be singly-dispositive cases: cases where the misleading higher-order evidence is dispositive with respect to the higher-order belief, but not the first-order belief. Let me spell out exactly why singly-dispositive cases of misleading higher-order evidence, if there are such cases, are instances of misleading total evidence about total evidence. Begin with a case in which your first-order evidence does not support believing p, but you have misleading higher-order evidence that your first-order evidence does support believing p. As I said above, the possibility of such a case, so far, should be uncontroversial. Now, if the higher-order evidence is singlydispositive, then it is dispositive with respect to the belief that your total evidence supports p—meaning that this belief is now supported by your total evidence. At the same time, if the higher-order evidence is singly-dispositive, then it is not dispositive with respect to believing p. So, since your first-order evidence doesn’t support believing p, and your higher-order evidence isn’t dispositive with respect to believing p, your total evidence doesn’t support believing p. Thus, any singlydispositive case of misleading higher-order evidence is one where your total evidence supports believing that your total evidence supports believing p, but where your total evidence doesn’t support believing p. That is, it is a case of misleading total evidence about total evidence. I’ve tried before (though not using the same terminology) to argue for the possibility of singly-dispositive cases (Worsnip ). But though I think they are on the right track, I’m not fully happy with the arguments I’ve provided previously. In this chapter, I am going to explore a slightly more formal argument, using a simple mathematical model. I believe this argument poses a significant challenge to those who deny the possibility of misleading total evidence, though as I will acknowledge, it still falls short of an incorrigible proof. As a final preliminary to my core argument, let me reiterate that those who affirm the possibility of misleading total evidence (about total evidence) do tend to acknowledge that misleading higher-order evidence has some evidential bearing on the relevant first-order belief.⁸ I concur here—evidence of evidence is (typically, some⁹) evidence—and my argument will not attempt to show otherwise. Indeed, it may even be that misleading higher-order evidence is often dispositive with respect to the

⁸ Cf., e.g., Kelly (); Pryor (, pp. –); Worsnip (); Lasonen-Aarnio (forthcoming). ⁹ See Fitelson () for a case where evidence that there is evidence for p does not itself provide any evidence for p. But Fitelson explicitly says (translating to my terminology) that this only works for evidence that is not dispositive. By his own lights, his case is not one where one has dispositive evidence that one’s evidence supports believing p. In fact, he says that he suspects that when one does have such dispositive evidence that one’s evidence supports believing p, this fact is always evidence for p (Fitelson , fn. ).

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relevant first-order belief. My view is simply that misleading higher-order evidence can sometimes be dispositive with respect to a higher-order belief, while not being dispositive with respect to the corresponding first-order belief (that is, that it is sometimes singly-dispositive). If that is so, then misleading total evidence (about total evidence) is possible. The plan for the remainder of this chapter is as follows. In section ., I will argue for the following claim: misleading higher-order evidence is often stronger evidence for the relevant higher-order proposition than it is for the relevant first-order proposition. Call this the Comparative Claim. I think that the Comparative Claim can be established fairly decisively, using the model that I provide. In section ., I will explore how we might get from this claim to the claim that there are singlydispositive cases: cases where the higher-order is dispositive with respect to the relevant higher-order belief without being dispositive with respect to the relevant first-order belief. This step might seem trivial, but it turns out not to be. Nevertheless, I will suggest that it is still fairly hard to resist this step of the argument. And if one accepts that step of the argument—if one accepts that there are singly-dispositive cases—it follows, as I’ve already shown in this section, that misleading total evidence about total evidence is possible.

. Arguing for the Comparative Claim As I stated it above, the Comparative Claim was vague. I will begin by arguing for the following claim: Comparative Claim₁: There are cases where one’s higher-order evidence supports the higherorder proposition that one’s first-order evidence supports believing p more strongly than it supports the first-order proposition p.

I will then argue from that claim to another, subtly different Comparative Claim, which is the one that is needed to subsequently argue to the possibility of misleading total evidence about total evidence.

.. Arguing for Comparative Claim₁ Let us work with a more concrete case. I’ve used this example before, in Worsnip ().¹⁰ In this case, Miss Marple, an expert detective, and her niece Mabel, a novice, visit a murder scene, where they survey the (first-order) evidence—the clues. (Suppose they gather exactly the same first-order evidence as each other, and they both know this.) This (first-order) evidence does not support any positive belief about who committed the crime. However, Miss Marple—who is generally an expert about what the evidence supports—makes an uncharacteristic mistake, and declares that the (first-order) evidence supports believing that the vicar did it. In fact, the first-order evidence supports suspending judgment about whether the vicar did it; in fact, let us suppose,¹¹ it doesn’t even on balance support the proposition that the vicar did it. ¹⁰ In turn, it’s based on other similar cases in Coates () and Horowitz (). ¹¹ And this is a further supposition; see section . for a reminder of the terminology.

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       ?



