NEW ESSAYS ON THE KNOWABILIT Y PARADOX
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Joe Salerno

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NEW ESSAYS ON THE KNOWABILIT Y PARADOX

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New Essays on the Knowability Paradox Edited by

JOE SALERNO

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Great Clarendon Street, Oxford ox2 6dp 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 in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With ofﬁces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © the several contributors 2009

The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2009 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, 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 book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Laserwords Private Limited, Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 978–0–19–928549–5 1 3 5 7 9 10 8 6 4 2

—for Rebecca

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Contents List of Contributors Acknowledgements

x xii

Introduction

1

Joe Salerno PA RT I : E A R LY H I S TO RY 1.

Referee Reports on Fitch’s ‘‘A Deﬁnition of Value’’

13

2.

Alonzo Church A Logical Analysis of Some Value Concepts

21

3.

Frederic B. Fitch Knowability Noir: 1945–1963

29

Joe Salerno PA RT I I : D U M M E T T ’ S C O N S T RU C T I V I S M 4.

Fitch’s Paradox of Knowability

51

5.

Michael Dummett The Paradox of Knowability and the Mapping Objection

53

6.

Stig Alstrup Rasmussen Truth, Indeﬁnite Extensibility, and Fitch’s Paradox

76

Jos´e Luis Berm´udez PA RT I I I : PA R AC O N S I S T E N CY A N D PA R AC O M P L E T E N E SS 7.

Beyond the Limits of Knowledge

8.

Graham Priest Knowability and Possible Epistemic Oddities Jc Beall

93 105

viii

Contents PA RT I V: E PI S T E M I C A N D T E M P O R A L O PE R ATO R S : AC T I O N S , T I M E S A N D T Y PE S

9. Actions That Make Us Know

129

Johan van Benthem 10. Can Truth Out?

147

John Burgess 11. Logical Types in Some Arguments about Knowability and Belief

163

Bernard Linsky PA RT V: C A RT E S I A N R E S T R I C T E D T RU T H 12. Tennant’s Troubles

183

Timothy Williamson 13. Restriction Strategies for Knowability: Some Lessons in False Hope

205

Jonathan L. Kvanvig 14. Revamping the Restriction Strategy

223

Neil Tennant PA RT V I : M O D A L A N D M AT H E M AT I C A L F I C T I O N S 15. On Keeping Blue Swans and Unknowable Facts at Bay: A Case Study on Fitch’s Paradox

241

Berit Brogaard 16. Fitch’s Paradox and the Philosophy of Mathematics

252

Ot´avio Bueno PA RT V I I : K N OWA B I L I T Y R E C O N S I D E R E D 17. Performance and Paradox

283

Michael Hand 18. The Mystery of the Disappearing Diamond

302

C. S. Jenkins 19. Invincible Ignorance

320

W. D. Hart

Contents

ix

20. Two Deﬂationary Approaches to Fitch-Style Reasoning

324

Christoph Kelp and Duncan Pritchard 21. Not Every Truth Can Be Known (at least, not all at once)

339

Greg Restall Bibliography Index

355 367

List of Contributors Jc Beall is Professor of Philosophy at the University of Connecticut, and Arché Associate Fellow at the University of St Andrews. Johan van Benthem is University Professor of Logic at the University of Amsterdam and Professor of Philosophy at Stanford University. Jos´e Luis Berm´udez is Professor and Director of Philosophy, Neuroscience and Psychology at Washington University in Saint Louis. Berit Brogaard is Research Fellow at the RSSS Philosophy Program and Centre for Consciousness at the Australian National University, and Associate Professor of Philosophy at the University of Missouri-Saint Louis. Ot´avio Bueno is Professor of Philosophy at the University of Miami. John Burgess is Professor of Philosophy at Princeton University. Alonzo Church was Associate Professor of Mathematics (without tenure) at Princeton University when his contribution to this volume was written in 1945. He was Professor of Mathematics and Philosophy at Princeton, 1961–7, and then Professor of Philosophy and Mathematics at UCLA until he retired in 1990. Church died in 1995. Sir Michael Dummett is Emeritus Professor at the University of Oxford. He was knighted for service to philosophy and racial justice. Frederic B. Fitch was Sterling Professor Emeritus of Philosophy at Yale University when he died in 1987. Michael Hand is Professor of Philosophy at Texas A&M University. W. D. Hart is Professor of Philosophy at the University of Illinois at Chicago. C. S. Jenkins is Arché Associate Fellow at the University of St Andrews, Associate Fellow of the Centre for Metaphysics and Mind at the University of Leeds, and Lecturer at the University of Nottingham. Christoph Kelp has recently completed his doctoral dissertation at the University of Stirling. Jonathan L. Kvanvig is Distinguished Professor of Philosophy at Baylor University. Bernard Linsky is Professor of Philosophy at the University of Alberta. Graham Priest is the Boyce Gibson Professor of Philosophy at the University of Melbourne, and Arché Professorial Fellow at the University of St Andrews. Duncan Pritchard is Professor of Philosophy at the University of Edinburgh.

List of Contributors

xi

Stig Alstrup Rasmussen has formerly held positions at the Universities of Edinburgh and Copenhagen. He obtained his habilitation at the University of Copenhagen in 2004. Greg Restall is Associate Professor of Philosophy at the University of Melbourne. Joe Salerno is Research Fellow at the Australian National University and Associate Professor of Philosophy at Saint Louis University. Neil Tennant is Humanities Distinguished Professor of Philosophy at the Ohio State University. Timothy Williamson is the Wykeham Professor of Logic at the University of Oxford.

Acknowledgements I am indebted to the contributors for their hard work and their patience with the editorial process. Each of them has furthered my understanding of the knowability proofs and other matters modal epistemic. I am extremely grateful to the graduate students that have served as my research assistants: Amy Broadway (Missouri), Heidi Lockwood (Yale), Julien Murzi (Shefﬁeld), Jonathan Nelson (Saint Louis) and Nick Zavidiuk (Saint Louis). They helped me enormously with various aspects of the project—among them, plundering archives, transcribing, compiling the bibliography, typesetting, and proof-reading for typographical and philosophical errors. Thanks also to my graduate students at Saint Louis University for stimulation, and to those grads at the Goethe University of Frankfurt who attended my seminar on the knowability paradox in May 2006. I thank the editor, Peter Momtchiloff, and supporting editors for their assistance and dedication to the project and to two anonymous Oxford University Press readers for their exceedingly helpful comments and suggestions. Many people have helped to improve the volume in one way or another, including Aldo Antonelli, Jc Beall, Scott Berman, Jim Bohman, Susan Brower-Toland, John Burgess, David Chalmers, Roy Cook, Judy Crane, Michael Della Rocca, Herbert Enderton, Saul Feferman, Bas van Fraassen, Anne-Sophie Gintzburger, John Greco, Nick Grifﬁn, Michael Hand, Monte Johnson, John Kearns, Jon Kvanvig, Bernard Linsky, Heidi Lockwood, Ruth Barcan Marcus, Robert Meyer, Gualtiero Piccinini, Graham Priest, Krister Segerberg, Wilfried Sieg, Roy Sorensen, Kent Staley, Jim Stone, Eleonore Stump, Neil Tennant, Achille Varzi and Ted Vitali. Berit Brogaard deserves special mention for being a constant source of feedback. My daughter, Rebecca, has been an endless source of sleepless nights but also inspiration, both of which were needed to complete this project. It is to her that I dedicate the volume. J.S.

Introduction Joe Salerno

T h e K n ow a b i l i t y Pa r a d o x In his seminal paper A Logical Analysis of Some Value Concepts (1963; reprinted, Chapter 2 of this volume), Frederic Fitch articulates an argument that threatens to collapse a number of modal epistemic distinctions. Most directly, it threatens to collapse the existence of fortuitous ignorance into the existence of necessary unknowability. For it shows that there is an unknown truth, only if there is a logically unknowable truth. Fitch called this ‘Theorem 5’, which usually is represented formally as follows: (Theorem 5 ) ∃p(p & ¬Kp) ∃p(p & ¬♦Kp) , where p holds a place for sentence letters; ♦ is normal possibility, read ‘it is possible that’; and K is the epistemic operator, ‘it is known (by someone [like us] by some means or other at some time) that’. The theorem rests on tremendously modest modal epistemic principles, which we will turn to shortly. The converse of Theorem 5 is modest as well. So Theorem 5 does the interesting work in erasing the logical difference between there being truths forever unknown and there being truths logically unknowable. The contrapositive of Theorem 5 is better known as the knowability paradox: (Knowability Paradox)

∀p(p → ♦Kp) ∀p(p → Kp).

If each truth is knowable in principle, then it follows logically that each truth is at some time known. That’s the result. It is thought to be paradoxical for a number of related reasons. First, it refutes all too easily interesting brands of anti-realism which are committed to the knowability principle, ∀p(p → ♦Kp). It refutes them since the knowability principle entails the obviously false omniscience principle, ∀p(p → Kp). The knowability principle has been claimed for a number of historic non-realisms, among them Michael Dummett’s semantic anti-realism, Hilary Putnam’s internal realism, the logical positivisms of the Berlin and Vienna Circles, Peirce’s pragmatism, Kant’s transcendental idealism, and Berkeley’s metaphysical

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idealism. How strange that the knowability principle, and every brand of nonrealism that avows it, are only as plausible as the exceedingly implausible, and obviously false, omniscience principle. An extension of Fitch’s result, found in Williamson (1992: 68), shows that a traditional strengthening of the knowability principle forecloses on the very distinction between what is possible and what is actual. Roughly, if truth is possible knowledge then possibility is actuality.¹ The paradoxicality is that sophisticated forms of anti-realism could be so easily refuted. A second reason to regard the proof as paradoxical is that it threatens to erase the logical distinction between the knowability principle and the omniscience principle. More speciﬁcally, the proof logically collapses the relatively moderate and plausible claim that each truth can be known into the apparently stronger and unbelievable claim that each truth is in fact known. The claims seem to carry distinct logical commitments, but they do not if Fitch’s result is valid. Fitch’s result presupposes the following principles. Knowing a conjunction requires knowing each of the conjuncts: (A) K (p & q) Kp & Kq Knowing entails truth: (B) Kp p Theorems are necessarily true: (C) If p, then p And, a necessarily false proposition is impossible: (D) ¬p ¬♦p The proof may be characterized this way:

At the top of the tree we suppose for reductio that the Fitch-conjunction, p & ¬Kp, is known. By (A), it follows that each conjunct is known. The third line demonstrates an application of factivity, (B), to the right conjunct of the second line. In the face of the ensuing contradiction, we discharge and deny our only assumption. By necessitation, (C), and then by (D), we conclude with the impossibility of our initial assumption—giving, ¬♦K (p & ¬Kp). Now suppose the knowability principle, ∀p(p → ♦Kp), and take the following instance: (p & ¬Kp) → ♦K (p & ¬Kp). This together with the above ¹ More carefully, Williamson shows this: if necessarily something is true if and only if it is knowable, then necessarily p is possible if and only if p.

Introduction

3

theorem, ¬♦K (p & ¬Kp), entails ¬(p & ¬Kp), which may be generalized to ∀p¬(p & ¬Kp). The classical equivalent is the omniscience principle, ∀p(p → Kp). At a glance:

In sum, if all truths are knowable, then all truths are known: ∀p(p → ♦Kp) ∀p(p → Kp). T h e Ge n e r a l i z e d Pa r a d o x Fitch generalized the knowability result, showing that any operator O that is both factive and closed under conjunction-elimination, generates the following aporia: ∀p(p → ♦Op)

.. .

∃p(p & ¬Op)

⊥ To prove this Fitch begins with Theorem 1, which holds of any factive operator O that is closed under conjunction-elimination: (Theorem 1) ¬♦O(p & ¬Op). Theorem 1 generates the above aporia.² Others have noted that it is not just factive, conjunction-distributive operators that validate Theorem 1 and generate the aporia. Belief, for instance, is closed under conjunction-elimination but is not factive. Yet arguably a belief-instance of Theorem 1 is provable, giving ¬♦B(p & ¬Bp).³

In this way the corresponding aporia is generated for the belief operator: ∀p(p → ♦Bp) ∃p(p & ¬Bp) .. . ⊥ ² To see how, substitute p & ¬Op for p in ∀p(p → ♦Op). By Theorem 1, it follows that ¬(p & ¬Op). This in turn may be may be generalized, giving ∀p¬(p & ¬Op), or equivalently ¬∃p(p & ¬Op). ³ See, for instance, Linsky (1986; and Ch. 11 of this volume).

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That is, the plausible notion that any truth could be believed is inconsistent with the truism that some truths are not ever believed. Such proofs about belief avoid unrestricted factivity principles in favor of restricted principles about the transparency of beliefs about one’s own beliefs. To take another example, a knowledge-version of the result may be derived without the conjunctiondistributivity principle (Williamson 1993). Most generally, then, a Fitch-aporia, or Fitch paradox, is generated for any operator O just when 1. The conjunction p & ¬Op is un-O-able: ∀p¬♦O(p & ¬Op); 2. The O-ability principle, ∀p(p → ♦Op), is plausible; and 3. Clearly, some truths are un-O-ed: ∃p(p & ¬Op). Operators that seem to generate Fitch-aporias include It is written truthfully on the board that Somebody brought it about that God brought it about that The laws of nature made it the case that It is believed that It is thought that So, for instance, the paradox of omnipotence may be seen, logically, as a special case of Fitch’s paradox. It says, roughly, that God can do anything that is in fact done, but only if God does in fact do everything. Another example: any truth can in principle be thought, but only if every truth is (at some time) thought. This latter result, like the result about belief, requires (in lieu of factivity) a principle that avows some minimal transparency of one’s thoughts about one’s thoughts. T h e Vo l u m e , C o n t r i b u t i o n s a n d L i t e r a t u re We here turn to some traditional and developing treatments of the paradox. The earliest version of the knowability proof appears in a 1945 referee report for the Journal of Symbolic Logic (printed here as Chapter 1). Its author, Alonzo Church, anonymously conveyed the proof to Fitch. The proof had the effect of undermining a certain deﬁnition of ‘value’ that Fitch was articulating—a deﬁnition that is trivialized if there are unknowable truths. So the proof originates in a context that is very different from the one in which we discuss the proof today. We think of the knowability paradox today either as an all-too-quick refutation of anti-realism or as a logical collapse of apparently distinct philosophical commitments. More on the more recent debate in a moment. Church offers a number of potentially promising ways to block the proof. He is most sympathetic to a rejection of closure principles for knowledge and belief, and a fortiori

Introduction

5

the principle that knowledge is closed under conjunction-elimination. This is principle (A) in our earlier presentation of the proof. So Church ultimately takes the knowability proof to be invalid—dare I say, paradoxical. However, Church’s proposal does not help Fitch, since Fitch is deeply committed to necessary logical connections between the relevant propositional attitudes. Church considers that one may alternatively appeal to Russell’s theory of logical types, which would have the effect of blocking special instances of the conjunctive distributivity principle—principle (A). The appeal to types foreshadows Linsky (Chapter 11 of this volume) and Hart (Chapter 19 of this volume). However, Church notes that the type-theoretic approach, like the rejection of closure principles, is antithetical to the goals of Fitch’s manuscript. Fitch had a very different kind of reply in mind. He responds to the referee report with a letter to the editor, in which he restricts the relevant class of true propositions to ones that it is ‘empirically possible’ to know. Fitch’s deﬁnition of value is thus resuscitated. His restriction strategy foreshadows Neil Tennant (1997), where we ﬁnd an analogous restriction to the class of true propositions that it is logically possible for somebody to know. We will discuss Tennant’s restriction in a moment. It should be noted here that Church was unimpressed with Fitch’s restriction strategy, and in a subsequent referee report (also in Chapter 1) attempted a version of the knowability proof that respects Fitch’s restriction. The debate between Fitch and Church is tracked in Salerno (Chapter 3). Church’s argument against Fitch’s restriction strategy, I argue, is critically ﬂawed. Part I of the volume is dedicated to this, the early history of the Church–Fitch paradox of knowability. Chapter 1 is the pair of referee reports from 1945. They record one side of a dialog between Fitch and the referee regarding the paper submitted by Fitch to JSL. Chapter 2 is Fitch’s seminal 1963 paper, shaped in no small part by those reports. Fitch’s paper has been the logical fuel or foil for the literature on the knowability paradox. Essay 3 is my understanding of the ﬁrst two essays. It offers an account of why Fitch included the knowability result in the 1963 paper. Part II is about Michael Dummett’s semantic anti-realism. The ﬁrst wave of reactions to Fitch’s 1963 paper, including Hart and McGinn (1976), Hart (1979), Mackie (1980), and Routley (1981), had a common theme. They all aimed to use Fitch’s proof to discredit various forms of veriﬁcationism, the view that all meaningful statements (and so all truths) are knowable.⁴ The knowability principle is commonly taken throughout the literature as a particularly clear expression of Hilary Putnam’s internal realism (1981) and Michael Dummett’s ⁴ An exception is Walton (1976), whose aim was to draw lessons in the philosophy of religion. For related discussion, see Plantinga (1982); Humberstone (1985); MacIntosh (1991); Kvanvig (1995, and 2006); Rea (2000); Wright (2000); Cogburn (2004); Bigelow (2005); Brogaard and Salerno (2005); and Moretti (2008).

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anti-realism (1959b; 1973, and elsewhere). The Fitch paper then threatens these forms of anti-realism. Since Williamson (1982) and Rasmussen and Ravinkilde (1982), however, we ﬁnd various proposals to vindicate at least Dummettian anti-realism. Fitch’s reasoning is classically, but not intuitionistically, valid. Speciﬁcally, the move from ¬(p & ¬Kp) to p → Kp (i.e., the ﬁnal step in our version of the proof ) is intuitionistically unacceptable, since it harbors an application of double-negation elimination—i.e., ¬¬p p. Leading developments in Dummettian anti-realism favor intuitionistic revisions to classical logic.⁵ As such, Dummettian anti-realism is said to evade the unwelcome classical consequences of Fitch. The proposal is further developed in Williamson (1988b; 1990; and 1992). An objection to the intuitionistic strategy is found in the view that the intuitionistic consequences of Fitch’s reasoning are as bad, or almost as bad, as the classical consequences. The objection is developed in Percival (1990).⁶ The main intuitionistic consequence is p → ¬¬Kp, which says that no truths are forever unknown. Some equivalent formulas include ¬(p & ¬Kp), which denies that there are unknown truths, and ¬Kp → ¬p, which says that anything forever unknown is false, and ¬(¬Kp & ¬K ¬p), which denies that there are any forever undecided statements. The potentially irksome consequence, which is a focus of Wright (1993a: 426–7) and Williamson (1994a), can be put this way. The intuitionistic anti-realist lacks the resources to express the apparent truism that there may be truths that never in fact will be known, formally ∃p(p & ¬Kp). That is because the inconsistency derivable from the joint acceptance of the knowability principle and ∃p(p & ¬Kp) is intuitionistically acceptable. In Chapter 4 Dummett embraces the intuitionistic consequences without regret. The paper defends p → ¬¬Kp as the best expression of semantic antirealism.⁷ In a letter to the editor of this volume, Dummett explains that the intuitionistic anti-realist as I conceive of him or her, does not think it irksome that the notion ‘never in fact’ cannot be expressed by the use of the intuitionistic logical constants. Rather he or she thinks that the only meaning that can be given to ‘never’ is that expressible by the intuitionistic logical constants. So there is no worry and no frustration. (Letter: September 27, 2005)

For insightful discussion of the intuitionistic use of ‘never’, see Williamson (1994a). Incidently, Dummett does not endorse the position articulated in his (2001), which proposes a restriction of the knowability principle to ‘basic’ or atomic ⁵ For alternative formulations of the anti-realist argument against classical logic, see Tennant (2000); Salerno (2000); and Wright (2001). ⁶ Important further discussion and a reply appears in DeVidi and Solomon (2001). ⁷ Cf., DeVidi and Solomon (2001), which offers a defense of this very position on behalf of the Dummettian anti-realist. Dummett embraces the truth of p → ¬¬Kp in much earlier work, including (1977: 339 [2000: 236]).

Introduction

7

sentences. Dummett tells me that he wrote that paper to dispel the myth that Fitch’s paradox is an objection to any form of anti-realism. In Chapter 5 Stig Rasmussen further investigates and defends Dummett’s newly favored knowability principle, p → ¬¬Kp. The centerpiece of the discussion is the ‘mapping objection,’ which points out that Gödel’s 1933 mapping of intuitionistic logic into S4 fails to preserve the original formulation of the knowability principle, and that this fact counts against the original formulation as an expression of intuitionistic anti-realism. In Chapter 6 José Bermúdez argues that the Dummett (2001) position is well-motivated. The position restricts the knowability principle to atomic statements, and deﬁnes intuitionistic truth inductively from there. Bermúdez offers an instructive account of Dummett’s development in (1990) and (1996). There Dummett attempts to clarify the notion of indeﬁnite extensibility of such concepts as set, natural number, and real number, and argues that only intuitionistic logic can illuminate a proper understanding of the notion. It is argued that if this is correct, then the Dummett (2001) theory of truth is well-motivated, and so, we have a principled solution to the knowability paradox. Part III is dedicated to paraconsistency and paracompleteness. The paraconsistent approach to the paradox is ﬁrst suggested in Richard Routley (1981). While considering the liar (‘This very statement is not true’), the knower (‘This very statement is not known’) and Fitch’s proposition, ♦K (p & ¬Kp), Routley entertains, but does not endorse, a uniform treatment: What the hardened paraconsistentist says is that [the liar] and ♦K (p & ¬Kp), though inconsistent, are nonetheless coherent, that this is how things are: some (but not too many) inconsistencies hold true. (1981: 112, n. 26)

Routley does not endorse the approach. His actual position is that Fitch’s result is valid and that it indicates a necessary limitation of human knowledge. Fitch’s result shows us that if there is in fact an unknown truth then there is a logically unknowable truth. On the assumption that our actual ignorance is a contingent matter, it is unclear whether the resulting unknowability is contingent or necessary. However, if necessarily we actually fail to know some truths, as Routley argues, then it follows by Fitch’s main argument and the closure of necessity (over necessary implication) that, necessarily, some truths are unknowable. The passing insight about paraconsistency emerges in the context of Routley’s more central discussion of the necessary limits of knowledge. The paraconsistent approach is ﬁrst defended in Beall (2000), where it is argued that the knower sentence provides independent evidence that knowledge is inconsistent. For the concept entails that Kp & ¬Kp, for some p. Further, it is argued that without a solution to the knower we should accept contradictions of this form and go paraconsistent. To this end Wansing (2002) deﬁnes a paraconsistent, constructive, relevant, modal, epistemic logic that evades Fitch.