Clearly, this case is one where Mabel has misleading higher-order evidence about what the first-order evidence supports. However, as the case was just stated, there seem to be few principled grounds on which to arbitrate whether the case is either a doubly-dispositive case, a non-dispositive case, or a singly-dispositive case—and it is only if it is a singly-dispositive case that it is a case of misleading total evidence (about total evidence). But remember that to begin with, I only want to argue that, at least given a few eminently possible stipulations, the Miss Marple case establishes the Comparative Claim—that Miss Marple’s testimony supports the relevant higherorder proposition more strongly than it supports the relevant first-order proposition.¹² Let F be the (first-order) proposition that the vicar did it, H be the (higher-order) proposition that the first-order evidence supports believing that the vicar did it, and T be the proposition that Miss Marple testifies that H. So, what we are investigating is whether there is a version of the Miss Marple case such that PrðHjTÞ > PrðFjTÞ. The value on the left-hand side—the probability that the evidence supports believing the vicar did it, conditional on Miss Marple’s testifying that the evidence supports believing the vicar did it—is essentially a matter of the reliability of Miss Marple’s testimony (assuming that Mabel knows how reliable Miss Marple is). So, for example, if Mabel knows that Miss Marple is % reliable with respect to relevantly similar matters—then PrðHjTÞ ¼ 0:9.¹³ This is so because Miss Marple’s testimony is directly about whether H. The value on the right-hand side is a little trickier to obtain, since Miss Marple doesn’t directly testify about whether F, but only about whether H. But it can be done. Given that Miss Marple testifies that the evidence supports believing that the vicar did it—that she testifies that H—there are exactly two possibilities under which the vicar did do it. () Miss Marple is right that the evidence supports believing that the vicar did it, and he did in fact do it. () Miss Marple is wrong that the evidence supports believing that the vicar did it, but as a matter of fact the vicar nevertheless did do it. Let’s assume for now, to simplify, that if it is already known whether H, T (i.e., Miss Marple’s testifying that H) does not change the probability of F—in other words, T and F are probabilistically independent conditional on whether H. (I’ll return to this assumption in a moment.) Given this assumption, the probability of the first possibility, given Miss Marple’s testimony, is in effect the probability that the evidence does support believing that the vicar did it, given that Miss Marple testifies that it does, multiplied by the probability that the vicar did it, given that the evidence supports believing that he did. More formally, it is: PrðHjTÞ:PrðFjHÞ. The probability of the second possibility, given Miss Marple’s testimony, is in effect the probability that the evidence does not support believing that the vicar did it, given that Miss Marple testified that it does, multiplied by the probability that the ¹² Thanks to the editors of this volume for helpful suggestions about how to present the formal details of the foregoing argument. ¹³ Of course, there may be other evidence bearing on H, as I’ll discuss in section ..; here we are just considering the probability of H conditional on T alone.

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vicar did it, given that the evidence does not support believing that he did. More formally, it is: Prð HjTÞ:PrðFj  HÞ. By adding the probabilities of each of the two possibilities together, then, we can get the probability that the vicar did it, conditional on Miss Marple’s testifying that the evidence supports believing that the vicar did it. In other words: PrðFjTÞ ¼ PrðHjTÞ:PrðFjHÞ þ Prð HjTÞ:PrðFj  HÞ Given this, we can now find out whether Pr(H|T) > Pr(F|T), for any case, as long as we have the values of the following three parameters: (A) PrðHjTÞ¹⁴ (B) PrðFjHÞ (C) PrðFj  HÞ And indeed, there are many values of (A), (B) and (C) such that PrðHjTÞ > PrðFjTÞ.¹⁵ For example, suppose PrðHjTÞ ¼ 0:9, PrðFjHÞ ¼ 0:9, and PrðFj  HÞ ¼ 0:5. Less formally: given that Miss Marple says the evidence supports believing the vicar did it, there’s a 0.9 probability that the evidence does support believing he did it; given that the evidence supports believing he did it, there’s a 0.9 probability that he did do it; and given that the evidence doesn’t support believing he did it, there’s a 0.5 probability that he did do it. There’s no reason to think that these values of (A), (B) and (C) aren’t co-possible. And given these values: PrðHjTÞ ¼ 0:9 PrðFjTÞ ¼ ð0:9Þ:ð0:9Þ þ ð0:1Þ:ð0:5Þ ¼ 0:86 Thus, in this version of the case, PrðHjTÞ > PrðFjTÞ. Note also that we can set possible (and, indeed, plausible) values for A, B, and C that create a much bigger gulf between PrðHjTÞ and PrðFjTÞ. In general, as parameter A [i.e. PrðHjTÞ] goes up, parameter B [i.e. PrðFjHÞ] goes down, and parameter C [i.e. PrðFj  HÞ] goes down, the gulf will get bigger. For example, suppose that PrðHjTÞ ¼ 0:95, PrðFjHÞ ¼ 0:8, and PrðFj  HÞ ¼ 0:2.¹⁶ Then: PrðHjTÞ ¼ 0:95 PrðFjTÞ ¼ ð0:95Þ:ð0:8Þ þ ð0:05Þ:ð0:2Þ ¼ 0:77 ¹⁴ Since Prð HjTÞ ¼ 1  ðPrðHjTÞÞ, we don’t need to separately know the value of the former, even though it occurs in the above formula for PrðFjTÞ. ¹⁵ Not every possible set of values for X, Y, and Z will yield the desired result, but again, we only need one case where they do yield this result to secure the possibility of misleading total evidence (about total evidence). ¹⁶ How could the probability that the vicar did it, conditional on the evidence not supporting believing that he did it, be anything other than .? Easily. First, the logical space may be partitioned such that the unconditional probability that the vicar did it is low (e.g., there may be lots of different suspects). Secondly, conditionalizing on the evidence not supporting believing that the vicar did it rules out the worlds in which the evidence supports believing that the vicar did it, leaving both the worlds in which the evidence support suspending judgment on whether the vicar did it, and the worlds in which the evidence supports believing that the vicar did not do it. So in general, one would expect the probability that the vicar did it, conditional on the evidence not supporting believing that he did it, to be lower than its unconditional probability.

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       ?