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The section of this volume on paraconsistency constitutes the most recent developments of the paraconsistent treatment of the problem. In Chapter 7, Graham Priest develops the Routley/Beall proposal by countenancing the mere possibility of truth-value gluts and appealing to a paraconsistent logic with excluded middle. Beall, in Chapter 8, compliments the development by exploring alternatives, some of which avoid the epistemic oddities of Priest’s framework. Beall’s centerpiece is a semantic framework that is paracomplete, but not paraconsistent, and avoids a commitment even to the mere possibility of truth-value gluts. Part IV is an exploration of temporal and epistemic analogs of Fitch’s reasoning. The strategy is to translate the modalities in the knowability principle into a favored temporal or epistemic logic, and to draw lessons from there about the plausibility of the knowability principle and the result in which it ﬁgures. Johan van Benthem (Chapter 9) does this by placing the result in a dynamic epistemic setting—a setting in which the truth values of our epistemic attributions vary over time with the performance of various actions, such as announcements. The essay is a more thorough development of van Benthem (2004). John Burgess (Chapter 10) translates the Fitch modalities into various Priorian temporal modalities. Each of these two approaches offers, not a rejection of Fitch’s proof, but an investigation of the problematic nature of the corresponding knowability principle. Bernard Linsky (Chapter 11) proposes that we block Fitch’s result by appealing to a theory of types in our account of epistemic and doxastic reasoning. Interestingly, this is one of the proposals that Alonzo Church (in Chapter 1) considers when entertaining objections to the knowability proof. Linsky shows that the theory of types systematically treats a wide variety of contemporary paradoxes of knowledge and belief. Part V is dedicated to Neil Tennant’s Cartesian restriction strategy. Tennant’s position is that intuitionistic logic alone will not free anti-realism from the grips of Fitch. His well-discussed proposal is to restrict the knowability principle to Cartesian propositions, that is, propositions that it is not provably inconsistent to know. Objections to the proposal include Hand and Kvanvig (1999), Williamson (2000b), and DeVidi and Kenyon (2003). For replies see Tennant (2000a; and 2000b). Further motivation for Tennant’s proposal can be found in Jon Cogburn (2004) and Igor Douven (2005). In Chapter 12, Williamson continues the debate, speciﬁcally against Tennant (2001a), and renews his pessimism about the prospects for a successful defense of semantic anti-realism (Cf. Williamson 2000b). Debate with Williamson continues in Tennant (forthcoming). Chapter 13 is Kvanvig’s renewed discontent with Tennant’s (and any other) restriction to the knowability principle. As he sees it, the real paradoxicality is not that Fitch’s result threatens anti-realism, but that it threatens to collapse the very distinction between the existence of unknown truth and the existence of unknowable truth. The section is completed

Introduction

9

by Chapter 14, which is Tennant’s current position—a modiﬁcation of the Cartesian restriction strategy. Part VI is about modal and mathematical ﬁctionalism. We learn in Brogaard (Chapter 15) that modal ﬁctionalism is threatened by Fitch’s paradox. Otávio Bueno (Chapter 16) evaluates the relevance of Fitch’s paradox in the epistemology of mathematics. He argues that the mathematical ﬁctionalist must contend with the unwelcome consequences of Fitch. Part VII, Knowability Reconsidered, includes papers that reconsider the antirealist thesis about the knowability of truth. There is a history of attempts to either refute or reformulate anti-realism in reaction of Fitch. I mentioned some refuters earlier. The reformulater is one who rejects, or offers an alternative to, the knowability principle as a characterization of anti-realism. They include Edgington (1985), Melia (1991), Wright (2000), Hand (2003), and Jenkins (2005), among many others. Michael Hand (Chapter 17) further develops his 2003 proposal that Dummett’s anti-realist is not committed to the knowability principle, owing to the fact that it carelessly blurs semantic conditions about veriﬁcation procedures with pragmatic conditions about the performance of such procedures. C. S. Jenkins (Chapter 18) agrees that the knowability principle fails as an expression of anti-realism. Her own statement of anti-realism (2005) is echoed here, but her primary concern is to take issue with Kvanvig (2006), in which it is argued that the real paradoxicality of Fitch’s proof is the modal collapse that occurs in the reasoning from the knowability principle to the omniscience principle. W. D. Hart (Chapter 19) takes Fitch’s proof to be evidence for realism. He argues that the prospects are not good for a solution coming from the theory of types. Christoph Kelp and Duncan Pritchard (Chapter 20) offer some hope for an anti-realism that endorses a justiﬁed believability principle in place of the knowability principle. They evaluate the thesis that, for all true propositions, it must be possible to justiﬁably believe them. An alternative weakening of the knowability principle is proposed by Greg Restall (Chapter 21). His principle states that, for every truth p, there is a collection of truths, such that (i) each of them is knowable and (ii) their conjunction is equivalent to p. Restall proves that this formulation evades the paradox, and draws lessons about the operant notion of possibility. I regret that the volume is incomplete. It includes no extensive discussion of Dorothy Edgington’s important 1985 proposal, in which the knowability principle is reformatted as a thesis about the knowability of actual truth. Important criticisms are found in Wright (1987a (2nd edn., 1993: 428–32)), Williamson (1987a; 1987b; 2000a, ch. 12), and Percival (1991). Developments of Edgington’s proposal are found in Rabinowicz and Segerberg (1994), Linström (1997), Rückert (2004), Fara (forthcoming), and Murzi (manuscript). Related proposals, that focus on the modal semantics of Fitch’s paradox, include Kvanvig (1995; 2006), Brogaard and Salerno (2006) and Costa-Leite (2006). These latter three approaches diagnose various modal fallacies. Kvanvig appeals to issues about

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when we are licensed to substitute into modal contexts. Brogaard and Salerno appeal to Stanley and Szabo’s (2000) theory of quantiﬁer domain restriction, according to which there is hidden structure in quantiﬁed noun phrases. CostaLeite appeals to the fusion of Kripke frames—the insight being that knowability is not to be understood compositionally out of one-dimensional possibility and knowledge operators.

Pa r t I E a r l y Hi s t o r y

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1 Referee Reports on Fitch’s ‘‘A Deﬁnition of Value’’ Alonzo Church

The referee reports transcribed below were handwritten by Alonzo Church to his co-editor, Ernest Nagel, of the Journal of Symbolic Logic. They were issued in 1945 in response to a paper by Frederic Fitch, ‘‘A Deﬁnition of Value,’’ which was not published. They contain the earliest formulations of the modal epistemic result today known as ‘‘Fitch’s knowability paradox.’’ The bracketed numerals, [n.], indicate the original page numbers. Our appendix is a list of the cited principles. They either originate in Lewis and Langford (1932) or are hypothesized by us to be the principles from Fitch’s original manuscript, which was not found. The original reports are located in the Ernest Nagel Papers, Box 1, Arranged Correspondence, Church, Alonzo. Rare Book and Manuscript Library, Columbia University. They are printed here in full by their permission and by kind permission of Alonzo Church, Jr. We are grateful to Nick Zavediuk for his assistance in the transcription process.

Fi r s t Re f e re e Re p o r t [1.] It seems to me that the role and meaning of Professor Fitch’s ‘SC’ is seriously in need of clariﬁcation. It is not sufﬁcient merely to take ‘SC’ as primitive and undeﬁned. It must be contemplated that ultimately there is either a deﬁnition of ‘SC’ or an elaborate set of empirical postulates about it; otherwise particular empirical necessitations such as ‘‘brakeless trains are dangerous’’ could not be decided. Perhaps Fitch means to say that there is one absolutely determined set of ‘‘all the valid laws of empirical sciences,’’ such that the currently accepted laws of the currently known empirical sciences are an approximation to a certain subset of this set, and that SC is something like the conjunction of all the laws of this set. But this belief in an ultimate set of absolutely valid empirical laws is held by hardly any contemporary empirical scientist. And I think that the recent history of the empirical sciences, especially physics, renders such a belief indefensible. The only alternative I see is to take a particular formalized system of empirical science (say a system which uniﬁes the empirical sciences as they are known today),

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and to take SC to be the conjunction of the primitive propositions of this system. This is acceptable, but it must be realized that it makes empirical necessitation relative to a particular system. Accepting this emendation, I go on to Fitch’s Def. 3. [2.] Following the line of Fitch’s thought, let me call a proposition empirically impossible if SC strictly implies its negation. (This makes empirical impossibility equivalent to the negation of empirical possibility and is therefore consistent with Fitch’s Def. 6.) Then it may plausibly be maintained that if a is not omniscient there is always a true proposition which it is empirically impossible for a to know at time t. For let k be a true proposition which is unknown to a at time t, and let k be the proposition that k is true but unknown to a at time t. Then k is true. But it would seem that if a knows k at time t, then a must know k at time t, and must also know that he does not know k at time t. By Def. 2, this is a contradiction. Now an empirically impossible proposition empirically necessitates every proposition. Therefore, the argument runs, by taking q in Def. 3 to be k , it may be inferred that everything is of value to a at time t. Thus Def. 3 is reduced to a triviality. In spite of the plausibility of the preceding argument I think Fitch has a good defense (but only one). This defense is that there is no law of psychology according to which one who believes a proposition must believe all its logical consequences; on the contrary, historical counter-examples are well known. To be sure, one who believes a proposition without believing its more obvious consequences is a fool; but it is an empirical fact that there are fools. It is even possible [3.] that there might be so great a fool as to believe the conjunction of two propositions without believing either of the two propositions; at least, an empirical law to the contrary would seem to be open to doubt. On this ground it is empirically possible that a might believe k at time t without believing k at time t (although k is a conjunction one of whose terms is k). Unfortunately this defense compels Fitch to abandon his Ax. 1. And, what is more serious, it lights the way to a second and opposite objection to Def. 3. If there is no empirical law according to which one who believes a proposition must believe its logical consequences, it would seem that by the same token there is no empirical law according to which a person’s desires must be in reasonable accord with that person’s beliefs. If someone desiring to recover from a certain disease, and knowing the one and only course of action which will lead to recovery, nevertheless does not desire that course of action, we may call that someone a fool; but again the fact is that fools there be. It is a historical fact that there have been persons who desired to avoid smallpox, who knew the medical efﬁcacy of vaccination as a preventive, and who nevertheless violently resisted vaccination (therefore presumably did not desire it). I conclude [4.] that there is no valid law of psychology according to which anything whatsoever about my desires may be inferred from the fact that I know so-and-so. It follows by Def. 3 that nothing

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is of value to a at time t, and again Def. 3 is reduced to a triviality. This is my second objection to Def. 3, and the one to which I attach the greater weight. Since Def. 3 is the core of the paper, I hold that the entire paper is therefore untenable. By this it is not to be understood that I disapprove of the idea of applying symbolic logic to value theory. However severe my criticism of Fitch for what I hold to be logical ﬂaws, my criticism is still more severe of those philosophers who offer similar deﬁnitions of value in vague verbal form without the attempt at even so much accuracy as may be attained by the use of certain elementary notations of symbolic logic. For these latter escape the kind of criticism I level against Fitch only by making their statement so vague as to render all criticism uncertain. The very fact that an attempt by Fitch to state formally what I take to be a rather ordinary sort of deﬁnition of value leads to these logical difﬁculties is an indication of the need for at least some elementary symbolic logic here. Let me say also that, in order to meet Fitch on his own ground, I have accepted uncritically what seems to be his notion of [5.] proposition, although it is well known that the notion of proposition is uncertain and in need of clariﬁcation. I am willing to concede, at least as a possibility, that one way to obtain clariﬁcation of the notion is to plunge directly into the use of propositions and to clear up individual difﬁculties as they arise. Finally, I note that Fitch makes a medical error on page 4 of the manuscript, in implying that quinine is the only cure for malaria. An entirely different drug, atabrine, is as a matter of fact also an effective cure. It seems to me that according to ordinary usage it would be said that quinine is valuable to the malaria sufferer, even if there does exist an alternative method of cure by means of atabrine. This observation may reveal another deﬁciency in Def. 3. But in view of more serious objections it seems unnecessary to go into this.

Se c o n d Re f e re e Re p o r t 1. It is not clear to me why Fitch thinks that, to quote his letter, ‘‘In order to show that a’s ignorance of k is empirically necessary, he would ﬁrst have to show that a’s ignorance of k is empirically necessary.’’ The fact is that the quoted statement is false. To enforce my point, let me put the matter quite formally: Assume: k. Assume also: ∼(a KNt k). Def.: k = (k & ∼ (a KNt k) ). By the foregoing Def., and Fitch’s Th. 3: (a KNt k ) EN (a KNt k). By Fitch’s Def. 2, and Lewis-Langford 11.2: (a KNt k ) SI k .

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Hence by the Def. of k , and Lewis-Langford 11.2, 11.6: (a KNt k ) SI ∼(a KNt k). Hence by Lewis-Langford 12.42: (a KNt k) SI ∼(a KNt k ). Hence by Fitch’s Th. 1: (a KNt k) EN ∼(a KNt k ). The transitive law for EN follows from Lewis-Langford 15.1, 16.2, 11.6. Hence from the foregoing step and step 4 above we get: (a KNt k ) EN ∼(a KNt k ). Hence by the law of double negation and Fitch’s Def. 5: ∼((a KNt k ) EC (a KNt k )). Hence by Fitch’s Def. 6: ∼EP(a KNt k ). However, it follows from our assumptions: k . From this now it follows (for the reason explained in my previous report, and as I [2.] understand Fitch to admit) that Fitch’s Def. 3 is untenable as it now stands, and must be altered if the paper as a whole is to be maintained. Now let us consider the revised form of Def. 3 which Fitch proposed in his letter of February 26 and afterwards abandoned. As I understand it this is: Def. 3R. (a VLt p) = (Eq) [q & EP(a KNt q) & [(a KNt q) EN (a DSt p)]]. I think that a reductio ad absurdum of Def. 3R is possible along the same lines as that I have given for Def. 3. At least, let me attempt it, and leave it to Fitch to say where if anywhere I have assumed something he would not admit. I shall show as a consequence of Def. 3R that instant death is of value to a at time t. In other words, if p is ‘‘a dies at time t,’’ I shall show that a VLt p . Assume that a does not desire instant death at time t (because in the contrary case, if we assume that it is empirically possible for one to know one’s own desires, the conclusion a VLt p is obvious). Nevertheless it is empirically possible for a to desire instant death at time t, both (1) because it is empirically possible that a should be insane, and (2) it is empirically possible that a’s external circumstances [3.] at time t might have been so dreadful as to compel even a sane man to desire instant death. Assume also that a is not omniscient, and let k be something which is true but unknown to a at time t. Deﬁne k as before. Then as before k is true but a’s ignorance of k is empirically necessary. Let q be the disjunction (a DSt p ) ∨ k . Then q is true. I suppose Fitch would admit (a KNt p) EN (a KNt (p ∨ q)). At least this seems to be entirely in the spirit of his Th. 3, and it is hard to see how he could maintain one and deny the other. Also I suppose that logical consequences of the empirically possible are empirically possible, and that it is empirically possible for one to know one’s own desires.

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Thus because a DSt p is empirically possible, therefore a KNt (a DSt p ) is empirically possible, and therefore a KNt q is empirically possible. Because (a DSt p ) EN (a DSt p ) and k EN (a DSt p ), therefore q EN (a DSt p ). Hence because (a KNt q ) EN q , it follows that (a KNt q ) EN (a DSt p ). Finally, taking, in Def. 3R, p to be p , and q to be q , we get the conclusion that a VLt p . The case of instant death is of course [4.] chosen only for illustration. In general, under Def. 3R, everything is of value to a at time t which it would be empirically possible for a to desire at time t under any empirically possible circumstances (however remote from the actual circumstances). This is little if any less disastrous than the situation under Def. 3, that everything whatever is of value to a at time t. Of course the foregoing refutation of Fitch’s deﬁnition of value is strongly suggestive of the paradox of the liar and other epistemological paradoxes. It may therefore be that Fitch can meet this particular objection by incorporating into the system of his paper one of the standard devices for avoiding the epistemological paradoxes. If this is possible it will involve a drastic rewriting of the paper, not just a footnote here and there. To my further objection—that there is no law of psychology according to which it can be inferred from the fact that a knows something that therefore a desires something—Fitch replies by pointing out that a might know that a desires p. If, however, Fitch consents to adopt one of the standard devices for avoiding the epistemological paradoxes, this reply will no longer be open to him. For example, on the basis of Russell’s original [5.] theory of types, ‘‘a desires p’’ is of higher order than p, whereas the two ‘‘something’’ ’s in my assertion must of course be understood as of the same order. On the basis of Tarski’s resolution of the epistemological paradoxes, the distinction between language and meta-language has roughly the same effect. I insist therefore that there is no known law of psychology according to which ‘‘a desires p’’ is ever a necessary consequence of ‘‘a knows q.’’ Moreover, in the light of every-day experience (summed up in the commonly heard conclusion, ‘‘Some people are utterly unreasonable’’), it seems unlikely that there is a valid law of psychology of that sort remaining to be discovered. If some of us think that there is a notion of value in spite of the fact that some people are utterly unreasonable, it is because we think we know how to distinguish between what is reasonable and what is unreasonable. The problem is whether this distinction between reasonable and unreasonable can be deﬁned in non-valuational terms, or whether this or some like value-theoretic concept must be accepted as primitive (undeﬁned). I do not think that Fitch has solved the problem. The assumption that there is an absolute set of valid empirical laws, SC, to which the accepted laws of the empirical sciences are [6.] approximations in some sense, is of course a piece of metaphysics. I have no objection to metaphysics per se.

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But this particular piece of metaphysics has more opponents than adherents, and it seems that a deﬁnition of value which presupposed it would be of interest only to a very restricted circle. I do not understand why Fitch objects to avoiding the metaphysical issue by making his deﬁnition of value relative to a particular (comprehensive) system of empirical science. But this is a side-issue, in view of the existence of more serious objections. As to the matter of quinine and atabrine: it seems that, according to Fitch, if a is a malaria sufferer who has equal access to quinine and atabrine, then both ‘‘quinine is of value to a’’ and ‘‘atabrine is of value to a’’ are false. Moreover it may be that there is some drug which is a quicker and more certain cure for malaria than either quinine or atabrine, which is easily accessible to a (if he only knew), but whose properties in this respect are still undiscovered. If so, then not even the disjunction, quinine or atabrine, is valuable to a. It seems to me that such a notion of value departs so far from the everyday notion that it is hardly justiﬁed to use the same word for it. Finally, Fitch’s plan of adding [6.] short postscripts to his paper commenting on particular objections by the referee does not seem to me a good one. So far as the objections either are valid or represent misunderstandings likely to be duplicated by others, they should be met (if that is possible) by alterations in the body of the paper.

Appendix Joe Salerno and Julien Murzi Fi t c h’s Op e r a t o r s

p SI q = p ≺ q = p strictly implies q p EN q = p empirically necessitates q EPp = p is empirically possible p EC q = p is empirically consistent with q aKNtp = a knows at time t that p aBtp = a believes at t that p aVLtp = a values at t that p aDStp = a desires at t that p L e w i s a n d L a n g f o rd ( 1 9 3 2 ) Deﬁ n i t i o n a n d T h e o re m s

(p ≺ q) =df ∼ ♦(p & ∼q) 11.2 (p & q) ≺ p

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11.6 ((p ≺ q) ≺ (q ≺ r)) ≺ (p ≺ r) 12.42 (p ≺ ∼q) ≺ (q ≺ ∼p) 15.1 ((p ⊃ q) & (q ⊃ r)) ≺ (p ⊃ r) 16.2 ((p ≺ q) & (p ≺ r) & T) ≺ (p ≺ (q & r)): T = ((q & r) ≺ (r & q)) Fi t c h’s De ﬁ n i t i o n s¹

Def. 2∗ Def. 2 is Fitch’s deﬁnition of knowledge. All we know from Church’s use is that it justiﬁes the principle that a’s knowing at time t that p strictly implies p: aKNtp ≺ p.² Def. 3∗ aVLt p =df ∃q(q & (aKNtq EN aDStp) ). Value is what one would desire given sufﬁcient knowledge: it is valuable to a at t that p if and only if there is a true proposition q, such that a’s knowing at t that q empirically necessitates a’s desiring at t that p.³ Def. 3R aVLt p =df ∃q(q & EP(aKNtq) & (aKNtq EN aDStp) ). Value is what one would desire given sufﬁcient knowledge: it is valuable to a at t that p if and only if there is a truth q that it is empirically possible to know and a’s knowing at t that q empirically necessitates a’s desiring at t that p.⁴ Def. 5∗ (p EN ∼ p) ≺ ∼ (p EC p). Necessarily, if p empirically necessitates ∼p, then p is not (empirically) consistent with itself.⁵ Def. 6∗ ∼ (p EC p) =df ∼ EPp. p is not (empirically) consistent with itself just in case p is not empirically possible.⁶ ¹ An asterisk, ‘∗ ’, indicates that the principle does not appear explicitly in the reports, and therefore, that we have hypothesized its content. ² Church’s applications appear in Report 1: 2 and Report 2: 1. ³ Our formulation of Def. 3 is based on Church’s trivialization argument against it. Compare Report 1: 2 and Report 2: 1–2. ⁴ Report 2: 2. ⁵ Report 2: 1. ⁶ Report 1: 2, and Report 2: 1.

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Alonzo Church Fi tc h’s Ax i o m s a n d T heo rem s

Ax. 1∗ (aBtp & (p EN q) ) ≺ aBtq Belief is closed under ‘‘empirically necessary’’ implication: necessarily, if a believes at t that p and p empirically necessitates q, then a believes at t that q.⁷ Th. 1∗ (p ≺ q) ≺ (p EN q) Strict implication strictly implies empirical necessitation: necessarily, if p strictly implies q then p empirically necessitates q.⁸ Th. 3∗ aKNt (p & q) ≺ (aKNtp & aKNtq) Knowing a conjunction strictly implies knowing the conjuncts: necessarily, if a knows at t that both p and q, then a knows at t that p and a knows at t that q.⁹ ⁷ The discussion at Report 1: 2–3 suggests that Ax. 1 is this closure principle for belief. Alternatively, it is an unrestricted closure principle for knowledge (viz., knowledge is closed under necessary empirical implication). ⁸ See for instance, Church’s use in Report 2: 1. ⁹ Report 2: 1.

2 A Logical Analysis of Some Value Concepts¹ Frederic B. Fitch

The purpose of this paper is to provide a partial logical analysis of a few concepts that may be classiﬁed as value concepts or as concepts that are closely related to value concepts. Among the concepts that will be considered are striving for, doing, believing, knowing, desiring, ability to do, obligation to do, and value for. Familiarity will be assumed with the concepts of logical necessity, logical possibility, and strict implication as formalized in standard systems of modal logic (such as S4), and with the concepts of obligation and permission as formalized in systems of deontic logic.² It will also be assumed that quantiﬁers over propositions have been included in extensions of these systems.³ There is no intention to provide exhaustive logical analyses, or to provide logical analyses that reﬂect in detail the usage of so-called ordinary language. This latter task seems impossible anyhow because of the ambiguities of ordinary language and the obvious inconsistencies and irregularities of usage in ordinary language. Furthermore, the term ‘ordinary language’ is itself rather vague. Whose ordinary language? Should English be preferred to Chinese? Various arguments that invoke English or Latin grammatical usage are seen to be without foundation from the standpoint of Chinese. Just as the concepts of necessity and possibility used in so-called ordinary language correspond in some degree to the concepts of necessity and possibility Chapter 2 was ﬁrst published in the Journal of Symbolic Logic 28/2, 135–42 (1963) and is reproduced by permission of the Association for Symbolic Logic. ¹ An earlier draft of this paper was presented as a retiring presidential address to the Association for Symbolic Logic; read before the Association at Atlantic City, New Jersey, December 27, 1961. ² For example see A. R. Anderson, The formal analysis of normative systems, Technical Report No. 2, Contract No. SAR/Nonr-609(16), Ofﬁce of Naval Research, Group Psychology Branch, 1956; also, by the same author, A reduction of deontic logic to alethic modal logic, Mind , n.s. vol. 67 (1958), pp. 100–3. ³ Such quantiﬁers can be introduced by methods analogous to those used in R. C. Barcan (Marcus), A functional calculus of ﬁrst order based on strict implication, this Journal, vol. 11 (1946), pp. 1–16; The deduction theorem in a functional calculus of ﬁrst order based on strict implication, ibid., pp. 115–18; and F. B. Fitch, Intuitionistic modal logic with quantiﬁers, Portugaliae mathematica, vol. 7 (1948), pp. 113–18. See also, R. Carnap, Modalities and quantiﬁcation, this Journal, vol. 11 (1946), pp. 33–64.

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used in modal logic, so too it is to be hoped that the ordinary language concepts of striving, doing, believing, desiring and knowing will correspond in some degree to the concepts that we will partially formalize here. Also, just as there are various slightly differing concepts of possibility and necessity corresponding to differing systems of modal logic, so too there are presumably various slightly differing concepts of striving, doing, believing, and knowing, having differing formalizations. We begin by assuming that striving, doing, believing, and knowing all have at least some fairly simple properties which will be described in what follows, and we leave open the question as to what further properties they have. First of all, we assume that striving, doing, believing, and knowing are twotermed relations between an agent and a possible state of affairs. It is convenient to treat these possible states of affairs as propositions, so if I say that a strives for p, where p is a proposition, I mean that a strives to bring about or realize the (possible) state of affairs expressed by the proposition p. Similarly, if I say that a does p, where p is a proposition, I mean that a brings about the (possible) state of affairs expressed by the proposition p. We do not even have to restrict ourselves to possible states of affairs, because impossible states of affairs can be expressed by propositions just as well as can possible states of affairs. In the case of believing and knowing, there is surely no serious difﬁculty in regarding propositions as the things believed and known. So we treat all these concepts as two-termed relations between an agent and a proposition. In a similar way, the concept of proving could also be regarded as a two-termed relation between an agent and a proposition. For purposes of simpliﬁcation, the element of time will be ignored in dealing with these various concepts. A more detailed treatment would require that time be taken seriously. One method would be to treat these concepts as a three-termed relation between an agent, a proposition, and a time. Another method would be to avoid specifying times explicitly, but rather to use a temporal ordering relation between states of affairs. This latter method might be more in keeping with the theory of relativity, in either its special or general form. As a further step of simpliﬁcation we will often ignore the agent and thus treat each of the concepts under consideration as a class of propositions rather than as a two-termed relation. For example, by ‘striving’ we will mean the class of propositions striven for (that is, striven to be realized), and by ‘believing’ we will mean the class of propositions believed, relativizing the whole treatment to some unspeciﬁed agent. But the agent can always be speciﬁed if we wish to do so, and we can replace classes by two-termed relations. A class of propositions (in particular such classes of propositions as striving, knowing, etc.) will be said to be closed with respect to conjunction elimination if (necessarily) whenever the conjunction of two propositions is in the class so are the two propositions themselves. For example, the class of true propositions is closed with respect to conjunction elimination because (necessarily) if the

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conjunction of two propositions is true, so are the propositions themselves. If α is a class closed with respect to conjunction elimination, this fact about α can be expressed in logical symbolism by the formula, (p) (q) [ (α[p & q] ) [ (αp) & (αq) ] ] , where ‘’ stands for strict implication. We assume that the following concepts, viewed as classes of propositions, are closed with respect to conjunction elimination: striving (for), doing, believing, knowing, proving.

For example, in the case of believing we assume: (p) (q) [ (believes[p & q] ) [ (believes p) & (believes q) ] ]. Here are some further concepts which are evidently closed with respect to conjunction elimination: truth, causal necessity (in the sense of Burks),⁴ causal possibility (in the sense of Burks), (logical) necessity, (logical) possibility, obligation (deontic necessity), permission (deontic possibility), desire for.

A class of propositions will be said to be closed with respect to conjunction introduction if (necessarily) whenever two propositions are in the class, so is the conjunction of the two propositions. If α is a class closed with respect to conjunction introduction, this fact about α can be expressed in logical symbolism by the formula, (p) (q) [ [ (αp) & (αq) ] (α[p & q] ) ]. Except for causal, logical, and deontic possibility, all the concepts so far regarded as closed with respect to conjunction elimination could perhaps also be regarded as closed with respect to conjunction introduction, or some varieties of them could. For present purposes, however, we do not need to commit ourselves on this matter except to say that truth and causal, logical, and deontic necessity are all indeed closed with respect to conjunction introduction. A class of propositions will be said to be a truth class if (necessarily) every member of it is true. If α is a truth class, this fact about α can be expressed ⁴ A. W. Burks, The logic of causal propositions, Mind , n.s. vol. 60 (1951), pp. 363–82.

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in logical symbolism by the formula, (p)[(αp) p]. The concepts truth, causal necessity, and logical necessity are clearly truth classes. It also seems reasonable to assume that doing, knowing, and proving are truth classes, and so we make this assumption. Thus, whatever is true or causally or logically necessary is true; and (as we assume) whatever is done, known, or proved is also true. The following two theorems about truth classes will be applied to some of the above-mentioned truth classes in subsequent theorems. Theorem 1. If α is a truth class which is closed with respect to conjunction elimination, then the proposition, [p & ∼(αp)], which asserts that p is true but not a member of α (where p is any proposition), is itself necessarily not a member of α. Proof. Suppose, on the contrary, that [p & ∼(αp)] is a member of α; that is, suppose (α[p & ∼(αp)] ). Since α is closed with respect to conjunction elimination, the propositions p and ∼(αp) must accordingly both be members of α, so that the propositions (αp) and (α(∼ (αp) ) ) must both be true. But from the fact that α is a truth class and has ∼(αp) as a member, we conclude that ∼(αp) is true, and this contradicts the result that (αp) is true. Thus from the assumption that [p & ∼(αp)] is a member of α we have derived contradictory results. Hence that assumption is necessarily false. Theorem 2. If α is a truth class which is closed with respect to conjunction elimination, and if p is any true proposition which is not a member of α, then the proposition, [p & ∼(αp)], is a true proposition which is necessarily not a member of α. Proof. The proposition [p & ∼(αp)] is clearly true, and by Theorem 1 it is necessarily not a member of α. Theorem 3. If an agent is all-powerful in the sense that for each situation that is the case, it is logically possible that that situation was brought about by that agent, then whatever is the case was brought about (done) by that agent. Proof. Suppose that p is the case but was not brought about by the agent in question. Then, since doing is a truth class closed with respect to conjunction elimination, we conclude from Theorem 2 that there is some actual situation which could not have been brought about by that agent, and hence that the agent is not all-powerful in the sense described. Theorem 4. For each agent who is not omniscient, there is a true proposition which that agent cannot know.⁵ Proof. Suppose that p is true but not known by the agent. Then, since knowing is a truth class closed with respect to conjunction elimination, we conclude from Theorem 2 that there is some true proposition which cannot be known by the agent. ⁵ This theorem is essentially due to an anonymous referee of an earlier paper, in 1945, that I did not publish. This earlier paper contained some of the ideas of the present paper.

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Theorem 5. If there is some true proposition which nobody knows (or has known or will know) to be true, then there is a true proposition which nobody can know to be true. Proof. Similar to proof of Theorem 4. Theorem 6. If there is some true proposition about proving that nobody has ever proved or ever will prove, then there is some true proposition about proving that nobody can prove. Proof. Similar to the proof of Theorem 4, using the fact that if p is a proposition about proving, so is [p & ∼(αp)]. This same sort of argument also applies to the class of logically necessary propositions, since this is a truth class closed with respect to conjunction elimination. Thus by Theorem 1 we have the result that every proposition of the form [p& ∼ p] is necessarily not logically necessary, and hence necessarily possibly false, where ‘’ denotes logical necessity. In other words, the proposition ∼ [p & ∼ p] is true for every proposition p.⁶ In particular, if p is a true proposition which is not necessarily true, then [p & ∼ p] is a true proposition which is necessarily possibly false. I now wish to describe a relation of causation, or more accurately, partial causation, which will be used in giving a deﬁnition of doing in terms of striving and a deﬁnition of knowing in terms of believing, as well as some other deﬁnitions. I will assume that partial causation, expressed by ‘C’, satisﬁes the following axiom schemata C1–C4: C1. C2. C3. C4.

[[p C q] & [q C r]] [p C r]. [p & [p C q]] q. [p & [[p & q] C r]] [q C r]. [[p C q] & [p C r]] ≡ [p C [q & r]].

(transitivity) (detachment) (strengthening) (distribution)

Here ‘p ≡ q’ is deﬁned as ‘[p q] & [q p]’. I will also employ an identity relation among propositions and will employ the following axiom schemata I1–I9 for this identity relation:⁷ I1. [[p = q] & (. . . p . . .)] (. . . q . . .). I2. p = p. ⁶ This result in slightly different form is to be found in the two papers by Anderson cited above. He uses it in constructing a model of deontic logic in alethic modal logic and attributes it to W. T. Parry, Modalities in the survey system of strict implication, this Journal, vol. 4 (1939), pp. 137–54, Theorem 22.8. ⁷ It is interesting to observe that I2–I9 may be used to serve as postulates for an algebra like Boolean algebra but somewhat weaker, provided that the identity symbol is regarded as a symbol for equality in such an algebra and that (in place of I1) there are added postulates to the effect that equality is symmetrical and transitive, and that the negates, conjuncts, and disjuncts of equal elements of the algebra are equal. Also, there should be a postulate to the effect that there are at

26 I3. I4. I5. I6. I7. I8. I9.

Frederic B. Fitch p = ∼∼ p. p = [p & p]. [p & q] = [q & p]. [p & [q & r ] ] = [ [p & q] & r ]. [p & [q ∨ r ] ] = [ [p & q] ∨ [p & r ] ]. p = [ [p & q] ∨ p]. [∼ p & ∼ q] = ∼ [p ∨ q].

Notice that we do not have such theorems as p = [p & [q ∨ ∼ q]] and p = [p ∨ [q & ∼ q]]. Only a few of the axiom schemata listed above will be directly relevant in what follows. The ones most relevant are C2, C4, I1, and I6. The property expressed by C3 reﬂects the fact that C is only partial causation. If C were total causation, then C3 would clearly be unacceptable. It should also be remarked that C need not be regarded as restricted to relating states of affairs that have space-time location, but may relate any state of affairs (e.g., a mathematical truth) to other suitable states of affairs. Otherwise, the sort of knowledge deﬁned below would be knowledge only of states of affairs that have space-time location. Using the relation C, a deﬁnition of doing in terms of striving will now be given. It is perhaps best to regard this deﬁnition merely as an axiom schema that provides a necessary and sufﬁcient condition for doing, and similarly in subsequent deﬁnitions. As before, reference to the agent and to time are omitted for simplicity. D1. (does p) ≡ ∃q[(strives for [p & q]) & [(strives for [p & q]) C p]]. This means that an agent does p if and only if there is some (possible or impossible) situation q such that the agent strives for p and q, and a result of this striving is that p takes place. Using I1, I6, C4, and properties of existence quantiﬁcation, it is easy to show that this deﬁnition gives the result that doing is closed with respect to conjunction elimination. A deﬁnition of knowing in terms of believing is now given: least two unequal elements of the algebra. Such an algebra provides an algebraic formulation for the Anderson–Belnap system of ﬁrst degree entailments with quantiﬁers omitted (A. R. Anderson and N. D. Belnap, Jr., First degree entailments, Technical Report No. 10, ibid., 1961, since the assertion that p entails q can be deﬁned as the assertion that p equals the conjunction of p with q, or equivalently as the assertion that q equals the disjunction of q with p. This algebra was suggested to me by a list of theorems on page 21 of my paper, A system of combinatory logic, Technical Report No. 9, ibid., 1960, and in part also by some discussions with Anderson. It also bears a close relation to the system of my paper, The system C of combinatory logic, Technical Report No. 13, ibid., 1962 (also forthcoming in this Journal). The system of ﬁrst degree entailment including quantiﬁers was also arrived at independently by Miss Patricia A. James and myself as a modiﬁed form of the system of my book Symbolic logic (New York, 1952) prior to the Anderson–Belnap formulation of that system. This alternative approach to the system of ﬁrst degree entailment is sketched on p. vii of Miss James’s doctoral dissertation, Decidability in the logic of subordinate proofs (Yale University, 1962).

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D2. (knows p) ≡ ∃q[p & q & [[p & q] C (believes [p & q])]]. This means that an agent will be said to know p provided that p and some (possibly other) situation q are both true, and provided that the fact that they are both true causes the agent to believe the fact that they are both true. Thus the known fact p must be causally efﬁcacious (as part of the conjunction [p & q]) in bringing about the agent’s belief that [p & q] is the case, and hence that p itself is the case, since belief is assumed closed with respect to conjunction elimination. It is easy to show that knowing, as thus deﬁned, is a truth class closed with respect to conjunction elimination. Ability to do can be deﬁned in the following way: D3. (can do p) ≡ ∃q[ (strives for[p & q] )Cp]. This deﬁnition can be shown to give the result that ability to do is closed with respect to conjunction elimination. Obligation to do can be deﬁned in terms of doing and the concept of obligation as expressed by the operator ‘0’ of deontic logic, as follows: D4. (should do p) ≡ 0 (does p). Obligation to do, as thus deﬁned, can be shown to be closed with respect to conjunction elimination and also with respect to conjunction introduction. I now wish to propose a deﬁnition of desire, as follows: D5. (desires p) ≡ ∃q[ (believes(can do[p & q] ) )C(strives for[p & q] ) ]. This means that an agent desires a situation p if his belief that he can achieve the conjunction of p with some (possibly other) situation causes him to strive for that conjunction of situations. Desire as thus deﬁned can be shown to be closed with respect to conjunction elimination. A concept of value, which I now wish to consider, can be deﬁned in the following way: D6. (value p) ≡ ∃q∃r [q & [ (knows q)C (strives for [p &r ] ) ] ]. This means that a situation p is a value for an agent if (and only if) there is an actual situation q and situation r such that if the agent knows q then he will strive for the conjunction of p and r. In knowing q the agent may be supposed to have all the knowable relevant information concerned with the effect of his striving for the conjunction of p and r, and if this knowledge causes him to strive for this conjunction, it must be because this conjunction, and in particular p itself, is of value to him. To see why q may be supposed to contain all the knowable relevant information for the purpose at hand, let us suppose, on the contrary, that q does not contain all such relevant information. Then there might be some additional information s such that knowledge of the conjunction of q and s would cause the agent not to strive for any conjunction of the form

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[p & t]. But in the hypothetical case that the agent knew [q & s], he would also know q because of the fact that knowing is closed with respect to conjunction elimination, and this knowledge of q, by assumption, would cause him to strive for [p & r]. Thus he would be caused to strive for [p & r] and also caused not to strive for [p & r], and the assumption that he could know such a proposition as [q & s] leads to an absurdity. Hence q may be regarded as containing all the knowable relevant information. It can be shown easily that value as thus deﬁned is closed with respect to conjunction elimination. The objection might be raised against the above deﬁnition of value that the agent must be assumed to be rational, since otherwise he might have all the relevant knowledge to enable him to make a choice in his own interest, and yet, being irrational, he would be caused by this knowledge to make some other choice and to strive for some outcome that would not be of value to him. One way, and perhaps the only way, to attempt to meet this objection is to maintain that all irrationality is due to lack of sufﬁcient knowledge, so that the having of sufﬁcient relevant knowledge already rules out any relevant amount of irrationality. According to this view, any sort of insanity would be curable simply by giving the patient sufﬁcient knowledge of himself and of the world around him. This view would not deny that in practice there might be insuperable obstacles that prevent the communication of this knowledge to the patient, but the existence of such obstacles would not prove that irrationality was not essentially a lack of knowledge. This deﬁnition of value of course does not guarantee that there are any values in this sense, though it seems to me not unreasonable to assume that there may be values in this sense. A more difﬁcult problem is the problem of the comparison of values, that is, the problem of greater and less among values. This problem will not be dealt with here. YALE UNIVERSITY

3 Knowability Noir: 1945–1963 Joe Salerno

The literature on the knowability paradox emerges in response to a modal epistemic proof ﬁrst published by Frederic Fitch in his famous 1963 paper, ‘‘A Logical Analysis of Some Value Concepts.’’ Theorem 5, as it was there called, threatens to collapse a number of modal and epistemic differences. Let ignorance be the failure to know some truth. Then Theorem 5 collapses a commitment to fortuitous ignorance into a commitment to necessary ignorance. For it shows that the existence of truths in fact unknown entails the existence of truths necessarily unknown. The converse of Theorem 5 is trivial (if truth entails possibility), so Fitch goes most of the way toward erasing any logical difference between the existence of fortuitous ignorance and the existence of necessary unknowability. More exactly, it is the contrapositive of Theorem 5 that is today referred to as the knowability paradox. The contrapositive tells us that any truth can be known but only if every truth is in fact known. As such it collapses sophisticated anti-realism into naive idealism—a philosophical difference we may wish to preserve even if we are not sympathetic to anti-realism. Further, and with slightly strengthened resources, Fitch’s proof threatens to dissolve the very distinction between what is possible and what is actual.¹ A special thanks to those who have assisted my archival research, including Aldo Antonelli, John Burgess, Michael Della Rocca, Herbert Enderton, Bernard Linsky, Heidi Lockwood, Ruth Barcan Marcus, Julien Murzi and Bas van Fraassen. An extra special thanks to Julien Murzi, who as my research assistant in the Fall of 2005 helped me to identify and think more clearly about the famous anonymous referee reports, which are central to the present paper. For discussion and/or assistance I am also grateful to many others, including Scott Berman, Berit Brogaard, Judy Crane, Susan Brower-Toland, David Chalmers, Solomon Feferman, Nick Grifﬁn, Michael Hand, Monte Johnson, Jon Kvanvig, Matthias Lutz-Bachmann, Robert Meyer, Andreas Niederberger, Gualtiero Piccinini, Graham Priest, Krister Segerberg, Wilfried Sieg, Roy Sorensen, Kent Staley, Jim Stone, Neil Tennant, Achille Varzi, Nick Zavediuk, anonymous readers for Oxford University Press, and audience members at the Paciﬁc APA in Portland (March 24, 2006), the Goethe University of Frankfurt (May 15, 2006), the Institute for Logic, Language and Computation at the University of Amsterdam (May 23, 2006), and the Namicona Epistemology Workshop at the University of Copenhagen (August 22, 2006). Thanks also to my department at Saint Louis University for granting time and resources to research and write the paper. ¹ See Williamson (1992: 68) for the proof. What Williamson shows, precisely, is this: (p ↔ ♦Kp) T +K ♦p ↔ p, which says that if, necessarily, a proposition is true just in case it is knowable,

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Fitch’s 1963 paper is an enigma in itself. Although much has been written about its knowability proofs, virtually nothing has been said about Fitch’s understanding of their signiﬁcance. That is because Fitch provides, but never comments on, the ﬁnding. Indeed, the paper appears to change subjects midway. In the ﬁrst half we ﬁnd some knowability proofs and general lessons about concepts that share certain logical properties with the concept of knowledge. In the second half we ﬁnd a logical analysis of a particular concept of value, which happens not to share the relevant logical properties with the concept of knowledge. Why does Fitch develop and include the knowability results in a paper whose primary goal is to articulate a logical analysis of value? It initially appears that the knowability considerations have nothing to do with Fitch’s ﬁnal analysis. The thesis of the present paper is that Fitch’s intent was to pinpoint a disruptive set of logical properties that lend themselves to the trivialization of conditional analyses. Or, at the very least, Fitch included the central theorems to demonstrate a sort of conditional fallacy that threatens, although not irredeemably, against his own analysis of value. If this is right, then Fitch does not take the knowability proofs to be paradoxical, but instead takes them to be a lesson about how intensional operators interact, surprisingly, to thwart the efforts of conditional analyses. Fitch’s demonstration of the knowability proofs may be understood as a logical lesson in how to avoid the so-called ‘‘conditional fallacy’’ in philosophical analysis. My reading of Fitch is based on unpublished papers archived at Yale, Columbia and Princeton. The important documents include a pair of reports from 1945 (Chapter 1 of this volume), in which an anonymous referee conveyed to Fitch the knowability proof. The handwriting of the draft to the editor gives away its author, which is unmistakably Alonzo Church. The subsequent debate between Fitch and Church paints a clearer picture of what Fitch, by 1963, perceived to be the philosophical signiﬁcance of the so-called paradox of knowability. The archival documentation puts us in a position, for the ﬁrst time, to articulate and evaluate a lost chapter in the history and philosophy of logic—the early history of the knowability paradox. T h e 1 9 6 3 Pa p e r : W h a t’s He Bu i l d i n g i n T h e re ? The published literature begins with Fitch’s 1963 paper. Here Fitch investigates intensional operators that are factive and closed under conjunction-elimination. An operator O is factive just when its application implies truth: (Factivity)

(Op → p)

then it follows in modal system T (augmented with minimal epistemic resources) that a proposition is possible just in case it is true.

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The formula says, necessarily, if Op then p. Factive operators include ‘it is true that,’ ‘it is known that,’ ‘it is perceived that,’ and ‘it is necessary that.’ By contrast, ‘it is believed that’ is not factive, since believing p does not require the truth of p. An operator is closed under conjunction-elimination (or is conjunctiondistributive) just when it applies to a conjunction only if it applies to the corresponding conjuncts: (&-E Closure)

(O(p & q) → (Op & Oq) )

Both knowledge and belief are conjunction-distributive in this sense, since knowing/believing a conjunction requires knowing/believing each of the conjuncts. Fitch’s concern primarily is with ‘knows,’ which is both factive and conjunction-distributive.² Fitch’s concern in the ﬁrst half of the paper is only with operators that satisfy these two principles and the theorems in which they ﬁgure. He proves six theorems. Their content is discussed below. The philosophical signiﬁcance of each theorem, if any, I for now leave open, since Fitch did not comment on their signiﬁcance. The ﬁrst two theorems are perfectly general. I paraphrase the ﬁrst: Theorem 1: for any factive propositional operator O that is closed with respect to &-E, ¬♦O(p & ¬Op). Fitch proves here that there is always an un-O-able proposition, when O has the aforementioned logical properties. The proof is well rehearsed in the literature for the case of knowledge. Substituting the knowledge operator K for O gives us a theorem about the unknowability of any Fitch-conjunction, p&¬Kp. The unknowability may be stated this way: ¬♦K (p & ¬Kp). The demonstration follows:³ (1) K ( p & Kp) (&-E Closure) Kp & K Kp (Factivity & trivial logic) K p & Kp (1) K (p & Kp) (Normal Modal Logic) K (p & Kp) At the top of the tree we suppose for reductio that the Fitch-conjunction, p & ¬Kp, is known. By the closure of knowledge under &-E, it follows that each conjunct is known. The third line demonstrates an application of factivity to ² There are few exceptions to the received view that ‘knows’ is both factive and conjunctiondistributive. Robert Nozick (1981), for instance, articulates a concept of knowledge that is not conjunction-distributive. ³ The Genzen–Prawitz notation is preferred throughout for the perspicuity of logical dependencies.

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the right conjunct of the second line. In the face of the ensuing contradiction, we discharge and deny our only assumption. By necessitation and the duality of the modal operators, we conclude with the impossibility of that assumption. Fitch-conjunctions are unknowable! And more generally, conjunctions of the form p & ¬Op are un-O-able, when O is factive and conjunction-distributive. Fitch’s second perfectly general theorem says this: for the aforementioned operators, O, if p is a true proposition that is not O-ed, then p & ¬Op is a true proposition that is un-O-able. Theorem 2: for any factive operator O that is closed under &-E, if p is true but un-O-ed, then that it is an un-O-ed truth is itself un-O-able. (p & ¬Op) → ¬♦O(p & ¬Op). The result follows trivially from Theorem 1. The remainder of the theorems, Theorems 3 through 6, are special cases or consequences of the above perfectly general results. Theorem 3: If an agent a is all-powerful in the sense that anything that is true could have been brought about by a, then everything that is true was brought about by a: ∀p(p → ♦aBp) → ∀p(p → aBp). B is the factive, conjunction-distributive operator ‘brought it about that.’ Theorem 3 follows from Theorem 1, substituting B for the operator variable, O. The next two theorems are the knowability proofs. Theorem 4 is credited by Fitch to an anonymous referee. Theorem 4: for each agent that is not omniscient, there is a true proposition that that agent cannot know: ∃p(p & ¬aKp) → ∃p(p & ¬♦aKp). Theorem 4 is the contrapositive of Theorem 3, replacing the knowledge operator, K , for B. The next theorem is Theorem 5. It or its contrapositive is most often equated with the knowability paradox. It is a modiﬁcation of Theorem 4. Theorem 5: If there is a true proposition which nobody knows (or has known or will know) to be true, then there is a true proposition which nobody can know to be true: ∃p(p & ∀a¬aKp) → ∃p(p & ∀a¬♦aKp). Theorem 5 strengthens both the antecedent and the consequent of Theorem 4. It does this by generalizing over subjects in both places. Theorem 5 is then slightly more interesting when we detach the consequent, since it commits us to the existence of a truth that cannot be known by anyone. When we suppress the quantiﬁers ranging over subjects, as is standardly done for ease of exposition, Theorems 4 and 5 say the same thing—viz., if there is an unknown truth then

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there is an unknowable truth. Although the referee conveyed to Fitch Theorem 4 and thereby informed this entire section of Fitch’s paper, the slightly more interesting Theorem 5 (i.e., the so-called knowability paradox) and the perfectly general theorems are owed, at least in part, to Fitch. Fitch’s ﬁnal result, Theorem 6, just is Theorem 5, replacing ‘knows that’ with ‘proves that’ and stipulating that our propositions p are themselves about proving. Theorem 6: If there is some true proposition about proving that nobody has proved or ever will prove, then there is some true proposition about proving that nobody can prove: ∃p(p & ∀a¬aPp) → ∃p(p & ∀a¬♦aPp), where our propositional variables range over propositions about proving. The set of six theorems in their own right constitute an interesting development in the logic of intensional operators and action, and they play a role in current developments of modal epistemic logic.⁴ Fitch, as we mentioned, does not comment on their signiﬁcance. If the ﬁrst half of Fitch’s 1963 paper is about the logic of un-O-ability, then what is its connection to the apparently unrelated subject that occupies Fitch in the second half of the paper? The second half of the paper is concerned to articulate a logical analysis of an informed-desire theory of value. Informed-desire says, roughly, that something is valuable to a subject just when she would desire it if she had all the relevant information.⁵ Fitch’s ﬁnal analysis in 1963 roughly is this: s values p just when there is a truth q, such that, necessarily, if s knows that q then s strives for p. The logical analysis appears as Deﬁnition 6 : (D6 ) Vp

iff ∃q(q & (Kq → Sp) ).⁶

D6 is the centerpiece of the 1963 paper, but Fitch develops analyses of other propositional operators as well. He offers, for instance, deﬁnitions of ‘knows,’ ‘does,’ ‘can do,’ and ‘desires.’ In each of these cases he employs the strict (causal) conditional in the analysis and ends with considerations about whether or not the main operator (or deﬁniendum) is factive and conjunction-distributive. It is only in the case of his causal deﬁnition of knowledge, D2, that we ﬁnd an ⁴ See, for instance, Rescher (2005) and van Benthem (Chapter 9 of this volume). ⁵ The counterfactual gloss appears in an earlier draft of the paper (1961) and is meant to capture a causal reading. Fitch borrows from the logical analysis of causation found in William Burks (1951). In so doing, Fitch (1961: 6a) explains that the relevant sense of ‘A causes B’ is strict implication in a modal system such as S2 or M. ⁶ Some liberty is taken here with the formalism. Fitch uses ‘C’ for ‘(partially) causes’ rather than the necessary conditional, although it is clear from the text and from the 1961 address that a strict conditional reading is adopted. (See n. 5.) Moreover, there are epicycles in Fitch’s analysis that involve other propositional variables. Not being relevant to the present discussion, they are suppressed.

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operator that is both factive and conjunction-distributive. The main attraction, i.e., the analysis of value, however, is conjunction-distributive but not factive. Importantly, ‘knows’, which, again, is the only factive, conjunction-distributive operator deﬁned in the second half of the paper, ﬁgures in Fitch’s deﬁnition of value. So, a desideratum for understanding Fitch is this: the signiﬁcance of the knowability theorems must carry a lesson about the role played by ‘knows’ in Fitch’s analysis of value. But which lesson? Why in a paper about how to articulate an informed-desire theory is Fitch concerned to prove the knowability results? The question can be answered more carefully once we have uncovered the lost history of the proofs. So we leave this section with the central question, to which we will return. What’s he building in there?