It follows that Comparative Claim₁ is true: we have a case where one’s higher-order evidence supports the higher-order proposition that one’s first-order evidence supports believing p more strongly than it supports the first-order proposition p. Let me now return to the simplifying assumption that I made, namely that T and F are probabilistically independent conditional on whether H. In general, this seems a plausible assumption. If it’s already known for certain that H (i.e., that the first-order evidence supports believing T), then it seems that T—Miss Marple’s saying that H— doesn’t add further evidential support for T. However, it might be challenged. Consider a case where it’s known that ~H—that the first-order evidence does not support believing that the vicar is guilty. ~H does not, in and of itself, say that the first-order evidence does not, on balance, support the proposition that the vicar is guilty to some degree; it merely says that the first-order evidence does not support this proposition to a sufficient degree to warrant believing that the vicar is guilty. Given this, it might be contended that Miss Marple’s testimony that H is still some evidence that F, even if we already know that ~H. After all, perhaps Miss Marple is more likely to have made a slight error—declaring that the first-order evidence sufficiently supports believing that the vicar is guilty when it does, on balance, provide some evidence for that proposition, but not enough—than to have made a more serious error—declaring that the evidence sufficiently supports believing that the vicar is guilty when it doesn’t even support that proposition on balance to any degree. If that is so, even knowing that Miss Marple’s testimony is false, it still provides some support for the proposition that the vicar is guilty—that is, for F.¹⁷ That is, T and F are not independent conditional on ~H. This is a good point, but we could make further stipulations about the case so as to neutralize it. Suppose Miss Marple is always in one of two states. In her sober state, she’s infallible with respect to what the evidence supports. But when she’s in the (unusual) state of having had three or more glasses of sherry, she’s no better than chance with respect to what the evidence supports. (Unfortunately, let’s add, Miss Marple gives no outward signs when she’s in the latter state.) Given those stipulations, the cases in which Miss Marple is wrong that the clues support believing that the vicar did it are guaranteed to be ones in which she’s drunk, and thus no better than chance. So, in this version of the case, it is no longer true that, conditional on ~H, T is evidence for F. And so the simplifying assumption is true at least in this case, which is enough to show that Comparative Claim₁ is true. Moreover, even when we relax the simplifying assumption, it’s still plausible that we’ll get some cases where PrðHjTÞ > PrðFjTÞ. In cases where T raises the probability of F even conditional on ~H, we need in our calculations to replace parameter (C)— PrðFj  HÞ—with what we could call parameter (C*)—PrðFj  H&TÞ. Recall that in my initial demonstration, I gave two examples where PrðHjTÞ > PrðFjTÞ. In the first, I supposed that C = 0.5; in the second, I supposed that C = 0.2; even the former supposition was enough to get the desired result. (See fn. 15 on why the value of C can fall well below 0.5.) In examples like the second, where the value of C is very low, the value of C* will plausibly still be quite low, lower (for example) than the

¹⁷ Thanks to an anonymous referee for making this point.

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 

value of C in the first example. And that, again, will be enough to deliver the result that Pr(H|T) > Pr(F|T).

.. From Comparative Claim₁ to Comparative Claim₂ So far, we have Comparative Claim₁, which, to repeat, is: Comparative Claim₁: There are cases where one’s higher-order evidence supports the higher-order proposition that one’s first-order evidence supports believing p more strongly than it supports the first-order proposition p. Contrast this with: Comparative Claim₂: There are cases where one’s higher-order evidence supports the higher-order proposition that one’s total evidence supports believing p more strongly than it supports the first-order proposition p. The argument directly establishes Comparative Claim₁, rather than Comparative Claim₂, because as we defined it, H is the (higher-order) proposition that the firstorder evidence supports believing that the vicar did it. Indeed, this is not accidental, for this is what Miss Marple testifies about in our case: she reports on what the clues support, not (directly) about what the totality of the evidence, including her own testimony, supports. However, for the argument for singly-dispositive cases (and thus, for the possibility of misleading total evidence about total evidence), we want Comparative Claim₂. After all, the endgame is to show that one’s total evidence can support believing falsehoods about itself, which requires us to focus on the higher-order belief that one’s total evidence supports believing p. Fortunately, for any normal case, it is a short step from Comparative Claim₁ to Comparative Claim₂. Recall that H is the proposition that the first-order evidence supports believing that the vicar did it. Let H* be the proposition that the total evidence supports believing that the vicar did it. Now, fortunately, in any normal case, Miss Marple’s testimony will be at least as strong evidence for H* as it is for H. Why is that so? Consider what Mabel’s total evidence consists in: the first-order evidence, plus Miss Marple’s testimony itself. Now, very plausibly, Miss Marple’s testimony (that the first-order evidence supports believing that the vicar did it) is some evidence that the vicar did it—and Mabel knows this. And at the very least, Miss Marple’s testimony is certainly not evidence that the vicar didn’t do it. So the difference between the first-order and total evidence here is just that the latter also includes a piece of evidence that very plausibly speaks in favor of believing, and manifestly doesn’t speak against believing, that the vicar did it. Given this, it seems that to whatever extent Miss Marple’s testimony supports H—the proposition that the first-order evidence supports believing that the vicar did it—it will also support H*—the proposition that the total evidence supports believing that the vicar did it—to at least the same extent. In order for it not to do so, Miss Marple’s testimony—that the first-order evidence supports believing that the vicar did it—would have to indicate of itself that it speaks against believing that the vicar did it (while all the while still being evidence that the first-order evidence

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       ?



supports believing that the vicar did it). It’s hard to see how this could possibly be so. And even if it were possible,¹⁸ it is evidently not the norm. So, it seems safe to assume that in at least the vast majority of cases, Miss Marple’s testimony is at least as strong evidence for H* as it is for H. But if Miss Marple’s testimony supports H more strongly than it supports F (as established in arguing for Comparative Claim₁), and it supports H* at least as strongly as it supports H (as I’ve just argued), then it supports H* more strongly than it supports F. That is, Comparative Claim₂ follows after all. Thus, I conclude that Comparative Claim₂ is true. There can be cases where misleading higher-order evidence supports a higher-order belief about one’s total evidence more strongly than it supports the corresponding first-order belief.