W h o Di s c ove re d Fi t c h’s Pa r a d o x ? Another curiosity of Fitch’s 1963 paper is the identity of the famous anonymous referee, to whom Fitch credits the ﬁrst of the two knowability results. Following Theorem 4, Fitch tells us, This theorem is essentially due to an anonymous referee of an earlier paper, in 1945, that I did not publish. This earlier paper contained some of the ideas of the present paper. (1963: 138, n. 5)

That is all that Fitch says on the matter. The present section reveals more. We ﬁnd that Fitch’s 1945 paper was titled ‘‘A Deﬁnition of Value’’ and submitted to the Journal of Symbolic Logic in January or February of 1945.⁷ And, although so many recent papers mention the anonymous referee (under that description), few have published speculation about his identity. According to Richard Routley (Sylvan), [Robert] Meyer conjectures, what seems to me unlikely, that Anon[ymous] = G¨odel. (1981: 110, n. 12)

Routley’s skepticism is not explained. Why think that G¨odel is an unlikely suspect for authorship of the result? G¨odel was not ofﬁcially an editorial consultant for JSL in 1945. More interestingly, as John Burgess noted to me, it was unlikely that the editors would have asked G¨odel to referee a paper, since his perfectionism would have prevented him from returning a report in a timely manner. As for Robert Meyer, he recently admitted that he does not recall having ever discussed Fitch’s paradox with Routley, but notes that he ‘‘would have been struck by the strong whiff of the G¨odel formula,’’⁸ which says of itself that it is true but unprovable. Meyer is recalling the ﬁrst incompleteness result, which ⁷ The title is mentioned in Nagel (1945b).

⁸ Personal correspondence.

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demonstrates that, for any consistent, sufﬁciently strong theory T in the language of arithmetic, there are truths unprovable in T . For any such theory T , we ﬁnd that there is a sentence p such that p is true but unprovable in T : p & ¬PTp . The resemblance to the anomalous Fitch-conjunction, p & ¬Kp, catches one’s attention here. The G¨odel-conjunction and the Fitch-conjunction are analogous epistemic claims. Both advocate that the truth of some proposition p cannot be established by certain means. An important difference of course lies, ﬁrst, in the self-reference that is indicative of the G¨odel sentence and, second, in the epistemic terminology. For G¨odel the terminology is ‘‘unprovable in T .’’ For Fitch the notion is, less formally but more generally, ‘‘unknowable.’’ G¨odel promises a truth that could never be proven in T . Fitch promises a truth that could never be known by any means. Wolfgang K¨unne (2003: 425, n. 159) brieﬂy considers the hypothesis that G¨odel was the originator of the knowability result but notes that Fitch’s result is ‘‘in one respect more ambitious’’ than G¨odel’s theorem. K¨unne’s suggestion, I believe, is a claim about the relative logical strength of the respective claims to unknowability. G¨odel shows us that there is a truth that cannot be proven in T , but of course this does not entail that the truth could not be proven by some other means. Whereas, via Fitch we may conclude that there is a truth, viz., the Fitch-conjunction, p & ¬Kp, that is unknowable, full stop. On the contingent assumption that p & ¬Kp is true, it does follow by Fitch’s result that p & ¬Kp is an unknowable truth. And so, it follows that it could not be proven in any consistent theory strong enough for arithmetic. The problem with taking the Fitch conclusion to be logically stronger is that the existence of unknowable truths depends on the existence of some ignorance, which arguably is a contingent matter. However, some have contended that the existence of some ignorance is logically necessary.⁹ If it is necessary, then the conclusion of the knowability result is in fact stronger than G¨odel’s ﬁrst incompleteness theorem. The G¨odel-hypothesis is the only candidate in the literature. In the Summer of 2005, however, an altogether different hypothesis emerges. The hypothesis was prompted by found correspondence between the 1945 coeditors of JSL. In a letter dated March 6, 1945 Ernest Nagel updates Alonzo Church:¹⁰ I made a copy of your report on Fitch’s ms. (on the assumption that his receiving your handwritten version would destroy your anonymity) and sent it to him with the statement that his ms. in its present form was not acceptable for publication by the JSL. He replied two days later—I enclose his letter; and yesterday he returned the ms. with another letter ⁹ See, for instance, Routley (1981) and Rescher (2005: Appendix 2). ¹⁰ Thanks to John Burgess for his assistance in searching the Alonzo Church Papers and for identifying this letter. Thanks to Herbert Enderton for bringing the Church Papers to my attention.

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appended. I do not think he has met either of your two fundamental objections—indeed, his reply to the second difﬁculty seems to me to evade the issue rather completely. I am sending you the material for any further comments you may wish to make. (Nagel 1945a)

The letter indicates a number of things. In 1945 Church refereed a paper written by Fitch; the author of the report was anonymous to Fitch; and Fitch’s paper was (at least, at this stage) not being accepted for publication. The evidence is circumstantial, but if this was the paper in question and there were no other referees on the job, then it would seem that Church was the anonymous referee who conveyed the knowability proof to Fitch in 1945. The Nagel letter led me to the Ernest Nagel Papers (Columbia University), where my research assistant, Julien Murzi, very quickly identiﬁed the referee report in October 2005. The document had not previously been identiﬁed. It was composed in Church’s trademark vertical handwriting, and thereby conﬁrmed that Church indeed was the referee. In the excerpt below we ﬁnd the earliest known formulation of the knowability proof. Church writes: it may plausibly be maintained that if a is not omniscient there is always a true proposition which it is empirically impossible for a to know at time t. For let k be a true proposition which is unknown to a at time t, and let k be the proposition that k is true but unknown to a at time t. Then k is true. But it would seem that if a knows k at time t, then a must know k at time t, and must also know that he does not know k at time t. By Def.2, this is a contradiction.¹¹ (1945; Report 1, p. 2)

In sum, if a person a is not omniscient (that is, if there is a truth unknown to a), then there is a truth unknowable to a. It is evident that this result becomes the ﬁrst knowability result, Theorem 4—the very result that Fitch credits to the anonymous referee. It is not surprising that Church was the author of the report and its main proof, which is often taken to be about the logical limits of knowledge. It would be understated to say that Church thought deeply about such matters. He formalized the concept of effective calculability (1936a) and proved the undecidability of ﬁrst-order logic (1936b). Possible inﬂuences on Church’s thought in 1945 include G¨odel’s work from the prior decade and the interactions the two philosophers had in Princeton in the years leading up to 1945. In JSL William Parry (1939: 140) had proved Theorem 22.8: ¬♦¬♦(p → ¬♦¬p), which is equivalent to ¬♦(p & ¬p) —the core of Theorem 4, replacing all occurrences of with K . There was also Moore’s Paradox (1942: 543), which reveals the peculiarities of propositions of the form, ‘p but I don’t believe p.’¹² The critical documents found in the Nagel Papers actually include two referee reports, which we will label chronologically, Reports 1 and 2 (or R1 and R2). ¹¹ Church refers here to Def. 2, which appears to be Fitch’s deﬁnition of knowledge. As can be seen from the context, Church employs it to exploit the factivity of knowledge. ¹² Thanks to Roy Sorensen for information about this related problem and its earliest source.

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They are included in their entirety as the ﬁrst chapter of this volume. Report 2 was written by the same hand as Report 1. The second, but not the ﬁrst, report was signed by the author. The originals were apparently seen only by Nagel in his capacity as editor, who typeset them to preserve Church’s anonymity. In large part they consist of a series of trivialization arguments against Fitch’s analysis of value. Some of these arguments utilize the knowability result quoted above. The next section evaluates the ideas central to Report 1. T h e Fi r s t Re f e re e Re p o r t

A trivialization of Fitch’s analysis The knowability result was developed by Church to trivialize Fitch’s analysis of ‘a values p at time t,’ which is referred to in Report 1 as ‘Def.3.’ No statement of Def.3 appears in the report, but the context allows us to reconstruct the deﬁnition as follows: (Def. 3)Vp iff ∃q(q & (Kq → Dp) ) The formula tells us that it is valued (or is valuable to a subject) that p just when there is a truth q, such that knowing q necessarily implies desiring p.¹³ The analysis tells us, for instance, that it is valuable to me that I take my migraine medication if it is true that the medication will stop the pain and knowing that it stops the pain leads me to desire that I take the medication. Church’s criticism of the analysis begins with the acknowledgment that we are non-omniscient—that there are some truths p that an agent a does not know (at time t). Formally, for some p, it is true that: (1) p & ¬Kp. By the familiar result it is impossible for a to know both that p is true and that p is not known by a (at t). (2) ¬♦K (p & ¬Kp). Conditionals with impossible antecedents are necessarily true. So, from (2), it follows that: (3) (K (p & ¬Kp) → r ) , where r is any proposition you like. ¹³ From Church’s report we learn that Fitch employs a notion of ‘empirical necessitation’ rather than ‘strict implication’ in the right-hand side of the deﬁnition and distinguishes between the two notions throughout his paper. Fitch’s strict implication is Lewis and Langford’s. The modality is governed by S2. Fitch’s empirical necessitation, by contrast, appears to be a weaker notion. At the very least, Fitch’s Th. 1 appears to be a principle stating that strict implication entails empirical necessitation. See, for instance, the application of Th. 1 in Report 2, page 1. I suppress Fitch’s distinction between strict implication and empirical necessitation throughout. The issues here do not hang on the decision.

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Let r be ‘It is desired by a that s’ or just ‘Ds’. Then: (4) (K (p & ¬Kp) → Ds). Hence, from (1) and (4) it follows that there is a truth q such that knowing q strictly implies desiring s: (5) ∃q(q & (Kq → Ds) ). Therefore, by Def.3, s is valued: (6) Vs. And since s was arbitrarily chosen, it therefore follows that everything is valued. In sum, if there is a truth unknown to a then a values everything. At a glance the result is this, where q is the true conjunction p & ¬Kp. q ∃q(q &

¬ Kq (Kq →Ds) (Kq →Ds)) (Def.3) Vs

Church’s argument illustrates the mistake in Fitch’s analysis. The mistake tends to occur when we deﬁne concepts in conditional terms. This, the so-called ‘‘conditional fallacy,’’ is not unrelated to the paradoxes of implication. Classical conditionals behave strangely when their antecedents are false or impossible. More speciﬁcally, but without attempting to characterize all and only cases of the fallacy, the conditional fallacy is a mistake that occurs just when the antecedent of the conditional deﬁniens is not always logically independent of the deﬁniendum. That is, instances of the analysis include cases where the deﬁniendum contradicts, entails or is entailed by the antecedent of the conditional deﬁniens. Such conditions will sometimes effect a surprising disparity in truth value between the deﬁniens and the deﬁniendum.¹⁴ This is what gets Fitch’s analysis into trouble. The conditional embedded in his deﬁnition, Vp iff ∃q(q & (Kq → Dp) ), has instances where the antecedent, Kq, is not logically independent of the deﬁniendum, Vp. And that is because there are instances of the antecedent that are logically impossible and so entail any proposition whatsoever. A fortiori, such instances necessarily imply the deﬁniendum. The mistake in Fitch’s analysis results from his failure to detect the logical anomaly of unknowable truth. For the existence of unknowable truth is the logical phenomenon responsible for the surprising trivialization of Fitch’s analysis. Fitch later takes to heart this lesson of philosophical analysis. The lesson will play a critical role in Fitch’s 1963 paper. ¹⁴ This understanding of the fallacy is informed by Shope (1978) and Wright (2000).

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In the second referee report Church considers blocking the above trivialization by appealing to Russell’s theory of types. In so doing Church foreshadows Linsky (Ch. 11, this volume) and Hart (Ch. 19, this volume). However, Church dismisses the option as contrary to Fitch’s purposes, since an employment of the theory of types would invalidate closure principles central to Fitch’s paper. As we will see in the next section, Church has independent reason for rejecting these closure principles.

Closure principles for knowledge and belief In the ﬁrst report Church foreshadows what he takes to be Fitch’s only good defense against the trivialization argument, and that is to question the validity of closure principles for propositional attitude operators. Speciﬁcally, he denies that there is a ‘‘law according to which one who believes a proposition must believe all its logical consequences’’ (Report 1: 2). Church questions here the validity of the principle that belief is closed under logical consequence. His intention, though, is to question the justiﬁcation for the principle that belief is closed under conjunction-elimination. Church writes: To be sure, one who believes a proposition without believing its more obvious logical consequences is a fool; but it is an empirical fact that there are fools. It is even possible that there might be so great a fool as to believe the conjunction of two propositions without believing either of the two propositions; at least an empirical law to the contrary would seem to be open to doubt. On this ground it is empirically possible that a might believe k at time t without believing k at time t (although k is a conjunction one of whose terms is k).¹⁵

Church denies that belief is necessarily closed under conjunction elimination. It is unclear, however, how this is supposed to help Fitch. The trivialization argument never utilizes a closure principle for belief. It utilizes, instead, a closure principle for knowledge. And, of course, it would be a fallacy of division to suppose that the concept of belief has a certain logical property P (e.g., closure under logical consequence) just because (1) belief is a component of knowledge and (2) knowledge has P.¹⁶ An alternative, non-fallacious reading of Church’s passage is that he simply means ‘‘knowledge’’ when he speaks of belief. In that case Church simply questions whether knowledge is closed under conjunction-elimination. However, this limited closure principle is harder to reject than the more general principle that knowledge is closed under logical consequence. That is because ‘‘knowing p’’ ¹⁵ R1: 2–3. ¹⁶ More recent instances of this very closure-fallacy in epistemology are detected by Ted Warﬁeld (2004).

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and ‘‘knowing q’’ are implicit in ‘‘knowing p & q.’’ So it is doubtful that Church has offered Fitch a ‘‘good defense’’ of the trivialization argument. But let us suppose that he has and consider one further point about the denial of closure in this context. Directly after his articulation of the problems with taking belief to be closed under conjunction-elimination and offering the defense on Fitch’s behalf, Church makes the following claim: Unfortunately this defense compels Fitch to abandon his Ax. 1. And, what is more serious, it lights the way to a second and opposite objection to Def. 3. If there is no empirical law according to which one who believes a proposition must believe its logical consequences, it would seem that by the same token there is no empirical law according to which a person’s desires must be in reasonable accord with that person’s beliefs. (R1: p. 3)

The consequence of rejecting closure principles for belief, according to Church, is that it invites a skepticism about other principles that express necessary connections between propositional attitudes, in particular between knowledge and desire. And without such necessary connections, the right-hand side of Fitch’s analysis is never satisﬁed, and so, the theory is trivialized in the opposite direction. Nothing is of value to anyone! Church’s point is overstated. Surely there may be laws about our propositional attitudes, even if belief/knowledge is not closed under logical consequence more generally. That is, for all we know, some principles other than the unrestricted closure principles justify necessary connections between our propositional attitudes. Fitch, in fact, gives the following example in reply to Church: necessarily, if it is known that I desire that p, then I desire that p. So sometimes knowledge does necessitate desire. The example is an instance of the principle that knowledge necessarily implies truth. With it Fitch proves trivially that there are some necessary connections between knowledge and desire. Fitch’s example is cited in Church’s second referee report (R2: 4). We learn from Nagel’s letter of March 6 that Fitch replied to the ﬁrst referee report with two letters and a revised manuscript. These documents, like Fitch’s initial submission, are yet to be found. In any case Nagel was unimpressed by them. Recall Nagel’s remark to Church: ‘‘I do not think [Fitch] has met either of your two fundamental objections—indeed, his reply to the second difﬁculty seems to me to evade the issue rather completely. I am sending you the material for any further comments you may wish to make.’’ The second difﬁculty, recall, was Church’s animadversions to closure principles and other ‘‘laws’’ relating propositional attitudes. I do not see that Fitch’s point—about factively knowing that one desires something—evades Church’s difﬁculty ‘‘completely,’’ but I will not pursue the issue further. Fitch’s very revealing reply to the other difﬁculty, i.e., the trivialization argument from unknowable truth, is summarized in the second referee report. We turn to that document next.

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T h e Se c o n d Re p o r t

Fitch’s Cartesian restriction strategy In reply to Nagel’s March 6 letter, Church issued a second referee report. From it we learn of Fitch’s reactions to the trivialization argument in Report 1. Fitch’s analysis said that it is valuable that p just in case there are truths that would, if known, necessitate the desire that p. Church showed us that, vacuously, there will be such truths, since there are truths that it is impossible to know. The natural reply is to restrict the class of truths to those that it is possible to know. Presumably it is only the knowable truths that should ﬁgure in causal relations between knowledge and desire. In reply to the ﬁrst report, Fitch endorses this insight by offering an alternative restricted theory of value, Def. 3R: (Def. 3R)

Vp iff

∃q(q & ♦Kq & (Kq → Sp) )¹⁷

The restricted analysis says that something p is of value to a subject a just when there is some knowable truth q that would, if known, necessitate a’s desiring that p. Let us call this ‘Fitch’s Cartesian restriction strategy,’ because it foreshadows Neil Tennant (1997), where the restriction is proposed under that name to block the knowability paradox. Tennant deﬁnes a Cartesian proposition p as one for which Kp is not provably inconsistent. Tennant (2001) considers versions of the restriction in terms of what it is metaphysically possible to know. The Cartesian restriction on the relevant class of truths blocks the problematic unknowable truth, ‘p & ¬Kp,’ from consideration. The fact that knowledge of it vacuously implies an arbitrary proposition becomes inconsequential.

Church’s objection to the Cartesian restriction strategy In the second report Church announces that ‘‘a reductio ad absurdum of Def 3R is possible along the same lines as that I have given for Def 3.’’ Church claims there is a Cartesian truth that trivializes Fitch’s restricted analysis. He begins by noting that, for some p, p is an unknown truth. So (i) Dp ∨ (p & ¬Kp) is true, for an arbitrary proposition p . That is, (i) follows from our nonomniscience. After all, if p & ¬Kp is true, then so is the weaker claim, Dp ∨ (p & ¬Kp). Church goes on to argue that: (ii) Proposition (i) is Cartesian. ¹⁷ The amended analysis appears in Report 2: 2. The above formulation of Fitch’s Def. 3R differs from Church’s in that I substitute ♦ for ‘EP’, which reads, ‘it is empirically possible that.’ Also, I continue to replace ‘EN’ or ‘empirically necessitates’ with the necessary material conditional and drop the variables ranging over subjects.

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We shall evaluate this and the next premise in a moment. The ﬁnal premise is that knowing proposition (i) strictly implies the desire that p : (iii) (K (Dp ∨ (p & ¬Kp) ) → Dp ) If premises (i), (ii) and (iii) are all correct, then it follows that there is a Cartesian truth q such that, necessarily, if q is known then p is desired. By Fitch’s Cartesian restricted theory of value, Def. 3R, it would follow that p is valuable, for arbitrary p . We may generalize. If an agent is non-omniscient, then everything whatsoever is valuable to her! Here we consider the premises of Church’s argument. Premise (i) is trivial. If p & ¬Kp is true for some p, then so is Dp ∨ (p & ¬Kp), by disjunctionintroduction. What about premise (ii)? It says that the awkward disjunction, given by (i), is Cartesian, i.e., can be known. Church begins his defense of this premise with the reasoning that any desire is possible (Report 2: pp. 2–3): (a) for any p , ♦Dp . But then knowledge of that desire is possible: (b) for any p , ♦K (Dp ). And so, by the closure of knowledge under disjunction-introduction: (c) for any p , ♦K (Dp ∨ (p & ¬Kp)). Church defends premise (a) by noting that anything, even one’s instant death, can be desired, since it is possible to be insane or in a position less fortunate than one brought about by instant death. By similar reasoning, we might argue further that even contradictions can be desired. So an arbitrary proposition can be desired. But how does Church get from premise (a) to premise (b)? He seems to be assuming that, necessarily, any possible desire is a knowable desire. That is, if it is possible for one to desire that p then it is possible for one to know that one desires that p . Note that there are some implicit principles being invoked. We uncover them by asking what it takes to justify the principle that any possible desire is a knowable desire? Perhaps Church believes that, necessarily, any desire can be known. So, necessarily, if p is desired, then it is possible to know that p is desired: (Dp → ♦K (Dp ) ) On this reading, Church assumes an unrestricted knowability principle about desire. Notice that this is not sufﬁcient to license the move from line (a) to line (b). For the assumption that it is possible to desire p , together with the above principle, by minimal normal modal reasoning, entails only that it is possible that it is possible that Dp is known: ♦♦K (Dp ).

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The S4 axiom is then needed to reduce this to ♦K (Dp ). So if the operant notion of possibility satisﬁes the S4 axiom and in fact, necessarily, any desire is knowable, then it follows that, necessarily, any possible desire is a knowable desire. With this latter principle in hand, premise (b) does in fact follow from premise (a). Premise (c) then follows from premise (b) on the assumption that knowledge is closed under disjunction-introduction. Kp K (p ∨ q) The principle, Church tells us, ‘‘seems to be entirely in the spirit of [Fitch’s] Th. 3.’’ (Report 2: 3) Th. 3 we may hypothesize to be the principle stating that knowledge is closed under conjunction-elimination. Church puts these principles on a logical par. Presumably he is onto the fact that both are instances of the principle that knowledge is closed under obvious logical consequence. There is, however, the objection that these two closure principles are not on a par. ‘a knows both that p and q’ and ‘a knows p and a knows q’ are implicit in one another. Arguably, they say the same thing. Such gives us reason to think that knowledge is closed under conjunction-elimination, despite the problems with thinking that knowledge is closed more generally under logical consequence. By contrast, ‘a knows p’ and ‘a knows p ∨ q’ are not implicit in one another. The latter is not implicit in the former, since q may embed concepts that are not grasped by one who understands ‘a knows p’. So it is not decisive that knowledge should be closed under disjunction-introduction, even if it is closed under conjunction-elimination. Now Fitch never commits himself to the S4 axiom, and need not be committed to the closure of knowledge under disjunction-introduction, even if he does accept its closure under conjunction-elimination. So Church’s argument for premise (ii) is not decisive. His logical assumptions are not trivial.¹⁸ We turn to Church’s justiﬁcation for premise (iii). How does Church prove that (K (Dp ∨ (p & ¬Kp) ) → Dp )? The reasoning here is troubling. It can be found on page 3 of Report 2. Church notes that, by the factivity of knowledge, knowing Dp ∨ (p & ¬Kp) entails Dp ∨ (p & ¬Kp). And further that each of these disjuncts implies Dp . So, by proof-by-cases, Dp . Therefore, if Dp ∨ (p & ¬Kp) is known, then Dp . Resting on no contingent assumptions, Dp ∨ (p & ¬Kp) necessarily implies Dp . Here is the reasoning at a glance: ¹⁸ Incidentally, there is a more modest defense of premise (ii). That is, it is metaphysically possible to know premise (i) for the following reason. It is possible to know Dp , recognize that Dp entails Dp ∨ (p & ¬Kp), and thereby come to know Dp ∨ (p & ¬Kp). If this is right, then it is after all possible to know Dp ∨ (p & ¬Kp). That is, premise (i) is Cartesian.

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Joe Salerno K (Dp ∨ (p &¬Kp)) Dp ∨ (p & ¬Kp)

(2) (1)

Dp Dp (2) K (Dp ∨ (p &¬Kp)) →Dp ( K (Dp ∨ (p &¬K p))→Dp )

(1) p &¬K p Dp (1)

If this is correct then there is a Cartesian truth, such that knowing it necessitates desiring p , for any proposition p . By Fitch’s Cartesian restricted theory of value, therefore, p is valued. The theory trivializes. If this is Church’s argument, then we have to reject it. Something went wrong in the proof-by-cases. The right disjunct p & ¬Kp does not imply Dp . Church may be confusing the proposition p & ¬Kp with K (p & ¬Kp). The latter strictly implies everything, since it is impossible. So obviously K (Dp ∨ (p & ¬Kp) ) strictly implies Dp . Perhaps that is what Church intended, and the mistake can be chalked up to a misprint. However, with the supposition of a misprint in the formulation of Church’s proof-by-cases one must make corresponding adjustments to the ﬁrst part of Church’s argument. The truth that must be shown to be Cartesian is now Dp ∨ K (p & ¬Kp). It is Cartesian. It is knowable because its left disjunct is. But it is not true. Or at least it is not true for an arbitrary desire, Dp . Therefore, the trivialization argument comes apart; it fails against Fitch’s Cartesian restricted theory of value.¹⁹ Other items that appear in the second referee report include (1) a more formal (Lewis and Langford style) proof of the central knowability result that appeared in the ﬁrst report; (2) some mention of the similarity of the trivialization arguments to the liar and set-theoretic paradoxes and the standard devices for resolving them; (3) a brief mention of a problem of accepting the factivity of knowledge while embracing a theory of types; (4) further discussion of Fitch’s concept of empirical necessity; and (5) some counterexamples to Fitch’s theory of value that do not hinge on the knowability theorems. I will not comment on these items. Fitch does not seem to have directly addressed Church’s ﬁnal trivialization argument against the Cartesian restricted theory. In Nagel’s last letter to Church ¹⁹ Jim Stone (in personal correspondence) constructs an argument for premise (iii) in Church’s spirit. It presupposes the transparency of desire (i.e., that all desires are known, Dp → K (Dp), and all failures to desire are known, ¬Dp → K (¬Dp) ). It also presupposes that knowledge is closed under disjunctive syllogism. It goes like this. Suppose K (Dp ∨ (p & ¬Kp)), and suppose for reductio that ¬Dp . By the transparency of desire, K (¬Dp ). Since a disjunction and the negation of the left disjunct are both known, it follows, by the closure of knowlege under disjunctive syllogism, that the right disjunct is known—giving K (p & ¬Kp). But that is impossible. So by classical reductio, Dp . Hence, by conditional proof, K (Dp ∨ (p & ¬Kp)) → Dp . So a defense of premises (iii) may be given and Church’s master argument may be rehabilitated, although the text does not warrant crediting this argument to Church. An even more modest master argument against Fitch’s restricted analysis is formulated in the ﬁnal section of this paper.