. From the Comparative Claim to the possibility of misleading total evidence But how do we get from Comparative Claim₂ to the possibility of misleading total evidence? The general shape of the strategy is as follows. Comparative Claim₂ asserts that there are cases where misleading higher-order evidence more strongly supports a higher-order proposition of the form my total evidence supports p than it supports the first-order proposition p. Now, not all of these cases will be singly-dispositive cases. It might be that even though the misleading higher-order evidence supports the higher-order proposition more strongly than the first-order proposition, it is still strong enough to be dispositive with respect to believing both propositions, or weak enough to be dispositive with respect to believing neither. However, the thought is this: there should be cases at the margins, where given that the misleading higher-order evidence supports the higher-order proposition more strongly than the first-order proposition, it is strong enough to be dispositive with respect to the higher-order belief, but not with respect to the first-order belief. Such cases are singly-dispositive, and as I already demonstrated in section ., any singly-dispositive case is a case of misleading total evidence (about total evidence). As stated, the above argument is suggestive but somewhat vague. One tempting way to precisify it might be as follows. Assume, as I’ve already suggested, that there is a threshold of degreed evidential support for a proposition above which belief in that proposition is evidentially supported in an on-off sense.¹⁹ For any reasonable threshold, we will be able to find values of parameters (A), (B), and (C) such that Pr(H|T)— and therefore, by the argument of §.(b), Pr(H*|T)—exceeds the threshold, but Pr(F|T) doesn’t. For example, if the threshold is somewhere between . and ., ¹⁸ Note, by the way, how odd it would be for someone who is trying to limit the capacity of evidence to mislead about itself to affirm this possibility. ¹⁹ Again, we need not assume the threshold is the same for all propositions and situations, though the argument as stated does assume that for any one situation, the threshold for H* and that for F is the same. This might be questioned. But even if the threshold for the higher-order belief is higher than that for the first-order belief, I’ve already shown that the gulf in how strongly Miss Marple’s testimony supports the two propositions can be quite large, and so it still seems likely that Pr(H*|T) could exceed the threshold while Pr(H|T) doesn’t.

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then the original set of values for the parameters are such that Pr(H*|T) exceeds the threshold but Pr(F|T) doesn’t. If the threshold is anywhere between . and ., then the second set of values for the parameters are such that Pr(H*|T) exceeds the threshold but Pr(F|T) doesn’t. However, it is too quick to conclude from this that the case is one where belief in H* is supported by the total evidence but belief in F isn’t. For this is a function of the evidential support provided for (or against) H* and F not only by Miss Marple’s testimony (T), but also by the remainder of the evidence: the first-order clues (call the totality of these first-order clues C). Again, it is tempting to argue as follows. By stipulation of the case, C does not support F. Therefore, C can’t possibly make up for the shortfall in support that T provides to F compared to that it provides for H*. Thus, T must in principle be capable of being dispositive with respect to believing H* without being dispositive with respect to believing F. Again, though, the argument is too quick. It overlooks the following inconvenient point: while C merely fails to provide evidence for F, arguably C actively provides evidence against H*. Remember what F and H* say. F is the proposition that the vicar did it. By stipulation of the case, C (the totality of the first-order clues), on balance, supports suspending judgment about F. H*, though, is the proposition that the total evidence supports believing F. Suppose that the clues enjoy at least some degree of rational self-evidence: that is, they support, at least to some degree, whatever proposition is true about the evidential support that they themselves provide (or do not provide). In that case, C, to at least some degree, supports the proposition that the first-order evidence does not support believing F—which is just the negation of H*. That is, C provides at least some evidence against H*. Given this, the opponent of misleading total evidence (about total evidence) might say that, to whatever extent that T supports H* more strongly than it supports F, this is cancelled out by the fact that C actively provides evidence against H*, while merely failing to provide evidence for F. Thus, it can never be the case that one’s total evidence supports H* more strongly than it supports F. So, even at the margins, it will never be the case that one’s total evidence on balance supports believing H*, without on balance supporting believing F. This is an important objection, and I do not have a knock-down response to it. Let me begin with a rather defensive point to clarify the dialectic here. It is still very much a conjecture on the objector’s part that the evidence that C provides against H* is always enough to make up for the difference in evidential support that T provides for H* as opposed to F. It’s unclear what justifies this conjecture. Remember that the defender of the possibility of misleading total evidence only needs there to be some cases in which the total evidence supports believing H* but does not support believing F. Given that the gulf between the support that T provides to H* and the support that it provides to F may in some cases be quite big, and given that (as I’ll also shortly argue) the evidence that C provides against H* may be quite weak, arguably the natural default position is still to hold that there is nothing to rule out the possibility of misleading total evidence about total evidence. Because of the vagaries of exactly how strongly C tells against H*, the argument here is going to fall short of an incorrigible proof. Still, I think it is a real step forward, in making a case for misleading total evidence, to have shown that T is stronger evidence for H* than it is

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       ?



for F. If T was always just as strong evidence for F as it is for H*, then—especially in light of the fact that C provides some evidence against H*—it really would be hard to see how T could ever result in a situation of misleading total evidence about total evidence. As things are, though, the opponent of misleading total evidence is now on the back foot: to defend her view, she is forced into a priori conjectures, which must hold as strict universal generalizations, about the relative strengths of different evidential relations, and it’s not clear what the argument for such conjectures would be. So, now, why is it that the evidence that C provides against H* may be relatively weak? I will discuss three reasons. The first is that, as clarified in the last section, H* is a proposition about what the total evidence supports believing. So, even if C provides even fairly weighty evidence about its own epistemic weightiness (or, in this case, lack of ), it does not follow that it is particularly weighty evidence about what the total evidence—including Miss Marple’s testimony—supports. Of course, by indicating that it itself doesn’t support the proposition that the vicar did it, it provides some evidence about what the total evidence supports, since it is part of the total evidence. But this need not be weighty. And this is so especially in light of the fact that it only indicates of itself that it doesn’t support H*, not that it is strong evidence against it. For virtually any proposition p, presumably there is some part of your evidence that isn’t evidence for p; learning this will often be very insignificant as evidence that your total evidence doesn’t support p. A second feature of the case that makes C’s evidential force against H* weak for Mabel is the fact that Mabel is a novice detective.²⁰ Though Mabel is directly acquainted with the clues, and though the clues have a particular property, namely that of supporting suspending judgment about whether the vicar is guilty, this doesn’t entail that Mabel is directly acquainted with the fact that the clues (have the property of ) support(ing) suspending judgment about whether the vicar is guilty. (Compare: you can be directly acquainted with a piece of music, and that music can have certain properties—e.g., the duration of the piece, the exact intervals between notes, etc.—without your being directly acquainted with the fact that the piece of music has those properties.) In order for the clues themselves to provide Mabel with evidence about their own properties (e.g., that they support suspending judgment about whether the vicar is guilty), she has to have some capacity to register the properties of the clues. (Again, just as, in order for the music to provide one with evidence about its own properties, one has to have some capacity to register to properties of the music.) Now, we can grant that Mabel does have this capacity to some degree. But it is also true that the strength of the evidence that the clues provide with respect to their own properties is sensitive to the degree of Mabel’s capacity to register these properties; to how good she is at doing so. (Once more, the better you are at registering the properties of a piece of music, the weightier the evidence that the music provides with respect to its own properties.) And since Mabel is a novice detective, her capacity to register the properties of the clues is not especially refined. That weakens

²⁰ See Kappel (forthcoming, esp. §.) for some points related to this second reply.

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the evidence that the clues supply to Mabel with respect to their own properties. Because the clues only provide Mabel with evidence about their own properties via Mabel’s capacity to recognize these properties, shortfalls in this capacity attenuate the strength of the evidence they provide.²¹ This is why, in coming to a view about H*, some other expert (say, Sherlock Holmes) will be justified in giving more weight to the clues than a novice (like Mabel) will be—as compared to the weight given to Miss Marple’s (countervailing) testimony. This point can be sharpened by attending to the distinction between the balance and the weight of evidence (see Joyce ). One’s evidence can be balanced quite strongly against p, in the sense that what evidence one does have all points against p, while still not being very weighty evidence against p, in the sense that it wouldn’t take much countervailing evidence to swing one’s total balance of evidence in the opposite direction. If we think in terms of credences, then the balance of the evidence determines what one’s credence in p should be, while the weight of the evidence determines how resilient that credence should be in response to future evidence. Importantly, there can be differences in the weight of two different evidential situations that nevertheless are balanced in the same way. (Compare a situation where there is no evidence concerning p whatsoever with one where there is a large amount of evidence for p and an equally large amount against it. In both situations, the evidence is precisely evenly balanced, such that one’s credence should be .. But in the latter, the evidence has much greater weight than in the former— such that one’s credence should be much more resilient, that is, it ought to take a lot more future evidence to significantly shift one’s credence away from ..) Now, compare the evidential situation of Mabel and Sherlock. For both of them, C is some evidence against H*. Moreover, C may in one sense be equally strong evidence against H* for the two of them, if by that we mean that the balance of the evidence, when C is the only evidence, is the same for the two of them. But if what I’ve suggested is right, C is less weighty for Mabel than it is for Sherlock. That is, even if C should (on its own) make Mabel’s credence in H* low, this low credence should nevertheless not be very resilient, relative to other potential evidence, such as Miss Marple’s testimony. By contrast, Sherlock’s credence should be at least somewhat more resilient in the face of Miss Marple’s testimony (though not completely so). If we focus only on the balance of the evidence provided by C, and not its weight, we miss this difference between Mabel and Sherlock, and so we miss a crucial sense in which the evidential force of C against H* for Mabel is weak.

²¹ Note that this isn’t to make the (somewhat) more controversial claim that the evidence provides support for first-order propositions only to the extent that one has the capacity to recognize that it provides support for those first-order propositions. The (generalized version of the) claim I’m making here is just that the evidence provides support for higher-order beliefs about which first-order attitudes it itself provides support for only to the extent that one has the capacity to recognize that it provides support for those first-order attitudes. The latter claim ought to be uncontroversial, even among those who dispute the former claim. Even if my being acquainted with the clues, plus the clues supporting a first-order proposition, suffices for my possessing the clues as evidence for that first-order proposition (regardless of my capacity to recognize them as such), this doesn’t change the fact that if I can’t recognize them as evidence for the first-order proposition, I won’t possess evidence for the higher-order proposition that the clues support the first-order proposition (or belief in it).

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       ?