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on the matter (April 13, 1945), we learn that Fitch has withdrawn his paper owing to ‘‘a defect in my deﬁnition of value’’ and because ‘‘the paper should be rewritten anyhow.’’

Fi t c h i n t h e Si x t i e s The 1945 Church–Fitch debate helps to explain some things about the 1963 paper. One question is about the intended signiﬁcance of the knowability results. Why does Fitch include them? Consider again the knowability theorems, which say, roughly, that there is an unknowable truth if there is an unknown truth. Fitch presents them in passing but does not comment on their signiﬁcance. Of course, that he demonstates the proofs without comment indicates that he takes them to be valid and not paradoxical. Moreover, it is obvious that there are unknown truths, and so, by the relevant theorems, it would seem that we are meant to recognize the existence of unknowable truths. The insight is intrinsically interesting, but the question regards its role in the paper. One interpretation is that Fitch is offering a refutation of veriﬁcationism, the thesis that all meaningful statements (and so, all truths) are veriﬁable. Indeed, this is how the early literature interprets Fitch.²⁰ To be consistent, this reading requires us to argue, analogously, that there are implicit conclusions that Fitch wishes us to draw from the other theorems. Recall that Theorem 3 shows that if there is an omnipotent being, then he has in fact done everything. Presumably, Fitch would expect us to conclude from this that there is no omnipotent being (or that he is not supremely good, or that there is no free will, or something of the sort). However this is an unlikely reading of Fitch’s intent, as it marks the theorems, including the knowability proofs, as a curious tangent from the paper’s primary goal. The 1963 paper is not a defense of any metaphysical position, not even a defense of the informed-desire theory of value. Rather, it aims to articulate the logical content of that theory. It would be odd, in a paper with that purpose, for Fitch to prove the absurdity of veriﬁcationism or disprove the existence of God. More to the point, this interpretation of the theorems tells us nothing about the factive, conjunction-distributive role played by ‘knows’ in Fitch’s analysis of ‘value.’ So it would seem that the key to understanding Fitch lies elsewhere. ‘Knows’, unlike the other concepts that Fitch deﬁnes in the second half of the 1963 paper (including the concept of value), is both factive and conjunctiondistributive. For this reason it gives rise to the existence of unknowable truth. Fitch wishes us to recognize the existence of unknowable truths for logical, not metaphysical reasons. Unknowable truth is the hallmark of the kind of trivialization that Fitch wishes to avoid in 1963—the very kind of trivialization ²⁰ See, for instance, Hart and McGinn (1976); Hart (1979); Mackie (1980); and Routley (1981).

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that Church wielded against him in 1945. Through the 1945 exchange Fitch recognizes that conditional analyses harbor grave pitfalls. When the dominant propositional operator of the antecedent of a conditional deﬁnition is factive and conjunction-distributive, then there will be an instance of the conditional analysis whose antecedent is impossible. But then the antecedent will not be logically independent of the deﬁniendum (whatever it is), and consequently, trivialization threatens. For the special case, the moral of the knowability theorems, is then to beware of this fallacy in the conditional understanding of the informed-desire theory. My explanation of why Fitch included the knowability theorems in the paper is supported by the fact that Fitch does in fact heed the warning by protecting against the fallacy. Directly following the formal articulation of his 1963 analysis of value, Vp iff ∃q(q & (Kq → Sp) ), Fitch explains that to avoid absurdity, q may be regarded as containing all the knowable relevant information. (Ch. 2, this volume: 28)²¹

We see here the very Cartesian restriction that Fitch attempted in 1945, although Fitch includes it here without much remark. It appears then that the reason that the knowability theorems are included in the ﬁrst half of the paper is to explain the need for the Cartesian restriction that emends the ﬁnal analysis in the second half. Recall that in 1945 Fitch attempted in this way to restrict his analysis in reply to Church’s ﬁrst referee report. But the attempt was met with an overwhelmingly negative second report. At the time Fitch decided to withdraw his paper, even though the second report, as we have seen, was critically ﬂawed. However, Fitch must have recognized the errors of Church’s second report. For by 1963 he was perfectly happy with a Cartesian-restriction in his ﬁnal analysis of value. Why did Fitch wait so long to publish the analysis? I believe that skepticism about non-trivial necessary connections between knowledge and desire kept Fitch sufﬁciently worried, and that it was not until Burks (1951) offered a logical analysis of causal conditionals that Fitch believed himself to have the logical resources to explain the relevant modal relation. This is supported by the fact that Fitch makes extensive use of Burk’s analysis in both his presidential address to the Association for Symbolic Logic (1961) and his 1963 publication. We have seen that the role of the knowability theorems in Fitch’s paper do carry lessons about the role played by ‘knows’ in Fitch’s ﬁnal analysis of value. These are the aforementioned lessons about whether and how to protect against the conditional fallacy. In 1963 what Fitch is building in there is an analysis of value that is sheltered from this fallacy. With this interpretation of the knowability proofs, we ﬁnd an account of the early history and initial, perceived signiﬁcance of the so-called knowability paradox. ²¹ See other mentions of the restriction on page 27.

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A g a i n s t Fi t c h’s C a r t e s i a n Re s t r i c t i o n Church’s result shows us that there are unknowable truths, and that these truths serve to trivialize Fitch’s 1945 theory of value. Fitch responds by restricting his theory to truths that it is possible to know. The problem with the restriction, as Church attempted to show in his second report, is that there are knowable truths that trivialize the restricted theory. The trick is to come up with a knowable truth q, such that q is weaker than the unknowable truth, p & ¬Kp, and such that q trivializes the theory. Church’s choice of such a proposition was Dp ∨ (p & ¬Kp). It is knowable, but as I argued it fails to do the job that Church set for it. And that is because knowing that proposition does not necessitate an arbitrary desire. Church fails to trivialize the restricted theory, but he was right in thinking it can be done. There are in fact knowable truths weaker than p & ¬Kp that serve to trivialize Fitch’s restricted theory of value. They are truths of the following form: (1) p & (Kp → q) which says both that p and that knowing p implies q. That will be true whenever there is an unknown truth—i.e., whenever: (2) p & ¬Kp is true. So whenever p &¬Kp is true for some sentence p, p & (Kp → q) will be true for an arbitrary sentence q. And that is because a false proposition (in this case Kp) materially implies any proposition.²² Therefore, for some proposition p, the following is true: (3) p & (Kp → Sq). Moreover, knowledge of its truth necessarily implies that q is strived for: (4) (K (p & (Kp → Sq) ) ) → Sq.²³ And ﬁnally, p & (Kp → Sq) is Cartesian. That is, it is possible to know p & (Kp → Sq), even though it is not logically possible to know the logically stronger proposition, p &¬Kp. In sum, if there is an unknown truth p, then p & (Kp → Sq) is a knowable truth, and, necessarily, knowing it necessitates striving for q. But then there is a knowable truth, p, that satisﬁes (Kp → Sq). Consequently, the right-hand ²² Such consequences of the Fitch-conjunction are used by Williamson (2000b: 110–12) and Brogaard and Salerno (2006: 266–7) against Tennant’s (1997) Cartesian restriction strategy. Interesting discussion also appears in Rosenkranz (2004), although his prescription is for the Cartesian restriction strategist to reject normal modal logic. ²³ The proof of (4) is straightforward. It requires that K be factive and closed under conjunctionelimination.

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side of Fitch’s theory of value is vacuously true. It follows by the restricted theory of value that q is valued, for arbritrary q. The theory collapses! The formalism below more perspicuously demonstrates the collapse of Fitch’s 1963 theory of value. After all, p & (Kp → Sq) is a knowable proposition; knowing it necessarily implies Sq; and it is true if p &¬Kp is true, for some p. p & ¬K p p & ( K p → Sq) (p & ( K p → Sq) & ∃q(q &

( K (p & (K p → Sq)) → Sq) (K (p & (K p → Sq)) → Sq) (K q → Sq)) (D6) Vq ∀qV q

Although the above argument trivializes Fitch’s theory of value, it does not uncover a conditional fallacy. The conditional’s antecedent K (p & (Kp → Sq) is logically independent of the deﬁniendum Vq. There is, however, a conditional fallacy that the 1963 analysis perpetrates. This is demonstrated by a different version of the above argument. Just replace all occurrences of the formula p & (Kp → Sq) in the above proof with ( ∗ )p & (Kp → (Vq &Sq) ). (∗ ) appears to be Cartesian; knowing it is not logically independent of the deﬁniendum, Vq; knowing (*) strictly implies Sq; and it succeeds in trivializing the theory, since Sq and its embedded proposition, q, were chosen arbitrarily. There are instances of Fitch’s deﬁnition of value where the antecedent of the relevant conditional is not logically independent of the deﬁniendum. If a lesson of Church’s 1945 result is not to commit the conditional fallacy in philosophical analysis, then by 1963 Fitch had appreciated the danger but his analysis had not satisfactorily protected against it.

Pa r t I I Dum m e t t’s C o n s t r u c t i v i s m

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4 Fitch’s Paradox of Knowability Michael Dummett

Fitch’s paradox of knowability runs as follows. The constructivist or (as I have been calling him) justiﬁcationist believes that every true statement is capable of being known to be true. This may be symbolized by: (A) p → ♦ Kp, where ‘K’ means ‘is, has been or will be known by somebody’. We should normally think that there are many true statements that will never be known to be true; favourite examples concern the parity of the number of a large set of objects. Replacing ‘p’ by ‘p & ¬Kp’ we obtain: (B)

(p & ¬Kp) → ♦K(p & ¬Kp).

But K(p & ¬Kp) is contradictory and hence impossible; hence ¬♦K(p & ¬Kp), and accordingly: (C) ¬(p & ¬Kp), whence no true statement will never be known: (D) p → ¬¬Kp, which by classical logic implies that every true statement will eventually be known: (E) p → Kp, contrary to our strong intuition. It follows that principle (A) cannot be maintained, and hence that the constructivist/justiﬁcationist is wrong. That is essentially Fitch’s reasoning. What is wrong with it? The fundamental mistake is that the justiﬁcationist does not accept classical logic. He is happy to accept principle (D), provided that the logical constants are understood in accordance with intuitionistic rather than classical logic. In fact, in line with the inspired suggestion of Bernhard Weiss, he will prefer (D) to (A) as a formalization of his view concerning the relation of truth to knowledge. When A is a mathematical statement, ‘¬A’ is usually explained as meaning ‘Given a proof

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of A, we could derive a contradiction’, where ‘we could derive’ is to be glossed by ‘using anything we have already proved’. It follows that ‘¬A’ may be read as ‘It is in principle impossible to prove A’. In a more general context, ‘¬A’ may be read as ‘It is in principle impossible for us to be in a position to assert that A’ or ‘There is an obstacle in principle to our being able to assert that A’, where ‘in principle’ is to be glossed by ‘in the light of all that we already know’. Hence ‘¬¬A’ means ‘There is an obstacle in principle to our being able to deny that A’, where denying that A is asserting that ¬A. It follows that ‘¬¬KA’ means ‘There is an obstacle in principle to our being able to deny that A will ever be known’, in other words ‘The possibility that A will come to be known always remains open’. That this holds good for every true proposition A is precisely what the justiﬁcationist believes. This is the principle expressed by (D); and (D) captures the relation which the justiﬁcationist believes to obtain between truth and knowledge. He is not concerned to deny that there may be true propositions which will in fact never be known. This is something that cannot be expressed by means of intuitionistic logical constants. We cannot capture this proposition by introducing a binary operator Kn (p, t ) to mean ‘it is known at time t that p’. Intuitionistically interpreted, ‘∀t ¬Kn (A, t )’ holds good only if there is a general reason why it cannot be known at each time t that A, that is, precisely if ¬KA. What the justiﬁcationist wants to deny is not that there are true propositions that will always happen to remain unknown, but that there are true propositions that are intrinsically unknowable: for instance one stating the exact mass in grams, given by a real number, of the spanner I am holding in my hand. With (D) interpreted intuitionistically and adopted in place of (A) as the principle connecting truth and knowledge, there is now no paradox. (C) indeed will hold good; but when understood intuitionistically, it is no longer contrary to intuition.

5 The Paradox of Knowability and the Mapping Objection Stig Alstrup Rasmussen

I According to Timothy Williamson, the Paradox of Knowability—or Fitch’s Paradox (Fitch 1963)—is not really a paradox from any point of view. Despite what is suggested by a cursory glance at the reasoning involved, the argument presents no genuine paradox to either semantical realist or semantical anti-realist (Williamson 1982, 1992, and 2000a, Ch. 12). I agree. However, Williamson’s various treatments of the alleged paradox leave open several issues. Furthermore, at least some alternative proposals deserve serious consideration. Originally, the paradox-generating piece of reasoning was launched—and later revived—as a refutation of any form of idealism adhering to the thesis that all truths are in principle knowable. The supposedly suitable formal rendering of the thesis is (1)

(∀p) (p → ♦Kp).

The (second-order) quantiﬁer ranges over propositions, the modal operator indicates possibility, and ‘K’ is rendered as (although the reading of ‘K’ turns out to be one of the trouble-spots): (K) Kp, if and only if it is currently known that p. Furthermore, we are not concerned with the bearer of any purported bit of knowledge. This issue is relevant to some questions, but these will not be ours. Thesis (1) is supposedly characteristically held by a Dummettian anti-realist. The realist, on the other hand, supposedly balks at (1), if (1) is thought of as holding unrestrictedly. Now Dummett nowhere seems to saddle his anti-realist with precisely (1). However, his various characterizations of the semantical anti-realist have usually been taken to imply a commitment, on the part of the anti-realist,

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to some such thesis.¹ In any case, Dummett’s Victor (Dummett 2001) or Neil Tennant’s ‘naive’ anti-realist (Tennant 2002: 135) are certainly thus committed; and, e.g., Crispin Wright will unavoidably be taken to underwrite a similar view, on behalf of anti-realism (Wright 1992: Chs 1–2). Indeed, (1) can seem to hold trivially, once truth is equated with warranted assertibility, or the like. This was among our (mistaken) contentions in Rasmussen and Ravnkilde (1982: 436–7 n. 77). It was even earlier that most of the above (rightly) thought it worth while to revive the thesis and the attendant putative paradox (Hart and McGinn 1976 and Hart 1979). The set of issues involved in the paradox has generated a good deal of discussion. Overall, the argument has always been thought of as an attempted reductio ad absurdum of, speciﬁcally, anti-realism-cum-idealism, albeit often as an unsuccessful one. Let us kick off by agreeing on what the argument is. Plausibly, there are facts not currently known. (2)

(∃p) (p & ¬Kp).

Also, as Plato is often credited with having pointed out, knowledge is factive. The usual formal rendering of this is: (F)

(∀p) (Kp → p).

And knowledge distributes over conjunctions: (D)

(∀p) (∀q) (K(p & q) → (Kp & Kq) ).

Now to the Basic Version (BA) of the troublesome derivation, which purports to show that (1), (2), (F) and (D) form an inconsistent set (We use Lemmon-style natural deduction, as presented in Read and Wright 1994. I ﬁrst published this version in Rasmussen 1997): 1 2 3 1 1,3 6 6 6 6 6 6

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

(∀p) (p → ♦Kp) (∃p) (p & ¬Kp) q & ¬Kq (q & ¬Kq) → ♦K(q & ¬Kq) ♦K(q & ¬Kq) K(q & ¬Kq) Kq & K¬Kq Kq K¬Kq ¬Kq Kq & ¬Kq

A (anti-realism?) A (trivial?) A 1 ∀E 3,4 →E A 6 SI (D) 7 &E 7 &E 9 SI (F) 8,10 &I

¹ Dummett’s latest statement of his ofﬁcial position concerning anti-realist truth known to me is in Dummett 2005, especially p. 673.

The Paradox of Knowability and the Mapping Objection 1,3 1,3 1,2

(12) (13) (14)

Kq & ¬Kq ⊥ ⊥

5,6,11 12 2,3,13

55 ♦E ¬E ∃E

Assumptions (1) and (2) are framed in a manner consonant with Fitch’s original formulation. The argument (BA) works classically, as well as intuitionistically. Often, the adherent to classical logic is presented as blaming the contradiction on (2): 1 1

(15) (16)

¬(∃p) (p & Kp) (∀p) (p → Kp)

2, 14 15

¬I SI

And therefore the anti-realist is supposedly in the position of having to conclude from his cherished thesis (1) that all truths are currently known; which is patently false. So, the thought is, (1) is false, and semantical anti-realism untenable. The move from (15) to (16) is, of course, intuitionistically invalid. This matters, as pointed out in Rasmussen and Ravnkilde (1982: 436–7 n. 77) and Williamson (1982), since the anti-realist is presumably committed to a logic no stronger than the intuitionistic one. Interestingly, the most compelling kind of proof of this appeals to the anti-realist’s commitment to (1) (Wright 2001: 66 n. 24).² So, we may note in passing that a principle very much like (1) seems to be ﬁrmly entrenched within the anti-realist way of thinking. However, the logical revisionism seemingly inherent in anti-realism merely exempts its adherent from endorsing the ﬁnal step in the (extended) argument (BA); and the derivation (1) through (14) may seem bad enough. At least one of (1) and (2) must go. If the derivation is regarded as a reductio of (1) speciﬁcally, then it is tempting to adopt the restriction strategy: the anti-realist can have (1), but only as applied to a suitably restricted set of propositions. Neil Tennant restricts the range of the quantiﬁer to so-called Cartesian propositions, i.e., propositions that admit of coherent anti-realist assertion (Tennant 1997/2002: ch. 8, and 2002). Dummett has proposed a more subtle restriction, according to which (1) can be taken to hold primitively for basic propositions only; whereas its fate in cases involving more complex substitution-instances is up for discussion (Dummett 2001). On the other hand, Bernhard Weiss has put forth a proposal according to which there should at least be no objection to taking the paradox as a reductio of (2), although he tinkers with (1) as well (cf. note 14 below). Williamson, however, takes the paradox as having been not properly stated. One crucial move in his restatements over the years, starting with Williamson 1982, is the introduction ² The argument is to the effect that, given (1), the law of excluded middle entails intuitionistically that all sentences are in principle decidable. A previous, less formal, version of the argument (Wright 1992: 41–3) misﬁres, since it shows only that the decidability in principle of all sentences follows from (1), combined with the universal applicability of bivalence. Cf. my discussion in Rasmussen 2002.

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of an explicit time-index on the epistemic K-operator. This measure has met with a favourable reception by others, as indeed it will in the present paper.

II Before turning to these various approaches, two points deserve mentioning. First, if the above basic version of the paradox is supposed to be effective against anti-realism, then it seems unavoidable that the semantical realist is equally vulnerable. True, the realist will not endorse (1), in general. However, if the quantiﬁer is restricted to, say, effectively decidable sentences of number theory, the case for his having to endorse (1), as thus restricted, is at least as strong as the case for holding that anti-realism resides in the general adherence to (1). At the same time, the realist has no quarrel with (2). On the contrary, to him (2) captures the truism that reality outstrips human knowledge. But if so, the basic version of the paradox is not suitable as a refutation of anti-realism, speciﬁcally. Furthermore, the point tends to reinforce the feeling that perhaps something has gone seriously wrong, probably with both (1) and (2), since the anti-realist is in any case not happy with (2). Second, and importantly for nearly everything in the sequel, once we realize that intuitionistic and classical logic are both in play, the question arises whether the paradox is supposed to be couched in one or the other. Since the derivation itself is valid in both, this might seem not to matter. But this is far from the truth, since the choice of logic has a bearing on the interpretation of the assumptions responsible for the paradox, viz. (1) and (2). Until 1982 it seems to have been presupposed that the logic is classical. But, as pointed out in the above, the anti-realist is likely to insist on an intuitionist construal of the logical constants. Once this is realized, the imputation to the anti-realist of (1) seems puzzling. Suppose attention is restricted to the ﬁeld of mathematics, where truth-values may be assumed to be monotonous.³ Assumption (1) appears to be an attempt to capture that, to the intuitionist, all mathematical truths are provable, in the ³ Famously, the defeasibility of empirical propositions gives rise to their anti-realistically conceived truth-values failing to be monotonous. Wright’s attempt to deal with this problem in terms of superassertibility (Wright 1993b: 414–16 and 1992: 48–61) seems to me to misﬁre. Either superassertibility is monotonous, in which case the notion appears to be uanavailable to the anti-realist. Or superassertibility does not generate monotonous appraisals. If so, superassertibility is hardly an improvement on garden-variety assertibility. The difﬁculty very nearly surfaces in Wright (1992: 54 n. 17). Wright seems to plump for the ﬁrst horn of the dilemma, since ‘p’ is supposed to be superassertible, only if we currently have a warrant for supposing that, as a matter of fact, a defeater will never turn up for the now assertible ‘p’; and Wright clearly assumes that, occasionally, such a warrant may be had. This notion is unlikely to curry favour even with the semantical realist. But he, of course, has no sleepless nights over the non-monotonicity of assertitibility, however construed, since he has available to him a nice monotonous notion of (possibly veriﬁcation-transcendent) empirical truth. Wright still seems to hold on to the notion of superassertibility being useful (Wright 2006: 56–9).

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absolute sense of ‘provable’ (i.e., not provable in some given formal theory, but quite generally). We may further assume that intuitionistic provability is in question. But Gödel has shown how to represent intuitionistic provability in classical modal logic S4. And now (1) looks suspiciously as a misguided attempt, using the Gödel–McKinsey–Tarski–Rasiowa–Sikorski mapping (Gödel 1933 and Troelstra 1986), to translate into classical modal logic S4 some sentence, ‘q’, which the anti-realist might wish to assert intuitionistically.⁴ But (1) maps no such ‘q’. Consequently, (1), or some descendant of (1) like (1∗ ) below, cannot be thought of as a rendering in classical logic of any intuitionistic claim—put forth, perhaps, for the beneﬁt of the realist, who might not otherwise know what to make of anti-realist sayings. But, equally, the point about the Gödel-mapping serves to alert us to the further fact that intuitionistic logical constants are already (weakly) intensional.⁵ This raises the question as to whether the anti-realist would wish to introduce intensional operators, as in (1), into an explicit, intuitionistically construed statement of his fundamental principle. If not, (1) fails, as a rendering of anti-realism, even when read intuitionistically. Furthermore, the fault seems to be structural in such a way as to tell against the aforesaid restriction strategy, as well as against Williamson’s overall response to the paradox, on behalf of the anti-realist. Call this dual worry the mapping objection. It should be stressed that the objection is not decisive against proposals such as Williamson’s (1992). However, the objection offers a diagnosis in terms of the genealogy of principle (1), which strongly suggests that the principle is an amphibious hybrid between the points of view of realism and of anti-realism.⁶ Let us rehearse what the mapping objection is. The anti-realist is supposed to face a paradox, preliminarily shaped as in (BA). Quite independently of what the anti-realist might make of assumption (2), we consider the fate of (1). The anti-realist has two general ways of taking assumption (1), according as the logical constants are read classically or intuitionistically. In consequence, his opponent ⁴ There are several such mappings, bearing out that ϕ is an intuitionistic theorem, if and only if the map, ϕ ∗ , is a theorem of classical S4. I use Gödel’s original preferred mapping: For atomic ϕ, ϕ ∗ = ϕ; (ϕ & ψ) ∗ = ϕ ∗ & ψ∗ ; (ϕ v ψ) ∗ = ϕ ∗ v £ψ∗ ; (ϕ → ψ) ∗ = ϕ → ψ∗ ; and (¬ϕ) ∗ = ¬ ϕ ∗ . The constants are intuitionistic, when ﬂanking ‘=’ to the left, classical when ﬂanking identity to the right. ⁵ Only weakly intensional, because although intuitionistic sentential logic is of course not truthfunctional, sequents such as the ‘paradoxes of material implication’ hold good intuitionistically. The S4-mapping (or the set of such mappings) brings out exactly how, and to what extent, intuitionistic constants are intensional, modulo our comprehension of the intensionality built into standard modal logic. ⁶ There is a question as to whether these considerations generalize to the non-mathematical case. There is no telling, as long as we do not have available to us an anti-realistically convincing non-monotonic logic for criterially based assertions, and for empirical generalizations. However, it is not unreasonable to expect that, once such a logic becomes available, it makes sense to search for a modal mapping into standard (not necessarily classical) logic analogous to the Gödel-mapping. There is even a research programme here: ﬁnd the non-monotonic logic that maps well into some reasonable modal logic.