Thirdly, and finally, to the extent that C does provide evidence against H*, it is possible that at least some of this evidential support is defeated by Miss Marple’s testimony (T). Following Pollock (), it’s standard to distinguish “rebutting” and “undercutting” defeaters. On one way of drawing that distinction, a “rebutting” defeater is one that defeats one’s justification for believing p by offering positive evidence that not-p, while an “undercutting” defeater is one that defeats one’s justification for believing p by in some way casting doubt on one’s existing evidence for p. Importantly, undercutting defeat can take place even in cases where one’s existing evidence for p is in fact (or at least, would be, in the absence of the defeater) good evidence for p. That is, even misleading evidence that some piece of evidence E is not good evidence for p can undermine E’s (would-be) support of p: the phenomenon is not restricted to cases where E only apparently supports p, and one’s higher-order evidence correctly indicates that it doesn’t. Given these characterizations, it is plausible to claim that T is both a rebutting and an undercutting defeater for the evidence that C provides against H*. It is rebutting because T is itself evidence for H*; but it is also an undercutting defeater because it directly casts doubt on the claim that C is itself evidence against H*.²² So in addition to providing evidence for H*, T may at least partially disarm or disable the evidential force that C has against H*. If that is so, C’s force against H* will be even weaker than it would have been without the presence of T. It might be thought odd that I am appealing to defeat to help me in this way. In general, defeat-talk is used more often by those who resist the possibility of misleading total evidence.²³ The idea is that when higher-order evidence makes it rational to believe that some first-order attitude is irrational, it also defeats the justification (or evidential support) for the first-order attitude itself, thus avoiding the possibility of misleading total evidence. However, as I’ve suggested before (Worsnip , pp. –), this strategy is very hard to extend to cases—like the one that we are presently discussing—where the first-order attitude supported by the first-order evidence is one of suspending judgment. If the clues (taken on their own) supported believing that the vicar did it, or at least on balance supported the proposition that the vicar did it, perhaps misleading higher-order evidence (that the clues do not support believing that the vicar did it) could defeat the evidence that the clues would have otherwise provided that the vicar did it. But in the case where the clues (taken on ²² Not all cases of rebutting defeat are also cases of undercutting defeat. Take an ordinary case where you have two fairly weighty pieces of evidence, for and against a proposition. For example, suppose that you’re trying to figure out whether Jones will be at the party. On one hand, there will be dancing at the party, which is evidence that Jones will be there (since Jones loves dancing); on the other hand, there will be singing at the party, which is evidence that Jones won’t be there (since Jones hates singing). Though these two pieces of evidence cut against each other in the sense that they support opposite conclusions, and certainly either piece of evidence makes it rational to be less confident than it would be rational to be if one had only the other piece of evidence, still neither piece of evidence casts any doubt on the other piece of evidence’s status as evidence for the opposite conclusion. (At least, this is so if we don’t develop the case such that all of Jones’s enjoyment of dancing goes away in the presence of singing, or that all of Jones’s hatred of singing goes away in the presence of dancing.) ²³ See, e.g., Skipper (this volume); González de Prado (forthcoming). On the other side, Lasonen-Aarnio () accepts the possibility of misleading total evidence but resists (to some degree) the phenomenon of (higher-order) defeat.

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

 

their own) do not tell one way or the other as to whether the vicar did it, it’s unclear what there is for misleading higher-order evidence (that the clues do support believing that the vicar did it) to “defeat,” at least in the undercutting sense. It’s not like this misleading higher-order evidence can make the clues themselves actually support believing that the vicar did it. Thus, the (rather complex) dialectical situation is this. I imagined my opponent pointing out that while the clues (C) merely fail to support F, they actively provide evidence against H* (such that even if T supports H* more strongly than it supports F, the total evidence might still fail to support H* more strongly than it supports F). My (third) reply was that to the extent that C provides evidence against H*, this evidential support may be at least partially defeated by T. At the same time, C’s failure to support F isn’t the sort of thing that can be defeated (in the undercutting sense) by T. So the defeating effect that T has will tend to neutralize the asymmetry of support that C has with respect to F and H*: it will bring us back toward a situation where C does not tell for or against either F or H*. Drawing together the three replies I’ve given here, I conclude that C is fairly weak evidence against H*, and that whatever evidence it does provide is arguably undercut, at least in part, by T. This makes it hard to believe that C’s evidential force against H* will always make up for the shortfall in support that T provides for F compared with its greater support for H*. And so, I think we still have reason to think that, in principle, there can be situations where one’s total evidence supports H* without supporting F—that is, in situations of misleading total evidence about total evidence.

. Conclusion In closing, let me briefly say a bit about why this matters.²⁴ Most epistemologists assume that, at least in some sense, you ought to follow your evidence: that is, to believe what your total evidence supports, and refrain from believing those things that your total evidence does not support. If there can be cases of misleading total evidence about total evidence, then this entails that sometimes, it is both the case that you ought to believe that your evidence supports believing p, but yet also that you ought not to believe p. But if you have this combination of states, you are “epistemically akratic.” To put it another way, your doxastic states exhibit a kind of incoherence across levels. It is also widely thought that such akratic or incoherent combinations of attitudes are irrational; or, even more strongly, that it is not possible to sustain such states in full reflective awareness of them. So, if there can be misleading total evidence about total evidence, we must revise some piece of orthodoxy: either the claim that you always ought to follow your evidence, or that epistemic akrasia is always irrational, or that there cannot be dilemmatic or “tragic” situations where the normative injunction to follow your evidence and the normative injunction not to be akratic come into conflict. Thus, the possibility of misleading total evidence bears crucially on utterly foundational issues in the theory of epistemic rationality. ²⁴ Here I’m very briefly summarizing the puzzle laid out in Worsnip () and in Lasonen-Aarnio (forthcoming).

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       ?



Acknowledgements This chapter is greatly indebted to a conversation with Miriam Schoenfield, and an objection that she made to my (). Though it was initially introduced for the purposes of this objection, the basic idea behind the model in this chapter is due to her. I am also grateful to Nilanjan Das, David James Barnett, Ram Neta, and Jim Pryor for helpful discussion, and to an anonymous referee and the editors of this volume for extremely helpful comments.

References Bergmann, M. (). “Defeaters and Higher-Level Requirements.” In: The Philosophical Quarterly (), pp. –. Coates, A. (). “Rational Epistemic Akrasia.” In: American Philosophical Quarterly (), pp. –. Feldman, R. (). “Respecting the Evidence.” In: Philosophical Perspectives (), pp. –. Fitelson, B. (). “Evidence of Evidence is not (Necessarily) Evidence.” In: Analysis (), pp. –. González de Prado, J. (forthcoming). “Dispossessing Defeat.” In: Philosophy and Phenomenological Research, Online Early View. DOI: ./phpr.. Horowitz, S. (). “Epistemic Akrasia.” In: Noûs (), pp. –. Joyce, J. M. (). “How Probabilities Reflect Evidence.” In: Philosophical Perspectives , pp. –. Kappel, K. (forthcoming). “Bottom Up Justification, Asymmetric Epistemic Push, and the Fragility of Higher Order Justification.” In: Episteme, Online Early View, DOI: ./ epi... Kelly, T. (). “Peer Disagreement and Higher Order Evidence.” In: R. Feldman and T. A. Warfield (eds), Disagreement, Oxford University Press. Lasonen-Aarnio, M. (). “Higher-Order Evidence and the Limits of Defeat.” In: Philosophy and Phenomenological Research (), pp. –. Lasonen-Aarnio, M. (forthcoming). “Enkrasia or Evidentialism?” In: Philosophical Studies. Pollock, J. (). Contemporary Theories of Knowledge, Rowman & Littlefield. Pryor, J. (). “Problems for Credulism.” In: C. Tucker (ed.), Seemings and Justification, Oxford University Press. Skipper, M. (this volume). “Higher-Order Defeat and the Impossibility of Self-Misleading Evidence.” In: M. Skipper and A. Steglich-Petersen (eds), Higher-Order Evidence: New Essays, Oxford University Press. Titelbaum, M. (). “Rationality’s Fixed Point (or: In Defense of Right Reason).” In: Oxford Studies in Epistemology , pp. –. Worsnip, A. (). “The Conflict of Evidence and Coherence.” In: Philosophy and Phenomenological Research (), pp. –.

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Index of Names Adler, J.  Alvarez, M.  Anderson, C. ,  Arpaly, N.  Arsenault, M.  Bader, R.  Barnett, D. ,  Bergmann, M. , , ,  Berker, S.  Bertsekas, D.  Blackburn, P.  Bonjour, L.  Briggs, R.  Bronfman, A.  Brössel, P. , , , –, ,  Buchak, L.  Carr, J.  Chisholm, R.  Christensen, D. –, , , –, , –, , –, , , , , , –, , , –, –, , , , , –, , , , –, , , , –, –, –, –, –, , , , ,  Coates, A. , , , –, , , ,  Comesaña, J. , –, –, , ,  Conee, E. , ,  Cresswell, M.  Das, N. ,  Dennett, D.  DiPaolo, J. , , –, – Dancy, J.  Daoust, M-K.  D’Arms, J.  Dorst, K. , , , , , , , , , –, , , ,  Dutant, J. ,  Easwaran, K. , , ,  Eder, A. M. , , , , –, , , ,  Egan, A. , , – Elga, A. , , , –, , –, , , –, , , , , , , –, , , , –, , ,  Feldman, R. , , –, , , , , , , , , , , –, , , , , , –, , , 

Field, C. ,  Field, H.  Fisher, J. ,  Fitelson, B. , , –, , , , , –, ,  Foley, R.  Frankish, K.  Friedman, J.  Garson, J.  Geanakoplos, J.  Gendler, T.  Gettier, E. ,  Gibbons, J. , ,  Goldman, A. – Gracely, E.  Greco, D. –, , , , –, , , , ,  Hansson, S. O.  Hare, R.  Harman, E. ,  Hawthorne, J. , , ,  Hazlett, A. ,  Hieronymi, P.  Hintikka, J. , ,  Horowitz, S. , , , , –, , , , , , , , , –, –, , –, –, , , , , , , , , , ,  Huemer, M. , ,  Hughes, G.  Hume, D.  Huttegger, S.  Irving, Z. C.  James, W.  Joyce, J. , ,  Kappel, K. , , , ,  Kelly, T. , , , , , , , , , , , , –,  Keynes, J. M. ,  Klein, P.  Kolodny, N.  Konek, J.  Koons, R.  Kopec, M. ,  Kornblith, H.  Korsgaard, C.  Kvanvig, J. 

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

  

Lackey, J. , ,  Lange, M.  Lasonen-Aarnio, M. –, –, , , , , , , , , –, –, , –, –, , , , , , , –, , ,  Leitgeb, H. – Levi, I.  Levinstein, B. ,  Lewis, D. , , ,  Littlejohn, C. , , , , , , , , –, , , , , – Lockhart, T.  Loeb, L.  Lord, E. , ,  Lyons, J.  MacAskill, W.  Magidor, O. ,  Mantel, S. ,  Marley-Payne, J.  Matheson, J. ,  McGrath, M.  Moon, A. – Moretti, L. –,  Mueller, A.  Nelson, M.  Neta, R. , , , , , ,  Olson, J.  Parfit, D. , –,  Pettigrew, R.  Pittard, J.  Pollock, J. , ,  Popper, K. ,  Pryor, J. , , , , , , ,  Quine, W.  Rabinowicz, W.  Rayo, A. –, ,  Raz, J. , –, ,  de Rijke, M.  Reisner, A. ,  Rinard, S.  Robinson, P.  Roche, W. –, ,  Rosenkranz, S.  Ross, J. ,  Rnnow-Rasmussen, T.  Salas, J. ,  Salmon, W.  Salow, B. , , , , , , –, –, , , , , , , 

Samet, D. , , ,  Santorio, P.  Saul, J. ,  Schechter, J. –, , , , , ,  Schoenfield, M. , –, , , , , –, , , , , –, –, ,  Schroeder, M. , , , – Schultheis, G.  Schulz, M.  Schwitzgebel, E.  Sepielli, A. ,  Shah, N. , , ,  Sharadin, N.  Sher, G.  Shogenji, T. ,  Silva, P. ,  Skipper, M. –, , , –, , , , , , , ,  Skyrms, B.  Sliwa, P. , –, , , , , –, , , , , , , ,  Smithies, D. , , , , , , , , –, , , –, – Sorensen, R.  Sosa, E. ,  Srinivasan, A.  Stalnaker, R. , , , , , , , , , ,  Steglich-Petersen, A. –, –, , , –, , ,  Streumer, B.  Sylvan, K. ,  Tal, E. , –, ,  Titelbaum, M. –, , , , , , , , , , , , , , , , , , , , , , , , , , , ,  Torpman, T.  Tsitsiklis, J.  Tucker, C.  Van Ditmarsch, H.  Van Fraassen, B. – Van Wietmarschen, H. , ,  Vavova, K. , , , ,  Venema, Y.  Vogel, J. ,  Way, J. , ,  Weatherson, B. , , , , , , , ,  Weisberg, J. ,  Wedgwood, R. , ,  White, R. , , , , , , , 