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has a choice as to which reading to impute as the one to which the anti-realist is committed. From the realist point of view, the anti-realist therefore faces a dilemma. The anti-realist now confronts this dilemma by denying that either horn presents him with an insurmountable difﬁculty. Common to the anti-realist response to both parts of the challenge is that he, as an anti-realist, works in intuitionistic logic. The mapping objection just is the anti-realist’s response to the putative dilemma, in those terms. III The positive part of the anti-realist mapping objection takes assumption (1) as having been stated in classical logic. The idea then has to be that (1), or something very similar to (1), captures in the terminology of classical logic some principle central to anti-realist concerns, when stated intuitionistically. So, in short, (1) is a putative translation into classical logic, as reinforced by intensional operators, of an intuitionistically stated principle with a claim to the adherence of the anti-realist. We assume, here, that the resources of intuitionistic logic include nothing more than the usual intuitionistic logical constants. But if this is the picture, we already know, at least in the mathematical case, how to translate intuitionistic claims into intensionalistic classical ones: the Gödel-mapping gives us precisely what is known and—with a reservation—needed. The reservation is that this gives us no purchase on the case of non-monotonous anti-realist truth. But it is safe to ignore this case. If the realist wishes to challenge the anti-realist, the challenge must work, even where the anti-realist has the strongest and best understood case. This is the one of mathematics. The crucial point is that (1) cannot be taken as a plausible case of a classical translation of anything the anti-realist is committed to claiming intuitionistically. IV The negative part of the mapping objection is this. The anti-realist is now asked to accept that he is committed to endorsing a principle very close to (1), as construed intuitionistically. The anti-realist could reasonably claim that, when read intuitionistically, and no matter how we construe the non-standard operators, the assumption now seems to self-destruct by intensional overkill. After all, the standard intuitionistic logical constants are already (weakly) intensional, the intensionality being of an epistemically constrained sort to boot. The attempt to impute to him a commitment to (1) looks, from this point of view, as really a possibly misguided call for providing a better semantical account of the constants of intuitionistic logic. The matter is not just one of applying Occam’s razor to ensure the utmost paucity of basic logical operators. The whole enterprise looks

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like an attempt to patch up a job badly done, when the anti-realist endorsed intuitionistic logic as his standard of reasoning in the ﬁrst place.⁷ Nothing so far said, or to be claimed in the sequel, constitutes a decisive case against adding overtly intensionalistic operators to standard intuitionistic logic, either in general, or in order to hit upon a version of (1) acceptable to the anti-realist. For instance, the claim will not be that Williamson’s admirable efforts to do so are incoherent. At least if it is, this claim would have to be made good by a different kind of argument, in fact a proof to that effect.⁸ The state of play is, therefore, that the negative part of the mapping objection is highly suggestive of the thought that thus improving the arsenal of intuitionistic idiom lacks motivation, if the latter is supposed to derive from a need, on the part of the anti-realist, to face the putative paradox of knowability. On the other hand, the positive part of the mapping objection is decisive against the attempt to refute anti-realism by appeal to a classically construed version of (BA). This, quite apart from the time-indexing consideration, undercuts the presentations of the paradox in Fitch (1963), Hart (1979), and Hart and McGinn (1976). It is worthy of note in passing that the intuitionistic readings of the epistemiclogical principles (F) and (D) under the S4-mapping become the classically construed (FI) and (DI), respectively: (FI) (DI)

Kp → p K(p & q) → (Kp & Kq)

Both of these are valid, if (D) and (F) are valid in epistemic logic based on classical logic. For instance, on the assumption of ‘Kp’, it follows by (F), MPP and ‘p → p’ that p. By necessitation (justiﬁed by the assumptions both being S4—modal—trivially so in the case of ‘Kp’, and plausibly so for (F), which is a conceptual truth, if it is a truth at all), we arrive at ‘p’; whence (FI) by Conditional Proof. The derivation of (DI) is similar. V Now let us brieﬂy consider the claim that the paradox ought to bother the realist as much as it should the anti-realist. The problem was that the realist seems to have to insist on (2), while he must have (1), once the quantiﬁer-domain is restricted to effectively decidable sentences. The only reasonable reply seems to ⁷ Cf. Note 20 for an alternative view of the negative half of the mapping objection. ⁸ One might speculate that if there really is a need for Williamson’s epistemic modal intuitionistic logic, (1∗ ) as taken to be a formula in his logic, must be a translation as to provability of a formula in a hitherto unknown anti-realistically acceptable logic, ARL, which does not itself contain overt modal operators. The anti-realist might then accept (1∗ ) as a rendering, in Williamson’s logic, of a claim he wishes to make in ARL. But now we are speculating.

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be that, strictly speaking, the realist has neither. This is shown by extracting and making explicit the time-indexing in the K-operator, which is plausible in any case. The force of (1) ought to be that if p is the case, then there comes a time, t, such that p is possibly known at t (initial second-order quantiﬁers are usually left out, in the sequel): (1∗ )

p → (∃t)♦Kt p.

The reading of ‘K’ in (2) is however different, viz. as in (K) above. The point here is that there are true propositions not known now (at to ): (2∗ )

p & ¬Kto p

Since t may differ from to , {( 1 ∗ ) , ( 2 ∗ )} is consistent (quite apart from the modal world shift involved), and this result survives Fitch’s diagonalization move (substituting (2∗ ) for ‘p’ in (1 ∗ )). Note that a uniform reading of ‘K’ cannot be obtained by taking the operator to be really ‘Kto ’, as it occurs in (1), or by thinking of the time-index as somehow absorbed into the possibility-operator. The point of (1) cannot be that of afﬁrming that, if p, then it is possible to come to realize that we now know that p, the ‘now’ being rigid across alethically possible worlds. True, the unfortunate Jones may come to realize, truly, that he has really known since last week that his wife is unfaithful. But such cases of ‘having known all along’ are not pertinent to present concerns. The ‘K’ occurring in (1) is therefore parametrically time-indexed. And since this automatically calls for a binding by means of a quantiﬁer, the time-index becoming explicit as a variable in the process, the plausible construal of (1) is as in (1∗ ). The possibility of reading ‘K’ univocally in (1) and (2) as ‘(∃t)Kt ’ is considered below but ignored initially (cf. Section VIII below). Substantially the same point about the realist’s best stance to the putative paradox, as it concerns a domain of effectively decidable sentences, may be put in terms of an answer to the following query. The point of this is to highlight what the time-indexing mechanism is supposed to achieve in contexts such as the one under consideration. The objection to saddling the realist with even the restricted version of the paradox is that, to make this stick, the restriction to effectively decidable sentences, ‘p’, does not sufﬁce. The paradox-generating reasoning will get no grip, unless relevant instances of (2), with decidable ‘p’, are themselves guaranteed to be effectively decidable by the decidability of ‘p’ itself. Since we are now in the business of making good the claim that the realist seems to have a prima facie problem over some domain of sentences, whatever they be, if the anti-realist does generally, we are free to stipulate restrictions on the domain of the quantiﬁers occurring in (1) and (2). So, assumption (1) concerns, for immediate purposes, effectively decidable sentences only. In fact, (1) is supposed to hold truistically, in the sense that it simply states that

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we are now considering only such sentences as pass the positive test of being effectively decidable. Whether the truism can really be put in this way may be a moot point. But we are committing no injustice to the realist by assuming this, unless he himself has sinned grossly by saddling the anti-realist with (1). Assuming, then, that ‘p’ is effectively decidable, does it follow that ‘p & ¬Kp’ is, too? This obviously depends on the reading of the K-operator. Since we are employing the technical notion of effective decidability, let us lay it down that we are concerned with sentences representable in Robinson Arithmetic. Suppose, further, that ‘Kp’ is interpreted as, roughly, that we have a proof of ‘p’. This either means that somebody has a proof right in front of him. Or that there is a proof occurring on a ﬁnite list of proofs stored somewhere suitable. Or it means that a proof occurs on such a list or can be recursively generated from whatever we already have accessible to us, and so occurs as a member of a recursively speciﬁable list of (potential) proofs. On this last and weakest reading, ‘Kp’ is tantamount to our having access to a recursively enumerable, but possibly inﬁnite, list of proofs. On any of these readings of ‘K’, ‘Kp’ is effectively decidable. This does not depend on the decidability of ‘p’ itself. But the decidability of ‘p & ¬Kp’ does: the latter is effectively decidable, if ‘p’ is, on any of the three suggested readings of ‘K’. There are no other plausible readings of ‘K’, in the present context. On any of them, the required decidability of instances of (2) emerges. To assert (2) is to assert that there are effectively decidable truths the proofs of which occur on no currently available recursively enumerable list of proofs. Any set of the kind considered ‘forces’ at time t whatever has a proof in the set, St . And Sto therefore ‘forces’ whatever has a currently available proof. The time-index serves the sole purpose of indicating states of information permitting the assertion of the decidables in question. Presumably these states will admit of an ordering in terms of monotonous extensions over time. It deserves note that, even if ‘p’ is effectively decidable, it does not follow that ‘(∃t)♦Kt p’ is. Whether there is a proof in some admissible extension of our current stock of proofs depends heavily on how we construe admissibility. The question of a given ‘p’ having a proof occurring as a member in the union of all admissible monotonous extensions of the set of currently acceptable proofs of any sentence of a similar kind seems destined to be undecidable, even assuming that the set of proofs at any given stage is recursively enumerable. But if ‘p’ is supposed to be guaranteed to admit of effective proof, then such a proof there must be; and this is all the restricted version of (1) is supposed to claim. The upshot is that there is no paradox, from the point of view of the restricted realist.⁹ There are as yet unproven truths of which we can guarantee that a proof may be produced. But it certainly looked as if there had to ⁹ The ‘restricted realist’ is of course not restricted as to his realism. The restriction concerns the set of sentences for which the realist might seem committed to (1).

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be one, since (BA) was supposed to create trouble for the anti-realist, and since the challenge is structurally the same to realist and anti-realist alike. We also pick up from the above that time has no role to play, except—but crucially—as an index along an ordering of states of information. In the mathematical case, these may be assumed to increase monotonously over the index.

VI In the above section, we assumed pro tempore that the anti-realist faces a paradox. Furthermore, if so, the restricted realist would be up against a structurally similar problem. But the restricted realist can meet the challenge head on. Now, if time-indexing solves the problem for the realist in the restricted case, might not the same strategy work for the anti-realist, quite generally?¹⁰ This is what happens in Timothy Williamson’s epistemic, modal intuitionistic logic: (1∗ ) is retained, while the paradox is blocked by jettisoning (2∗ ) (Williamson 1992). Williamson provides a semantics for his sophisticated and complicated logic. The logic has a few somewhat weird features. However, I shall not discuss the details, merely put on record that I ﬁnd no incoherence in Williamson’s resulting position. On the contrary, Williamson’s 1992 proposal survives as a live option, for all I have to say. The worry is, simply, that he may be barking up the wrong tree, since the negative part of the mapping objection applies to (1∗ ), if it applies to the original (1). To spell this out more fully, one has to wonder about the motivation for introducing epistemic and alethic modal operators into an anti-realistically acceptable version of the claim that truth is epistemically constrained, once it is realized that the anti-realist will put his statement of the constraint in terms of intuitionistic logic. To repeat, the intuitionistic ‘standard’ logical constants are already intensional. Furthermore, the intensionality is inherently of an epistemic kind. To attempt anything along the lines of (1) therefore seems to be intensional overkill. Furthermore, the Gödel-mapping shows this up as not innocuous. (1) and its time-indexed descendants appear to be attempts to reﬂect in the object-language what is better left to meta-linguistic sayings. This is related to the superstition that the anti-realist, in contradistinction to the realist, has some special problem with the equivalence thesis for truth (ET):¹¹ (ET)

It is true that p, if and only if p.

¹⁰ Finn Guldmann took this line, as an unofﬁcial opponent at the oral defence of my Danish Doctoral Thesis (Habilitation) at the University of Copenhagen, October 2004. My response was mainly in terms of what I now call the mapping objection, which remains the basis of my quarrel with, inter alia, this approach. ¹¹ Cf. Rasmussen 2002.

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He is supposed to do considerably better, if he adopts, instead, the more anti-realist seeming equivalence thesis for warranted assertibility (EA):¹² (EA)

It is warrantedly assertible that p, if and only if p.

But ‘warranted assertibility’ is not, in general, a monotonous property. It is not, when we are dealing with, for instance, defeasible assertions. And so, the right-to-left reading of (EA) fails. And the converse implication may look ﬁshy as well, if we ignore the fact that the anti-realist will of course insist on an intuitionistic reading of the conditional. Suppose, then, that we lay it down that we are dealing with monotonous (assertions of) ‘p’, and that the bi-implication in (EA) is to be construed intuitionistically. It is notable that the right-to-left reading of (EA) now looks like something one might express more precisely, or at least symbolically, in the shape of (1∗ ), as read intuitionistically. If so, the anti-realist is in trouble over (EA), unless Williamson’s 1992 proposal is acceptable. However, the negative part of the mapping-objection suggests that the proposal is not in order. Furthermore, the worry seems to arise primarily from the attempt to represent in the intuitionistic object-language the meta-linguistic property of assertibility. In fact, it is difﬁcult not to read (EA) as not already an attempt to do just that. In the light of this it seems misguided to insist that (EA), in particular, captures the essence of anti-realism. It plainly does not constitute an improvement on (ET), even from the point of view of the anti-realist. On the contrary, anti-realist and realist alike can have (ET) as a regulative principle, with the usual (mostly Tarskian¹³) caveats. Their differences will come out in their (meta-linguistic) remarks concerning ‘truth’. But why cannot these differences be reﬂected at the same linguistic level, at some level? The mapping objection provides, or at least attempts to summarize, the answer. There is no doubt that the anti-realist wishes to say something amounting to ‘truth’ being epistemically constrained. So, following Wright (1992: 41), it seems safe to impute to the anti-realist adherence to (EC): (EC)

If p, then it is warrantedly assertible that p.

However, this is exactly what (1) and its descendants were supposed to capture all along. The positive part of the mapping objection points out that if the implication is taken classically, (EC) translates no claim the anti-realist would wish to put forth. On the other hand, if the implication is read intuitionistically, the sense of (EC) is, at best, that if ‘p’ is warrantedly assertible, then it is warrantedly assertible that ‘p’ is warrantedly assertible. That looks rather like ¹² My discussion to some extent draws on, without echoing, the one in Wright 1992: chs 1–2. Wright arrives at results quite different from mine. ¹³ Later developments (cf. the excellent discussion in McGee 1991) seem to me to be irrelevant to present concerns. In view of Dummett 1991: 72, I think it likely that Dummett would concur.

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an instance (for ‘It is assertible that . . .’) of the characteristic axiom of modal logic S4—p → p—and is hardly to be taken seriously as a shot at any speciﬁcally anti-realist insight. The negative half of the mapping objection is a way of saying that this kind of worry is unavoidable, however we tinker with (1). In addition to the worry over (1) in Williamson and in general, a problem arises over the anti-realist standing of (2) and its cognates. The latter worry is best discussed in connection with a proposal put forth by Bernard Weiss, which has the immediate advantage over its competitors that, on the proposal, (1) is rewritten in a way such as to pre-empt the mapping objection.

VII Weiss proposes, on the anti-realist’s behalf, to substutute (1W) for (1): (1W)

p → ¬¬Kp

The logical constants are of course to be construed intuitionistically. From (1W) the negation of (2) follows easily in intuitionistic logic: (N2)

¬(p & ¬Kp)

So, anti-realism, sc. (1W), and (2) trivially lead to contradiction, and no appeal to the Fitch substitution is called for in the proof. The proposal has a good deal to recommend it. First, it is remarkably simple. Second, the mapping objection fails to get a grip, since (1W) exploits the intensionality of intuitionistic constants. No need for modal operators, as intuitionistic negation is already intensional. Third, the anti-realist certainly does wish to claim that if p, then it is absurd to rule out that ‘p’ might one day admit of proof (become ‘forced’); and this appears to be the content of (1W). Yet all is not well with the proposal.¹⁴ The need for disambiguating ‘K’ by time-indexing is still in force. Once these indices are introduced, we arrive at: (1W∗ )

p → ¬¬(∃t)Kt p

and (N2∗ )

¬(p & ¬Kto p)

(N2∗ ) does not, however, follow from (1W∗ ). It is true that (N2∗∗ ) does follow: (N2∗∗ )

¬(p & ¬(∃t)Kt p)

¹⁴ I should stress that I know of Weiss’s proposal only from a brief account in a letter from Michael Dummett, dated 5 June 2005. It has been suggested to me that Williamson broached a similar idea somewhere. That may be. For the purposes of the present paper, I take Williamson’s position to be the one emerging in those of his contributions adverted to in the text.

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The latter is however not the intended reading of (N2).¹⁵ So, we seem to be back to relying on the time-indexing for doing most of the work for the anti-realist struggling to establish that his position is at least coherent. To forestall a possible objection to the above, suppose that Weiss’s proposal ought to read that (1W∗∗ ) leads to (N2∗ ). (1W∗∗ )

p → ¬¬Kto p

This has the merit that (1W∗∗ ) really does entail (N2∗ ) intuitionistically. However, it seems unlikely that the precisiﬁcation (1W∗∗ ) is the intended reading of (1W). Its import is that if ‘p’ is true, then it is refutable that ‘Kto p’ will never turn out to be ‘forced’. So, it is incoherent to rule out now that we didn’t know that p all along. To revert to the case of Jones and his unfaithful wife, it is plausible to claim incoherently that he will never come to realize that he really knew now (rigidly construed). For instance, Jones may come to realize on Thursday the truth that his wife has been unfaithful since Tuesday. And, supposing that his wife really is unfaithful, it is probably refutable on Tuesday to rule out that Jones will not realize the fact on Wednesday. For there must be evidence, concerning both the wife and Jones himself, which cannot be ruled out to come within the ambit of Jones’s cognizance. As before, such cases are however of no relevance to present concerns. A case closer to those is the following. Suppose that the intuitionist mathematician accepts that Andrew Wiles ﬁrst proved Fermat’s Last Theorem. So he accepts that the theorem is true, but that this was not known in, say, 1960. In fact, it is decidably false that the theorem was known in 1960. But if so (1W∗∗ ) and the facts generate a contradiction: it is both the case that ‘¬K 1960 p’ and (by modus ponens) that ‘¬¬K 1960 p’, with the relevant substitution for ‘p’. It will perhaps be objected that this application, with modus ponens, of (1W∗∗ ) was not available in 1960, on the ground that the relevant instance of ‘p’ was not then assertible. True, but in those days the consequent of (1W∗∗ ) was decidably false, as indeed it remains to this day. The fate of the Fermat conditional would then seem to depend on the status of ‘p’, i.e., of the truth status of Fermat’s theorem itself. In 2008 the theorem is taken to be true, and the conditional consequently false. In 1960, by contrast, the conditional had no truth-value. In conclusion, (1W∗∗ ) is not what the anti-realist wants; and (1W∗ ) ought to be adopted as our ofﬁcial reading of (1W). Now, Weiss plainly is quite happy with anti-realism entailing (N2), by which he presumably means (N2∗ ). But it is far from clear that the intuitionist wishes to endorse this position. Let us review what the anti-realist has to say about (2∗ ), (N2∗ ), and (16∗ ): (16∗ )

p → Kto p.

¹⁵ The proposal of adopting this reading is nevertheless discussed below, in Section VIII.

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And bear in mind the following: it is impossible to say within the ordinary language of intuitionistic mathematics that a [true, SAR] proposition has not yet been proved. (Dummett 1991: 78)

Dummett’s word ‘say’ turns out to be a weasel-word, in the present context. It might seem that (2∗ ), existentially quantiﬁed, expresses exactly that there are as yet unknown truths. I think we should allow that it does. For the quantiﬁed (2∗ ), construed constructively, translates into (G2∗ ): (G2∗ )

p & ¬Kto p

on the Gödel-mapping introduced above (Section II). (G2∗ ) is neither contradictory nor nonsensical. So, the proposition expressed by (2∗ ) is intuitionistically coherent. But (2∗ ) can never be asserted by anti-realist lights, for any interesting ‘p’. On Heyting’s informal semantical account of the intuitionistic constant, we can assert a conjunction, only if we are in a position to assert both conjuncts simultaneously (Heyting 1971: Ch. VII; and Dummett 2000: Section 1.2); and we patently cannot afﬁrm that ‘p’ is not now known, while also maintaining that ‘p’ is now assertible. It is no use tinkering with not presently known assertibles, as is clear from the mathematical case. There is no way of coherently maintaining that we have an assurance of the provability of a mathematical proposition, of which we can rule out that we have current knowledge, sc. proof. And so, the intuitionist cannot, and does not, assert that there are stable counter-examples to (16∗ ), although he obviously has no wish to assert that principle. By the same token, he will not wish to assert Weiss’s (N2∗ ), i.e. the negation of (2∗ ). Notice that the intuitionistic non-assertibility of (2∗ ) does not ensure that (N2∗ ) is also non-assertible. (2∗ ) is not intuitionistically contradictory. The point about (N2∗ ) may therefore perhaps be put in terms of Dummett’s notion of the ingredient sense of a sentence (Dummett 1991: 47–50). The ingredient sense of (2∗ ), as it occurs as the antecedent in ‘If (2∗ ), then q’ (for any sentence ‘q’) is perfectly coherent. The conditional might well be intuitionistically assertible, for a suitable choice of ‘p’ and ‘q’. The same then goes for the negation of (2∗ ), since the negation of ‘p’ is deﬁnable as ‘p →⊥’. So, even if (2∗ ) is never intuitionistically assertible, its negation might well be. It should be borne in mind that the interesting cases of sentences, ‘p’, are those that are anti-realistically problematic, i.e., sentences such that we can issue no guarantee that we shall ever be in a position to decide them either way. Bivalence, conceived of as the schema (BIV) (BIV)

Either ϕ is determinately true or ϕ is determinately false

is not applicable to such sentences, according to the anti-realist. On the other hand, the adherent to anti-realism does not envisage that the sentences in question constitute permanent, stable counter-examples to (BIV). Semantical anti-realism resides in agnosticism as regards the applicability to such sentences, non-KED

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sentences, using an abbreviation for ‘not-Known-to-be-Effectively-Decidable’ suggested by Jens Ravnkilde (Rasmussen and Ravnkilde 1982: section 3 and ‘Appendix’; and Rasmussen 1990: section 2). What is involved here is of course not the Church/Turing notion of effective decidability, or not just that notion. If a sentence is effectively decidable in that sense, it is KED in our terms. The converse implication however fails. We follow what we take to be Dummett’s view: a sentence is KED, just in case we are in a position to issue a present guarantee that nothing debars us, in principle, from deciding its truth-value; and the sentence is non-KED if and only if we have no present guarantee that we shall ever be able to decide it, yet are not in principle debarred from doing this.¹⁶ In the case of the anti-realistically problematic sentences, we are dealing with non-KEDs. Suppose, then, that ‘p’ is an anti-realistically problematic sentence, a non-KED sentence. Since we are able to decide, at each given point in time, whether or not we know that p, at that time, ‘Kto p’ is decidable. In fact, the latter is ex hypothesi false, in the case before us. But then the fate of (2∗ ) as to decidability is determined exclusively by that of ‘p’. In consequence both (2∗ ) and its negation (N2∗ ) are non-KED sentences. Among the entailments of this is that Weiss’s principle (N2∗ ) does not hold in the relevant cases. This, by the way, does not bring in its train that (1W∗ ) must be abandoned as well. For, although (1W∗∗ ) entails (N2∗ ), (1W∗ ) does not.

VIII It was claimed above that (N2∗∗ ) is not the intended reading of (N2). This is historically correct. However, suppose we stipulate (N2∗∗ ) as the intended reading. This is tantamount to taking (2), in the basic version of the paradox, as really amounting to (2∗∗ ): (2∗∗ )

p & ¬(∃t)Kt p

¹⁶ Occasionally, Dummett has treated of sentences of which we can presently rule out that they will ever be decided, as if they were of the sort to make the anti-realist frown upon attributing to them a determinate truth-value. But such sentences are not non-KED. Rather they monotonously lack both truth and falsity, hence occupy the territory of either possessing a third stable truth-value (in the manner of past-tense sentences with Lukasiewics) or lacking a determinate meaning (the Continuum Hypothesis of set theory would be a candidate). The sentences that engage our interest are however fully determinate as to meaning and lack a determinate truth-value. I believe that I am in genuine disagreement with Dummett over how best to capture what characterizes the kind of sentence of special interest to the Dummettian anti-realist. This may appear a bit steep. Surely, Dummett knows best what is a Dummettian anti-realist? The trouble arises because there are two ways in which to characterize the Dummettian anti-realist. The positive way, which is the one I adopt, is that of regarding the anti-realist as a generalized intuitionist, the latter thought of in Dummettian terms. The negative way is the one often taken by Dummett, especially since around 1980. According to this account, the anti-realist is anybody who deviates from Fregean semantic realism. However, when it comes to the question of how to deal with sentences of speciﬁc kinds, there is probably no substantial disagreement between Dummett and myself. Pursuing these matters will in any case take us too far aﬁeld.