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

   Whiting, D. , , , –, , – Williams, B. ,  Williams, M.  Williamson, T. , , , , –, –, –, , –, , , , –, , –, , , , , , , , , 



Worsnip, A. –, , , , , –, , , , , , , , , –, , –, , , , –, –, , , –, – Wright, C. ,  Zhao, J. 

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

Index of Subjects accuracy –, , , , , , –, , ,  akrasia –, , –, , , , , –, , , –, , , –, , –, , ,  see also akratic, enkrasia, enkratic akratic –, , , , , , –, –, , –, –, ,  see also akrasia, enkrasia, enkratic trilemma , – belief –, – principle , –,  state , , , ,  attenuator – Bayesian , , –, –,  agent ,  confirmation –,  Bayesianism , , –, , , ,  bias , –, , , , –, –, –,  bootstrapping , ,  bracketing , , , , ,  bridge principle ,  calibration , –,  coherence , , , , , , –, , ,  conciliatory –, , , , ,  see also conciliationist, conciliationism, steadfast conciliationist , , , ,  see also conciliatory, conciliationism, steadfast conciliationism , ,  see also conciliatory, conciliationist, steadfast conditional belief , , , , ,  credence , , , , – doxastic attitude – probability –, –, ,  conditionalization , , , , , , , –, –,  disagreement , , –, , , , , , –, , , –, , –, , , –, , , –,  see also peer disagreement dogmatic ,  see also dogmatism, modest, immodest, self-doubt dogmatism , ,  see also dogmatic, modest, immodest, self-doubt

doxastic resilience , , ,  see also resilience justification , , , , , , – defeater  disposition – Dyadic Bayesianism , , , –, ,  easy knowledge , , – EEE Slogan , – see also evidence of evidence enkrasia , , ,  see also akrasia, enkratic enkratic  see also akrasia, enkrasia principle , , –, – intuition , , , – requirement , –, – epistemic access , –, –,  see also positive introspection, negative introspection akrasia see also akrasia, akratic, enkrasia, enkratic dilemma , , , –, –, , – logic , –, , , –, ,  plan – norm , , , , , , – rationality , , , , , , , –, –, –, ,  peer , , ,  see also peer disagreement reason ,  rule , , , – justification  evidence of evidence –, –, –, –, , , –, – see also EEE Slogan Evidential bearing , , , ,  weight –,  evidentialism , , –, –, –, –,  see also evidentialist evidentialist –, , , , , , , –, –,  see also evidentialism evincibility –, –, – expected accuracy ,  see also accuracy utility , –,  value ,  reliability –, , –

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi

   externalism  see also externalist, internalist, internalism externalist , , , , , ,  see also externalism, internalist, internalism first-order bearing – belief , , –, , –, , –, ,  opinion , –, ,  proposition , , , , , , – consideration , ,  reason , , , –, ,  credence , –, , –, , , ,  defeat ,  support , , , , ,  way , ,  uncertainty  Fixed Point Thesis , , – fragmentation , , , –, , – higher-order belief , , –, , , –, – defeat –, –, , –, , , , , , , –, ,  credence , , , ,  opinion , –, , , , –, ,  proposition , , , –, –, – justification  reason ,  support  way , ,  uncertainty , , –, –, ,  hypothetical conditional credence ,  immodest , , – see also modest, dogmatic, dogmatism, self-doubt independence principle –, –, ,  independent hypothetical credence , , –, ,  internalism , , , ,  see also internalist, externalist, externalism internalist , , –, , , , , ,  see also internalism, externalist, externalism intensifier  introspection , , , – see also positive introspection, negative introspection level-connection ,  see also bridge principle level-splitting –, –



luminous  see also luminosity luminosity , ,  see also luminous misleading evidence –, , , , , , , , , –, , –, , –, –, –, , ,  see also selfmisleading evidence, predictably misleading evidence higher-order evidence –, , , , –, , , , , –, , , , , –, , – total evidence – see also selfmisleading evidence modest , , –, , –, –, , – see also immodest, dogmatism, dogmatic, self-doubt modifier ,  motivational constraint ,  Moorean belief  proposition  negative introspection , , , – normative evincibility see evincibility significance –, –, – peer disagreement , , , , –, , , , , , , , , , , –, – see also disagreement permissive –,  see also permissivism permissivism  see also permissive predictably misleading evidence , ,  positive introspection , –,  propositional justification , , , , – defeater  question-begging –, , –, , , – rational reflection , – see also reflection principle enkrasia  see also akrasia, akratic, enkrasia, enkratic dilemma , , , –, , ,  see also epistemic dilemma rebutting , ,  reasons for belief , ,  normative ,  object-given – practical , , , ,  second-order – state-given –,  reflection principle , , –,  see also rational reflection

OUP CORRECTED PROOF – FINAL, 24/9/2019, SPi



  

reliability assessment , –, ,  estimate ,  evidence –, , , ,  resilience –, –, , – see also doxastic resilience Right Reason , , –, –, – screening-off  self-doubt –, , ,  see also modest, immodest, dogmatic, dogmatism

self-misleading evidence –, , , –, – see also misleading total evidence stability theory of belief –,  steadfast , , , –,  transparency requirement  undercutting , –, , , , – Über-rule , , –, 