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This interpretation would be in line with the perception in some quarters that the upshot of the paradox is to the effect that if all truths are knowable, then all truths are actually known at some stage by somebody. Now the introduction of the bearers of knowledge is, as before, just inconsequential clutter. Also, even if the paradox is seen as a reductio of (2∗∗ ), there is no intuitionistic transition to (16∗∗ ): (16∗∗ )

p → (∃t)Kt p

Consequently, if there is a threatening version of the paradox in the ofﬁng here, it has to be because of the fate of (2∗∗ ). Two questions present themselves. (i): What is the intuitionistic status of (2∗∗ ) and (N2∗∗ )? And (ii): Is (2∗∗ ) actually inconsistent with the anti-realist’s preferred version of (1)? Now if (1) is interpreted along the lines of Weiss’s amended (1W∗ ), then (N2∗∗ ) unquestionably follows. This is all right, it seems, since the intuitionist can hardly claim that we can know, of a speciﬁc ‘p’, that nobody will ever know that p; and afﬁrm, in the same breath, that p. So we may rule out the intuitionistical assertibility of (2∗∗ ); and, as noted above, the intuitionist certainly would wish to assert (1W∗ ) in any case. In all of this, no semblance of a paradox surfaces. If, on the other hand, we proceed from assumption (1∗ ) and (2∗∗ ), no formal contradiction emerges, give or take the Fitch-substitution. The worst we get is the conjunction of ‘(∃t)♦Kt p’ and ‘¬(∃t)Kt p’, and these evidently form a consistent set, unless we interpret ‘possibility’ as ‘realized at some accessible actual time’. However, if we were to interpret the diamond according to this suggestion, (1∗ ) would amount to (16∗∗ ). But the whole point of the paradox was to point out trouble for the anti-realist he did not realize he faced. If he is supposed to adopt (16∗∗ ) as an assumption, the paradox becomes pointless. Parallel remarks apply to the realist who combines adherence to (1∗ ), as restricted to the case of decidable sentences, with an endorsement of (2∗∗ ). Occasionally, it is suggested that the intuitionist, perhaps the anti-realist generally, is in any case in trouble over (2∗∗ ). Perhaps all the above shows impeccably from intuitionistic principles that (2∗∗ ) can be ruled out. Still, the suggestion goes, there are cases where this must be wrong. The matchbox right in from of me contains a number n of matches. This is perfectly decidable. Hence, classical logic applies: either the number of matches equals n, or it does not. But nobody ever bothers to count the matches. Instead, a demented child dispenses with the matchbox forever by throwing it into the open ﬁre. A body-count of matches is henceforth out of the question. Yet it is pre-incident true that there is n matches in the box, yet nobody will ever know the size of n. So, we have a clear-cut case of an instance of (2∗∗ ) on our hands; and even the anti-realist ought to be able to assert as much.¹⁷ ¹⁷ I owe the example to an anonymous referee. Dummett’s well-worn example of the late Jones who never in his lifetime had an opportunity to acquit himself with bravery, or otherwise, would

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It deserves note in passing that, if this were all so, anti-realism would be in trouble independently of the alleged paradox under consideration. But it is not so. Realist intuitions have been allowed to interfere with the description of the alleged trouble-case. There is, here, an anti-realistically illicit appeal to the distributivity of counterfactual antecedents through disjunction; or an appeal to truth-value links of dubious anti-realist merit. Possibly both. Of course the anti-realist is not going to allow the derivation from¹⁸ If anybody had counted, then the number of matches would have turned out to be either n or not, to If anybody had counted, the number of matches would have turned out to be n or If anybody had counted, the number of matches would have turned out different from n. serve equally well (Dummett 1963: 147–50). Can we now assert that Jones was brave? That he was not? Dummett unfortunately comes down on the side of committing the anti-realist to the latter, on the ground that we will never know. The anti-realist should sternly remain agnostic about the truth and falsity of either claim, because we cannot rule out that we will come to know of Jones’s bravery (or the opposite) indirectly. If we can rule this out, the proposition that Jones was brave is not anti-realistically problematic, just decidably false (or true). Incidentally, there are even worse cases. Consider the case of a cat in a box, the latter being wired in such a way that the (potential) cat drops out through the bottom whenever the lid is raised. Even supposing that there now is a cat in the box, we shall never know by any obvious means (cf. Percival 1990 and Melia 1991 for similar examples). The proper anti-realist response to such purported straight counter-examples to (1) as ‘There is a cat in the box’ is, as before, that if the sentence is to be thought of as non-KED, it just lacks a truth-value. If we decide that the predicament would appear to be permanent, we must introduce a third stable truthvalue. I have elsewhere (Rasmussen 1997: 149) called such sentences veriﬁcationally inconsistent: the very attempt to verify them turns them into falsities, or at least inﬂuences whatever truth-value they might have possessed unexamined. Such sentences are generally of little interest anti-realistically, although nice questions of a related sort turn up in connection with the interpretation of quantum mechanical measurements. However, in the quantum-mechanical kind of case the mechanism responsible for affecting the truth-value seems to be not just unknown, but perhaps absent. ¹⁸ This sort of case has been well researched, in mathematical as well as in empirical contexts. Vide Dummett 1963, 1973, 2000: 267–9; Wright 1980: ch. XII; Rasmussen and Ravnkilde 1982: Section 3; and Williamson 1988a. The counterfactual fallacy involved receives attention quite independently of issues to do with anti-realism in Lewis 1973: Section 3.4. However, Conditional Excluded Middle surely fails anti-realistically. The paramount concern in contexts such as the matchbox case is to avoid turning non-KED sentences into presently decidable falsities. The temptation is obvious: since the child burned the matchbox, we shall never know the number of matches. But if this it what we think, the sentence is anti-realistically false, hence not problematic. It has been relegated to the domain of the decidable. The truth of it is, however, that we cannot now rule out future availability of evidence about the already discarded box, which will support that the number of matches contained in it was likely to have amounted to some speciﬁc number. For instance, we cannot rule out that some EU bureaucrat has kept a record that will some day turn up. Or some such thing. The story does not itself have to be likely for the sentence to stay anti-realistically problematic.

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And so, the requisite ‘p’ (‘The number of matches equals n’) admits of neither anti-realist assertion nor denial in the circumstances. Nothing like an assertible instance of (2∗∗ ) is in the ofﬁng. The presumably only alternative way of making obligatory the anti-realist assertibility of the requisite instance of (2∗∗ ), would be via truth-value links between the past and present tense statements. Since it is now true that the matchbox contains either n matches or not, it must be true tomorrow that the now discarded box yesterday contained either n matches or not. This too has been looked into.¹⁹ The tense-theoretic anti-realist plainly has a problem with defending the truth-value links customarily accepted. The usual tack adopted is either to hold this against the anti-realist, or to struggle to defend the anti-realist’s entitlement to the habitual links. Although I cannot here enter into the matter, my line would be that the anti-realist cannot in general expect to maintain these links, just as he cannot adopt the principle of bivalence. For present purposes, this means that there is no reason why the anti-realist should be committed to the assertion of anything to do with the number of matches in yesterday’s discarded box of matches. It is decidably true now that the number of matches in the box equals n, say. So, bearing in mind that our concern is now with tense, rather than with mathematical decidability, had we counted, we would now have come up with that result. But we did not count, and the box has been discarded. The number is therefore henceforth no longer a decidable issue. Nobody will ever know, at least not by the direct expedient of counting. But if so, we can no longer claim that there must be a fact as to what the number was. The requisite ‘p’ has become devoid of truth-value on temporal grounds, despite the mathematical decidability of the original question. Yet again, no anti-realistically assertible instance of (2∗∗ ) threatens. Summing up some of the above, (2∗ ) and (N2∗ ) are (complex) non-KEDs. And therefore (N2∗ ) is too strong for the anti-realist’s purpose of ruling out counter-examples to (16∗ ). The impression that this must be so is reinforced by the following consideration. On their own, neither (2∗ ) nor (N2∗ ) is intuitionistically incoherent. This is easily shown by running classical S4-trees on their classical S4-maps. There must be something wrong with the strategy of trying to frame an anti-realist principle along the lines of (1), which is supposed to capture Heyting’s informal semantical account of the intuitionistic constants, and yet is such as to contradict (2). Once again, (N2∗ ) is too strong for the anti-realist’s purpose. Note the surviving point that the antirealist will wish to assert (N2∗∗ ), on grounds not connected with the paradox. This means that if a paradox is still in the ofﬁng, it must have among its assumptions (2∗ ). ¹⁹ Cf. Dummett 1969 and McDowell 1978.

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IX Turning now to the restriction strategy, we shall brieﬂy consider two brands of this broad approach. The two agree in putting restrictions on the antirealist’s chosen variety of assumption (1)—(1∗ ), say—in such a way that the Fitch substitution is inadmissible. To consider ﬁrst Neil Tennant’s proposal, his restriction is to the effect that (1∗ ) holds anti-realistically only for so-called Cartesian ‘p’. A sentence, ‘p’, is Cartesian, just in case it is not anti-realistically incoherent to assume that ‘p’ is assertible. Crucially, (2∗ ) is not Cartesian. Note, ﬁrst, that Tennant’s position is of course vulnerable to the mapping objection, just as was Williamson’s. In addition, the position has speciﬁc weak points of its own. Tennant’s claim that (2∗ ) is not Cartesian seems compatible with the above discussion: (2∗ ) really is not anti-realistically assertible. However, Tennant is concerned to reﬂect a kind of intuitionistic assertibility in the object-language, and therefore proposes the following as a formal rendering of Cartesianism in sentences (Tennant 2002: 136) (T): A Cartesian proposition is a proposition p—of any syntactic complexity—such that Kp is [intuitionistically, SAR] consistent. But ‘K(p & ¬Kto p)’ does not appear to generate a formal contradiction, unless the preﬁxed ‘K’ is construed as ‘Kto ’. The intended reading, however, ought to be ‘(∃t)♦Kt ’, since otherwise Tennant fails to engage with the ﬁrst assumption in the paradox-generating reasoning, which we agreed to construe as in (1∗ ). For this is the version of the assumption which occurs conditionally asserted in the ﬁrst proof-line. But is not all of this refuted by Tennant’s explicit formal derivation of a contradiction from ‘K(p & ¬Kto p)’ (Tennant 1997/2002: 259–60)? No. Tennant ﬁrst derives a contradiction from the joint assumption of ‘p & ¬Kp’ and ‘K(p & ¬Kp)’. He then applies his rule (I) to the effect that ‘, Kp ⊥’ is derivable from ‘, p ⊥’, to arrive at ‘¬K(p & ¬Kp)’. Tennant’s particular way of displaying this within the framework of Prawitz-style natural deduction may cause some confusion. For clarity, here is what happens, written out in full sequents: (S1) K(p & ¬Kp), p & ¬Kp ⊥ (D) and & E (S2) K(p & ¬Kp), K(p & ¬Kp) ⊥ (S1) (I), with = {K(p & ¬Kp)} (S3) K(p & ¬Kp) ⊥ (S2) Deletion of repetitions This seems somewhat roundabout, but at least the proof nowhere commits Tennant to claiming that ‘p & Kp’ itself is contradictory; and he (perhaps deliberately) avoids appeal to the factivity of knowledge, (F). The latter is appealed to in the more direct derivation of a contradiction from ‘K(p & ¬Kp)’

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by (D), &E, and (F). In any case, suppose someone were to apply Tennant’s medicine to his own case. His rule (I) seems plausible enough: surely, we cannot know a contradiction. (By the way, this is probably because of factivity of knowledge.) It might be argued, however, that the rule should be restricted to Cartesian assumptions, ‘p’. The rationale behind this is a wish to eliminate trivial applications of (I). Obviously, any contradiction derivable from the joint assumption of ‘Kp’ and ‘p’ must be ascribable to ‘Kp’. It is not apparent why this line of thinking should be less plausible than Tennant’s restriction on (1). Indeed, since the latter would be plausible only as a general restriction on admissible assumptions in proofs to Cartesian ones, perhaps we are dealing with the very same restriction in an only slightly different proof-context. If this were the only worry raised by Tennant’s proof, it would be alleviated by the fact already adverted to that a contradiction is derivable from ‘K(p & ¬Kp)’ in ways not vulnerable to the objection just ﬁelded. The salient point is that Tennant fails to take due account of the fact that the fate of ‘K(p & ¬Kp)’ is irrelevant to the paradox, unless it is construed as ‘(∃t)♦Kt (p & ¬Kto p)’, in which case it does not entail a contradiction. Impressed by the above, Tennant might rescind from the attempt to reﬂect ‘assertibility’ in the object-language and make do with the informal (C): A Cartesian proposition is a proposition p, such that it is intuitionistically coherent to assert that p. The proposal has a certain air of ad hocness about it. This would have to be remedied by a case being made for the claim that, quite generally, an assumption, ‘p’, is inadmissible in intuitionistic natural deduction, if the assertion of ‘p’ is intuitionistically incoherent. This must be so even in cases, such as the present one, where the assumption is possibly to be discharged by reductio ad absurdum. Since RAA is perfectly respectable in intuitionistic logic, an assuption is of course not to be dismissed as inadmissible, on the sole ground that it gives rise to a contradiction, either in isolation or in the context of the surrounding argument. It is, rather, the intuitionistic non-assertibility of the assumption that is responsible for its inadmissibility. However, this is a puzzling suggestion. As the Gödel-mapping vividly reveals, the intuitionistic derivation of a contradiction from assumption ‘p’ corresponds exactly to the classical S4-derivation of a contradiction from the assumption that it is intuitionistically provable that p. And, surely, provability is the strongest form of assertibility in the market. Hence, if—as is the case—the intuitionist cannot derive a contradiction from (2∗ ), he seems to be excused for regarding that assumption as perfectly Cartesian, in any sense congenial to Tennant’s proposal. This bit of reasoning suggests that, to give the requirement of Cartesianism the right sort of bite in the case under consideration, there is a need to represent the notion of intuitionistic provability, or assertibility, in the intuitionistic

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object-language. Indeed, Tennant’s (T) might be regarded as an attempt to do something of the sort. However, as pointed out, (T) does not appear to be equal to the task. Finally, we should not let drop out of sight that the time-indexing of the K-operator seems to remove altogether the need for Tennant’s restriction on (1); or that, give or take the restriction, (1) and (1∗ ) are seen to be suspect in the light of the mapping objection.

X Dummett’s brand of restrictionism is more subtle (Dummett 2001). I guess his point of departure is that just as a semantical realist is under no compulsion to frame his theory of meaning in the straightforward manner of imposing the Tarskian T-scheme universally on every sentence of the object language; so the anti-realist is not forced to adopt the principle encapsulating his epistemic constraint on truth directly on each and every such sentence. On the contrary: pursuing this strategy is apt to ensue in a rather uninformative account of meaning (Dummett 1991: 25–7). We may, however, assume something akin to (1) for basic sentences. The generalization to all meaningful sentences would have to be earned recursively. Now, as has been pointed out by Tennant, Dummett is not clear about what is meant by a ‘basic’ sentence (Tennant 2002). His invocation of the recursiveness of the expected overall theory of meaning strongly suggests that he means basic sentences to be atomic. It is however not clear whether all atomic sentences are to count as basic. However that may be, nothing with a structure akin to (2) may be counted as basic. Consequently, Fitch’s derivation fails to take off. Dummett’s account clearly does not fall foul of the requirement of non-ad hocness. However, it is not clear that his strategy will be effective against what is assumed to be the otherwise paradox-generating assumption (1). The anti-realist seems to face a dilemma. Either he thinks he can recursively earn an entitlement to the fully general ﬁrst assumption in the paradox-generating argument, in which case he is back where he started. Or he envisages the possibility that (1), or some suitable descendant of (1), will not be forthcoming. This would have dramatic repercussions for the entire positive anti-realist programme in the theory of meaning (Wright 1993a: Section V), as well as for the signiﬁcance of that programme for metaphysics. The former concerns how to systematically implement in a theory of meaning the idea that truth is epistemically constrained. Concerning the latter, the anti-realist programme for the theory of meaning has no immediate metaphysical bearing, save through the conception that the meaning of non-KED sentences is fully determinate, despite the fact that such sentences are not determinate as to truth-value because of epistemic constraints on the concept of truth.

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XI The upshot, then, is this. The realist, restricted or not, can hold on to there being hitherto unknown truths. But he can make his position clear only by using timeindexing or some device tantamount to this. It is tempting for the anti-realist to attempt a similar ploy. This just might work, along the lines suggested by Williamson. But all such attempts must face the charge of implausibility derived from the negative half of the mapping objection. The restriction strategy, in the manner of Tennant or Dummett, turns out to be most likely a distraction, from the anti-realist point of view. Put differently, the anti-realist status of (2∗ ) is that of being a non-KED sentence, i.e., (2∗ ) is not assertible, but nor is (N2∗ ). On the other hand, (N2∗∗ ) is intuitionistically assertible. In view of how the Gödel-mapping works, whatever exact assumption one puts for (1), the reading of it must be intuitionistic. But the mapping objection presents an obstacle to some proposed versions of that assumption, although the objection does not in itself prove these incoherent. Weiss’s suggestion, as improved to (1W∗ ), is not vulnerable to this particular objection and in fact remains a principle that the intuitionist would wish to adopt, quite independently of the intricacies surrounding the Paradox of Knowability. The relevant reading of (N2) does not now follow; but this is just as well. Moreover, the specimens of the restriction strategy brieﬂy considered in the above display weaknesses of their own. Consequently, (1W∗ ) and the rightly construed (1∗ ) survive as at least coherent candidates for the status as the best, sc. strongest, anti-realist descendant of (1). Notably, both crucially involve time-indexing of the K-operator. The dialectical ﬂow in the above, especially as it concerns the mapping objection, has been as follows. The realist, in the guise of Fitch and others, originally presented the anti-realist with the obstacle of somehow overcoming the apparent paradox resulting from assumptions (1) and (2). The anti-realist was supposed to share common ground with the realist over (2), that is, over the assumption of there being currently unknown truths. The trouble for the anti-realist must then arise over assumption (1), which is in any case supposed to encapsulate the gist of anti-realism. The mapping objection is presented as part of the anti-realist’s defence against the challenge that (1) gives rise to a paradox. The anti-realist thought is that (i) either (1) or some suitable descendant in the vein of (1∗ ), with or without restrictions, is supposed to be couched in classical terms. But if so, the challenge has no bite, unless the classical formula translates something he wishes to claim intuitionististically. The positive part of the mapping objection points out that (1), and its cognates, map nothing the anti-realist would wish to claim. So far, then, the realist is arguing beside the point by attempting to saddle the anti-realist with adherence to a thesis he does not hold.

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Or (ii), the realist intends the anti-realist to accept (1), as construed intuitionistically. Since it will have to be agreed on all hands that time-indexing is needed in any case, the realist is now imputing to the anti-realist adherence to the intuitionistically construed (1∗ ), with or without restrictions. This is where the negative part of the mapping objection comes to the fore. The anti-realist will quite naturally claim that in adopting intuitionistic, as opposed to classical, logic as his base logic, he has already intensionalized logic to the extent necessary to state any claim central to his concerns. He does not, that is, wish to be forced to introduce additional modal machinery for the purposes of meting the realist’s challenge to his central tenets. He is not, therefore, forced to accept (1) or (1∗ ), even when construed intuitionistically. He would not, then, face the paradoxical challenge, even supposing that the paradox survives when transposed in a time-indexed, intuitionistic version. As Williamson has pointed out, and as transpires from the above considerations concerning the anti-realist status of (2), it is even doubtful that the paradox does survive, when thus transposed. The point, as far as the negative part of the mapping objection is concerned, is that even if the paradox did survive, the anti-realist should not worry unduly over this: he most likely does not wish to endorse (1∗ ) anyway. So, the anti-realist has successfully rejected the challenge posed by the apparent paradox.²⁰ The question remains as to how, then, he should characterize his position, in his own terms. I have offered no reason to think that he could not coherently do so by means of Williamson’s (1∗ ). But the negative part of the mapping objection strongly suggests that he should not wish to do so. An anternative way would be Weiss’s amended (1W∗ ). The anti-realist is committed to this, certainly. However, (1W∗ ) seems to be too weak to capture all the anti-realist is out to say about the epistemic constraint on truth. The suggestion, in the above, is that whatever else is to be said is best left to meta-linguistic sayings. ²⁰ There is another way of taking the proposals of Williamson, Tennant, and Dummett, as seen in the context of the mapping objection. I suspect many will see them as attempted straightforward anti-realist replies to the challenge raised by the paradox. This point of view is not altogether implausible, on the assumption that the anti-realist is in any case committed to something like (1). But is he? Be that as it may, the overarching conclusion will stand up: the restriction strategies are distractions, and Williamson’s proposal is a non-mandatory, albeit a seemingly coherent, anti-realist response to the original challenge. Furthermore, the latter proposal ought not to hold much prima facie attraction to the anti-realist, for the reason encapsulated in the negative half of the mapping objection. It is however true that, from the perspective of this overall view of the debate, the negative half of the mapping objection will be seen as a part of the realist challenge to the anti-realist’s proposals for straight solutions, rather than as a contribution, along with the positive half of the objection, to the anti-realist’s defence against the realist’s onslaughts.

6 Truth, Indeﬁnite Extensibility, and Fitch’s Paradox Jos´e Luis Berm´udez

I Fitch’s original presentation in Fitch (1963) of the line of argument that has come to be known as Fitch’s paradox begins with the notion of a truth class of propositions. A class α of propositions is a truth class just if, as a matter of necessity, every member of α is true. That is, (1) ∀p[p ∈ α ⇒ p] Suppose that α is a truth class closed under conjunction elimination and consider the proposition (2) p & ¬(p ∈ α) asserting that p is a true proposition that is not a member of α. Assume, for reductio, that (2) is included in α: (3)

[p & ¬(p ∈ α) ] ∈ α

Since α is closed under conjunction elimination we have (4) p ∈ α and (5)

[¬(p ∈ α) ] ∈ α.

Since α is a truth class, (5) gives (6) ¬(p ∈ α) Thanks to two anonymous referees for comments on an earlier draft.

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which contradicts (4) and shows that, as a matter of necessity, (3) cannot be true. That is (7) ∀p ¬♦[ (p ∧ ¬(p ∈ α) ) ∈ α] This is Fitch’s Theorem 2. Now, let α be the class of known truths, where q is a member of the class of known truths just if there is a time at which q is known by somebody. Plainly α is a truth class, so that (1) holds. On this interpretation (2) states that p is an unknown truth—i.e. that there is no time at which someone knows p. Suppose we assume, as seems highly plausible, that there is at least one unknown truth. Let that be p. It follows that it is true that [p & ¬(p ∈ α)], which is our (2). But Theorem 2 shows that it is contradictory to suppose that there is a time at which someone knows that [p & ¬(p ∈ α)]. So, not only is there at least one truth that is unknown, but at least one proposition that is unknowable. What has come to be known as Fitch’s paradox derives essentially from Theorem 5 in the same paper. Theorem 5 states that, provided we accept the existence of an unknown truth, it cannot be the case that all truths are knowable. This is supposed to be paradoxical because there are well-established philosophical positions that maintain precisely the claim that Fitch shows to be incoherent, namely, that it is true both that there is at least one unknown truth and that all truths are knowable. Any form of veriﬁcationism or semantic anti-realism appears to be committed to the general principle that all truths are knowable, while no anti-realist or veriﬁcationist is likely to accept that all truths are known. However, accepting that all truths are known seems to be the only alternative to denying that all truths are knowable. There is a familiar response to Fitch’s paradox. It has been pointed out by a number of authors (originally in Williamson 1982) that the argument from the knowability principle that all truths are knowable to the omniscience principle that all truths are known is not intuitionistically valid. Suppose we formulate the knowability principle as (8) ∀p [p ⇒ ♦(p ∈ α) ]. We can substitute the assumption that there is at least one unknown truth into (8) to give (9) ∀p [ [p ∧ ¬(p ∈ α) ] ⇒ ♦( [p ∧ ¬(p ∈ α) ] ∈ α) ]. We recall Theorem 2 (7) ∀p ¬♦[ (p ∧ ¬(p ∈ α) ) ∈ α] Trivially, (7) and (9) jointly yield (10) ∀p ¬[p ∧ ¬(p ∈ α) ].

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The inference from (10) to the omniscience principle (11) ∀p [p ⇒ (p ∈ α) ] is classically, but not intuitionistically, valid. From an intuitionistic point of view we are entitled only to move from (10) to (12) ∀p [p ⇒ ¬¬(p ∈ α) ]. Nonetheless, as it stands this response is hardly satisfying. Although the appeal to intuitionistic logic may well block the move from the knowability principle to the omniscience principle, one can plausibly ask why we should adopt an intuitionistic logic at all. It is true that anti-realists such as Dummett have argued that anti-realism stands or falls with an intuitionistic revision of classical logic. But at the very least, if the appeal to intuitionism is not to be question-begging, we need some independent reason for thinking that classical logic should be revised in the way the intuititionist suggests. The aim of this paper is to take a step back from the details of Fitch’s argument and the particular rules of inference on which it depends in order to explore a line of argument that holds the promise both of undercutting Fitch’s enterprise as a whole (as opposed to simply the deployment of Theorem 5 against anti-realism) and, as a corollary, of explaining why intuitionistic logic is appropriate in this context. This argument has its roots in the notion of indeﬁnite extensibility, as discussed by Michael Dummett in a number of writings (most extensively in chapter 24 of Dummett 1990). Dummett uses the argument to try to motivate anti-realism about mathematics. In particular, he deploys the putative indeﬁnite extensibility of such concepts as set, natural number, and real number to argue for the rejection of classical logic in the relevant domains. Only an intuitionistic logic, he thinks, can do justice to indeﬁnite extensibility. The problem arises when we try to quantify over indeﬁnitely extensible domains. Quantiﬁcation over indeﬁnitely extensible domains does not always, Dummett thinks, yield statements with a determinate truth-value. When we make an existential quantiﬁcation over an indeﬁnitely extensible domain what we are really doing is claiming to be able to cite an instance, and when we make a universal quantiﬁcation we are claiming to have an effective operation that is universally applicable. And this requires that the quantiﬁers be understood intuitionistically rather than classically. How might the notion of indeﬁnite extensibility be applied to the issues about knowability raised by Fitch? Suppose it is the case that the concepts proposition and true proposition are indeﬁnitely extensible, so that there is no deﬁnite totality of (true) propositions of which we have a deﬁnite grasp. If Dummett’s claims about indeﬁnitely extensible concepts are along the right lines then at the very least we need to inquire into the status of the universal quantiﬁcations that are at the heart of Fitch’s reasoning. What is the status of the claim that all truths are knowable, which is required to derive the so-called paradox from Theorem 5?

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And, for that matter, what is the status of the claim that deﬁnes the notion of a truth class? It may turn out that the indeﬁnite extensibility of the concept proposition renders these universal quantiﬁcations problematic in ways that blunt the force of Fitch’s work and the conclusions that have been drawn from it. II In order to explore this terrain, however, we need to begin by clarifying the basic notion of indeﬁnite extensibility and the connection that Dummett sees between indeﬁnite extensibility and intuitionistic logic. Dummett gives the following characterization of an indeﬁnitely extensible concept in ‘What is Mathematics About?’. An indeﬁnitely extensible concept is one such that, if we can form a deﬁnite conception of a totality all of whose members fall under that concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it. (Dummett 1996: 441)

Elsewhere he mentions as an antecedent Russell’s diagnosis of the set-theoretic paradoxes in terms of what he (Russell) terms self-reproductive classes. Russell states: The paradoxes result from the fact that there are what we may term self-reproductive processes and classes. That is, there are some properties such that, given any class of terms having such a property, we can always deﬁne a new term also having the same property. (Russell 1906: 144)

Indeﬁnite extensibility is a property of concepts, while it is classes that are selfreproductive. In both cases the problematic phenomena emerge from features of (certain) inﬁnite totalities. The totalities of which we have indeﬁnitely extensible concepts are all inﬁnite and the properties that generate self-reproductive classes are properties deﬁning inﬁnite collections. For Dummett, the indeﬁnitely extensible concepts include the concepts set, natural number, real number, and ordinal. Russell’s list of self-reproductive classes would no doubt be similar, although I doubt that he would have counted the property of being a natural number as generating a self-reproductive class. The fact that, for Dummett, indeﬁnitely extensible concepts invariably characterize inﬁnite totalities does not mean either that indeﬁnite extensibility is really a property of inﬁnite totalities (so that we can distinguish deﬁnite totalities from indeﬁnitely extensible totalities) or that every concept that characterizes an inﬁnite totality is thereby indeﬁnitely extensible. We can appreciate both these points, together with the general deﬁnition of indeﬁnite extensibility, through examples. One example, which Dummett discusses frequently, is the concept of an ordinal number. One interesting feature of the indeﬁnite extensibility of the

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concept ordinal is that it reveals itself in paradox (in this case the Burali-Forti paradox). Suppose that there is a set α of all ordinal numbers. This set is transitive (since every member of it is an ordinal and every member of an ordinal number is itself an ordinal number) and it is well-ordered by ∈.¹ Since every transitive set well-ordered by ∈ is an ordinal number (Enderton 1977: 191), it follows that α is an ordinal number. But then α would be a member of itself, which no ordinal can be. What generates the paradox is the simple fact that any collection of ordinal numbers gives rise to an ordinal number that is not in that collection (viz. the order-type of the collection). This ﬁts Dummett’s description perfectly. For any collection of ordinals α we can characterize a larger collection, which is the union of α and the order-type of α. The concept set is itself indeﬁnitely extensible. This can be seen in a number of ways. For any given collection β of sets we can characterize a larger collection γ ⊃ β, all of whose members are sets. This is ℘ (β), the power set of β—since, by Cantor’s theorem, card (℘ (β) ) = 2 card (β) . But the indeﬁnite extensibility of the concept set can be shown with far less machinery. For any set δ we can construct a set that is not a member of δ. Let A = {x ∈ δ : ¬(x ∈ x)}. By construction we have A ∈ A ⇔ A ∈ δ ∧ ¬(A ∈ A). Hence, if A ∈ δ, then we have the obviously contradictory A ∈ A ⇔ ¬(A ∈ A). So we can characterize our larger totality by taking δ ∪ {A}—in fact, in a set theory with no self-membered sets, δ = A and the larger totality is δ ∪ {δ}. This is sufﬁcient to show that assuming a set containing all sets leads to paradox. But indeﬁnite extensibility does not entail or require paradox. The indeﬁnite extensibility of the concept real number is revealed by Cantor’s proof of the nondenumerability of the set of real numbers, but there is nothing paradoxical about the fact that the real numbers cannot be put into a one–one correspondence with the set of natural numbers. What Cantor’s proof shows is that any putative enumeration of the set of real numbers will yield a real number that does not feature in the enumeration. Adding that new real number to the original enumeration will give the ‘larger totality’, all of whose members fall under the concept real number. In fact, according to Dummett, the domains of all fundamental mathematical theories are given by indeﬁnitely extensible concepts because, in a claim that has puzzled many commentators, he holds that the concept natural number is indeﬁnitely extensible.² The indeﬁnite extensibility of the concept natural number derives from what Dummett terms the intrinsic inﬁnity of the totality of ¹ A set A is transitive if it is closed under set membership—i.e. if, whenever x ∈ A and y ∈ x, then y ∈ A. ² Dummett’s thinking on this developed signiﬁcantly between ‘The philosophical signiﬁcance of Gödel’s theorem’ (1978) and Frege’s Philosophy of Mathematics (1991). In the earlier article Dummett stopped short of claiming that the concept natural number is indeﬁnitely extensible. There he placed the burden of indeﬁnite extensibility with respect to our understanding of the natural numbers at the door of the Gödel phenomenon. What he says there is that the concept of a

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natural numbers. The totality of natural numbers is intrinsically inﬁnite because, whatever totality of natural numbers we are given, we always have a means of ﬁnding another element of the totality—an element characterized in terms of the elements we already have. Indeﬁnitely extensible concepts are problematic, according to Dummett, because their extensions do not form deﬁnite totalities. This is not because their extensions are in some sense hazy or vague. There is no vagueness in the concept set. There are no entities that we would place on the borderline between things that are sets and things that are not sets. Nor is there any indeterminacy about what is to count as an ordinal number or a real number. The problem comes, Dummett thinks, because it is, strictly speaking, misleading to think of them having extensions in any straightforward sense at all. As he evocatively puts it, indeﬁnitely extensible concepts have ‘an increasing sequence of extensions: what is hazy is the length of the sequence, which vanishes in the indiscernible distance’ (Dummett 1990: 317). Each member of the sequence (each putative extension of the concept natural number or real number is perfectly deﬁnite. But the sequence can be indeﬁnitely extended. This means that we do not have determinate conceptions of the relevant domains of quantiﬁcation for statements about objects falling under those concepts. In fact we cannot have determinate conceptions of the totality of mathematical objects (be they sets, ordinal numbers, or real numbers) over which we are quantifying—as soon as we try to form such a determinate conception we are led inexorably to the conception of a totality that is a superset of the totality with which we began. The problem is not to be avoided by familiar strategies such as the distinction between sets and proper classes. If, as in von Neumann-Bernays set theory, we allow there to be collections that are not sets (because they cannot be members of any collection), then this gives us the means to name the extensions of indeﬁnitely extensible concepts. We are in a position to include terms such as ‘On’ (denoting the proper class of all ordinals) in our set theory, but the totalities thereby denoted are no less indeﬁnitely extensible. The power to name the extension of an indeﬁnitely extensible concept can hardly be thought to eliminate its indeﬁnite extensibility. In any event, we do not need proper classes to deﬁne the extension of the concept real number, which is a perfectly respectable set (on one way of constructing the real numbers it is the set of Dedekind cuts)—and we already have the name ‘ω’ for the extension of the concept natural number, indeﬁnitely extensible concept though it is (according to Dummett). We can now see how there can be concepts of inﬁnite totalities that are not indeﬁnitely extensible—and, indeed, how the same inﬁnite totality can be characterized both by an indeﬁnitely extensible concept and by a perfectly property well-deﬁned over the natural numbers is indeﬁnitely extensible. The argument in the text follows the presentation in the later book at pp. 318–19.

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deﬁnite one. The set of natural numbers is a good example. We can consider the set of natural numbers either as the extension of the concept natural number or as the extension of the concept member of the ﬁrst limit ordinal (since ω, the ﬁrst limit ordinal, has as members all the ﬁnite ordinals). The ﬁrst way of thinking about the set of natural numbers yields an indeﬁnitely extensible concept, for the reasons sketched out earlier. The second way does not, however. It is certainly true that if I form a conception of the totality of members of the ﬁrst limit ordinal then I can characterize a larger totality. I can, in the standard manner, extend the totality by taking the union of all the members of the totality. But the number that I thereby generate, ω, is not the extension of the concept member of the ﬁrst limit ordinal. It is the extension of the concept member of the successor of the ﬁrst limit ordinal. Dummett’s interest, then, is not with inﬁnite totalities per se, but rather with inﬁnite totalities given by indeﬁnitely extensible concepts. These include, he thinks, the domains of the basic mathematical theories, such as number theory and analysis. The fact that these domains are given by indeﬁnitely extensible concepts has deep implications for our understanding of those mathematical domains. Of course, we do manage to quantify meaningfully over mathematical domains that are given by indeﬁnitely extensible concepts—and we do, correspondingly, have some sort of a grasp of the relevant domains of quantiﬁcation. But this is very different from our grasp of totalities given by deﬁnite concepts. In both cases we have, as mentioned earlier, clear and unequivocal criteria for determining, of any particular object, whether it falls under the relevant concept—and for determining when an object falling under the concept but given in one way is identical to an object given in another way. But only for deﬁnite concepts is this enough to ﬁx a determinate totality as the extension of the concept—and hence enough to give us a clear understanding of the extension of the concept. For indeﬁnitely extensible concepts we need something more. We need, ﬁrst, a clear collection of objects that canonically satisfy the relevant criteria, and, second, a principle of extendibility that shows us how the domain is to be extended beyond the canonical base. In the case of the indeﬁnitely extensible concept natural number the principle of extendibility is the fact that every number has a successor. Things are slightly more complicated for the concept ordinal number. We have, of course, the analogous principle that the successor of every ordinal number is an ordinal number, but we also have the further principle that, if A is a set of ordinals, then the least upper bound of A is also an ordinal. The ﬁrst principle gives us the successor ordinals, while the second gives the limit ordinals. The crucial step in Dummett’s argument, and the one that has bafﬂed most commentators (e.g. Clark 1998: 61; and see Potter 2004: 29–30 for further references), is the argument from the indeﬁnite extensibility of key mathematical concepts to the rejection of classical quantiﬁcation. It is, Dummett maintains, quite simply not the case that every quantiﬁcation over a mathematical domain

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given by an indeﬁnitely extensible concept has a determinate truth-value. In trying to understand this we would do well to begin with those quantiﬁcations that Dummett does think acceptable. Any deﬁnite totality of ordinals must therefore be so circumscribed as to foreswear comprehensiveness, renouncing any claim to cover all that we might intuitively recognize as being an ordinal. It does not follow that quantiﬁcation over the intuitive totality of all ordinals is unintelligible. A universally quantiﬁed statement that would be true in any deﬁnite totality of ordinals must be admitted as true of all ordinals whatever, and there is a plethora of such statements, beginning with ‘every ordinal has a successor’. Equally, any statement asserting the existence of an ordinal can be understood, without prior circumscription of the domain of quantiﬁcation, as vindicated by the existence of an instance, no matter how large. (Dummett 1990: 316)

It must be recognized that here, as in many places, Dummett is talking about what it is to understand particular statements—as opposed, for example, to what might make them true. The question of what makes a universal quantiﬁcation over all ordinals true has a simple and uninformative answer, namely, that the statement hold true of every ordinal. The question of understanding is, unfortunately, rather trickier—although of course we cannot understand a statement without understanding what it would be for that statement to be true. What we understand when we understand a universal quantiﬁcation over all the ordinals is the fact that the statement holds true of every deﬁnite totality of ordinals—a fact that we grasp by grasping that the statement holds true of any arbitrarily chosen collection of ordinals. Dummett is somewhat elliptical here, but we can reconstruct his reasoning with the example that he himself gives—the statement that every ordinal has a successor. Let α be an arbitrary ordinal number and α+ the successor of α. By deﬁnition, α+ = α ∪ {α}. The statement says, then, that α ∪ {α} is an ordinal number. Plainly α ∪ {α} is a deﬁnite totality of ordinals (composed of α together with all the members of the members of . . . α). So, for Dummett, what we understand when we understand the statement that every ordinal has a successor is the claim that every deﬁnite totality of ordinals of the form α ∪ {α} is an ordinal. This claim is perfectly intelligible, Dummett claims (on my reconstruction), because we can understand it as the assertion that there is a procedure for showing that any such deﬁnite totality is an ordinal (namely, by noting that α+ is a transitive set all of whose members are ordinals). We can, I think, put Dummett’s point in a more general way as follows. When we are dealing with a universal quantiﬁcation of the form ∀x ϕx claimed to hold over an indeﬁnitely extensible totality, there is (by assumption) no deﬁnite domain of which we can say ∀x ϕx is true just if ϕ holds for every object in the domain. Instead, there is an indeﬁnite sequence of domains and ∀x ϕx is true just if ϕ holds of every object in every domain. But of course the indeﬁnite sequence is not itself a deﬁnite totality, which means that we cannot understand this implicit quantiﬁcation over the indeﬁnite sequence of domains in the standard manner.

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What we are really doing when we assert ∀x ϕx is asserting that the fact that ϕ holds of every object in a given domain is transmitted across the principle of extendibility that creates a new and more inclusive totality from any given deﬁnite totality. But what is it to make such a claim? According to Dummett, when we claim that universal ϕ-ness is transmitted across the principle of extendibility what we are really claiming is that there is a way of showing that, if ϕ holds of all the members of any particular deﬁnite totality in the sequence, then it holds of all the members of the totality to which that deﬁnite totality might be extended Matters are somewhat obscured here by the fact that our example is itself one of the principles of extendibility governing ordinal numbers, but we can still see what is going on by considering the other principle of extendibility. Let ω be the least upper bound of the ﬁnite ordinals. Our second principle of extendibility tells us that ω is itself an ordinal. Part of what is asserted when we assert that ∀x ∈ On ∃y (y = x + ) is that the very same means by which we show that any ﬁnite ordinal has a successor can be extended to the inﬁnite ordinal that is the least upper bound of the set of ﬁnite ordinals. And this in fact is the case. The very same line of reasoning that shows that α ∪ {α} is an ordinal when α is a ﬁnite ordinal will equally show that ω ∪ {ω} is an ordinal. An opponent of Dummett will most likely object at this point that Dummett is confusing proof and truth. On this view, what we assert when we assert ∀x ∈ On ∃y (y = x + ) is simply that every ordinal, be it zero, a successor ordinal, or a limit ordinal, has a successor. Although a proof of this claim will no doubt have to cite just such an operation, its truth depends simply upon there being a successor for every ordinal. We should understand assertion in terms of truth-conditions, not in terms of the operations discovery of which will convince us that the truth-condition holds. Here we come to the nub of the issue, because Dummett’s point (as I understand it) is precisely that we cannot grasp the truth-conditions for quantiﬁcations over indeﬁnitely extensible totalities except in terms of the type of transmissibility sketched out in the previous paragraph. And we can only grasp the possibility of such transmissibility through the idea that there is something that secures it—there is no such thing as transmissibility simpliciter. Of course, there would be no need for this were we dealing with deﬁnite totalities, where the truth-conditions for universally quantiﬁed statements can be understood in the normal manner. But we cannot treat domains deﬁned by indeﬁnitely extensible concepts as if they were deﬁnite totalities. Dummett’s position, then, is that the truth-conditions for universal quantiﬁcations over domains given by indeﬁnitely extensible concepts must be understood in terms of operations that secure transmissibility in the manner discussed. To assert such a universally quantiﬁed statement is to assert that such operations exist. But this has the inevitable consequence that we must abandon classical logic. From the fact, for example, that it is not the case that it is not the case that every x in some indeﬁnitely extensible domain is F it by no means follows that there is an operation that will secure the transmissibility of ϕ-ness throughout

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the increasing sequence of extendible totalities. We might be able, for example, to give a reductio of the thesis that ∀x Fx can be reduced to absurdity. But this would hardly give us the required operation. Nor will the standard quantiﬁer interchange rule ¬∀x ¬Fx ⇔ ∃x Fx be valid. Even if it is absurd to suppose that there is an operation securing the transmissibility of F not holding in an appropriate domain, this by no means provides an instance of something in the domain that is F. Dummett is surely right that if the truth-conditions of universal quantiﬁcations over indeﬁnitely extensible totalities make ineliminable reference to operations securing transmissibility, then we cannot understand the logic of such statements classically. III We return now to Fitch’s paradox. The excursion into the philosophy of mathematics in the previous section has shown that it is possible to argue (with some plausibility, in my opinion) for the thesis that quantiﬁcation over mathematical domains given by indeﬁnitely extensible concepts should be understood intuitionistically rather than classically. Plainly, applying this to Fitch’s paradox depends upon construing Fitch’s paradox as making ineliminable reference to indeﬁnitely extensible totalities in a way that will support the type of argument canvassed in the previous section. Exploring whether this is indeed the case is the task of this section. But suppose for the moment that we can apply a Dummett-style argument in this domain. There are two ways in which this holds promise for dealing with Fitch’s paradox. Most obviously, as we saw earlier, the omniscience principle can only be derived by an inference that is classically but not intuitionistically valid. If it can be shown that we are dealing with an indeﬁnitely extensible totality over which quantiﬁcation must be understood intuitionistically then we have a plausible case for denying the validity of this inference. The appeal to intuitionistic logic becomes more than simply a technical ﬁx. But there is a more subtle way in which an argument from indeﬁnite extensibility might get a grip here. The logic governing statements that quantify universally over indeﬁnitely extensible domains is intuitionistic because those quantiﬁcations have to be understood in a constructivist manner—that is, in a manner that appeals to the existence of effective operations securing transmissibility across increasing sequences of domains. To assert a universal quantiﬁcation is essentially to assert that there is such an operation. But then it is very natural to wonder whether a prudent anti-realist really ought to commit themselves to the knowability principle in the form that we have given it (that is, as the universal quantiﬁcation (8)). In any event, let us begin at the beginning. Are there good reasons for thinking that the collection of all propositions forms an indeﬁnitely extensible totality?

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A straightforward argument shows that there can be no set S of all true propositions (cf. Grim 1991: Ch. 4). We take a proposition to be an abstract entity, such that S can be non-denumerably inﬁnite, and we assume that S does not contain the proposition corresponding to the sentence ‘0 = 0’. Since S is a set it has a cardinality κ. Consider the power set ℘(S). Any arbitrary member si of ℘(S) is a set of true propositions. As such there is a true proposition corresponding to the sentence ‘0 = 0 ∈ / si ’. Let that proposition be pi . Since S is the set of all true propositions, a subset axiom permits us to form the set P of all such pi . Since P can be put into one–one correspondence with ℘(S) we have card (P) = card (℘ (S) ) = 2 κ > κ = card (S). Since it is impossible for a set to have a subset of greater cardinality there can be no set of all true propositions. With minor alterations this argument can be used to show that there is no set of all propositions. (For each si we take pi as either ‘0 = 0 ∈ / si ’ or ‘0 = 0 ∈ si ’, as appropriate.) Of course, the argument just sketched out contains many hostages to fortune. Without a precise understanding of what a proposition is it is hard to know how to understand sets of propositions. Nor do we have a proper deﬁnition of set P. But we can put these problems to one side. We have enough to go on to see how the case might be made for the indeﬁnite extensibility of the concept proposition. Certainly, Dummett’s deﬁnition seems to be satisﬁed. Given any deﬁnite totality S of propositions we can deﬁne a totality S∗ = S ∪ P such that S ⊂ S∗ and S∗ is itself a totality of propositions. The real question is not whether the concept proposition is indeﬁnitely extensible in this sense, but whether it is indeﬁnitely extensible in a way that permits the type of argument for intuitionistic logic sketched out in the previous section? That line of argument rests upon certain claims about what it is to grasp the truth-conditions of quantiﬁcation over totalities given by indeﬁnitely extensible concepts. The argument, in essence, is that universal quantiﬁcation over a totality given by an indeﬁnitely extensible concept must be understood in terms of principles of transmissibility that secure the holding of the relevant property across an increasing sequence of domains. We know, when we are dealing with indeﬁnitely extensible concepts in the mathematical sphere, that any given deﬁnite totality of a given type of object will generate a larger totality of the same type that includes it. So, what we assert when we assert some statement to be universally true within the domain given by such an indeﬁnitely extensible concept is that, if the statement holds true of any given deﬁnite totality, it will hold true of the larger totality to which the original totality can be expanded—and so on through the increasing sequence of domains. It is for this reason, Dummett thinks, that universal quantiﬁcation over indeﬁnitely extensible domains incorporates ineliminable commitment to the existence of effective operations, thereby requiring an intuitionistic logic. We saw how this way of thinking about universal quantiﬁcation makes sense in the context of the ordinals. But is it mandated by quantiﬁcation over the intuitive

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totality of propositions? In one sense it is hard to see how it could not be. After all, the argument just canvassed shows that there is no deﬁnite set of all propositions and so, even though we are assuming that we have clear criteria of identity and individuation for propositions (which may, of course, be a vague hope), we cannot assume that those criteria will determine a truth-value within that deﬁnite totality. And so one might well feel justiﬁed in arguing with Dummett that we can only make sense of the truth-conditions for statements quantifying over all propositions in terms of operations that secure the transmissibility of the relevant property. But this does not really get to the heart of the matter. What we really want to know is what those operations would look like. In the case of quantiﬁcation over the ordinals we saw an example of how transmissibility across principles of extendibility might be achieved. In order to see how something comparable might work in the case of quantiﬁcation over all propositions we need a clear idea of what the relevant principle of extendibility might be. We need to know what, in the case of propositions, plays the role that is played for ordinals by the twin principles that every ordinal has a successor that is an ordinal and that the least upper bound of any collection of ordinals is itself an ordinal. Let us revert to Dummett’s original characterization of how we grasp domains given by indeﬁnitely extensible concepts. A necessary but not sufﬁcient condition is that we have clear and unequivocal criteria for determining, of any particular object, whether it falls under the relevant concept—and for determining when an object falling under the concept but given in one way is identical to an object given in another way. When we are dealing with indeﬁnitely extensible concepts we also need, ﬁrst, a clear collection of objects that canonically satisfy the relevant criteria, and, second, a principle (or principles) of extendibility that shows us how the domain is to be extended beyond the canonical base. The domain given by an indeﬁnitely extensible concept is closed under the relevant principle(s) of extendibility. It is straightforward to apply this general model to quantiﬁcation over all propositions. We begin with the distinction between simple and compound propositions, where a simple proposition is one that does not contain any quantiﬁers or truth-functional connectives and a compound proposition is constructed from simple propositions with quantiﬁers, truth-functional connectives or other operators. Plainly the totality of all propositions is the closure of the set of simple propositions under the operations of conjunction, disjunction, and so on—just as the totality of all ordinals is the closure of the empty set under the successor and limit operations. So, we may conclude that the principles of extendibility for the totality of propositions are, in effect, the rules governing the truth-functional connectives, quantiﬁers, and other operators. We can now see what form must be taken by the principles of transmissibility for quantiﬁcation over the totality of all propositions. The principles of transmissibility for a universal quantiﬁcation of the form ∀p Fp must show that

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the property of F-ness is transmitted across the logical operations whose closure gives the totality of propositions. By the same token, the truth-condition for the statement ∀p Fp is, in essence, that all simple propositions are F and that F-ness is transmitted in the appropriate manner across the relevant logical operations. To assert that ∀p Fp is to claim that F-ness is transmitted, and it is this claim that one understands when one understands the statement that ∀p Fp. It appears, therefore, that the concept proposition qualiﬁes as indeﬁnitely extensible by Dummett’s lights, even though it is not (in any obvious sense) mathematical. How does this affect the reasoning that leads to Fitch’s paradox?

IV We note ﬁrst that the set of true simple propositions is a deﬁnite totality. We can run arguments such as that canvassed above on many different proper sub-totalities of the totality of all propositions. If we assume, for example, that ‘0 = 0 ∈ / si ’ expresses a proposition of mathematics (where si is a set of mathematical truths), then a parallel line of reasoning shows that there is no set of true mathematical propositions—and hence that the concept mathematical proposition is an indeﬁnitely extensible concept. No such argument can work, however, for the totality of true simple propositions, since the propositions required to run the argument are not simple propositions. Arguments of this type contain an ineliminable use of negation. They also hinge upon propositions that have simple propositions as constituents. Since the concept true simple proposition is deﬁnite we can quantify over simple propositions unproblematically. We can say, for example, that every true simple proposition is knowable. This might be formulated using restricted quantiﬁcation as follows (where S is the set of true simple propositions): (13) ∀p ∈ S [p ⇒ ♦(p ∈ α) ] By Dummett’s lights this principle has a determinate truth-value and can be asserted and understood in the standard manner. The antecedent effectively restricts us to the deﬁnite totality of simple propositions. The same does not hold, however, for what we can term the unrestricted knowability principle—i.e. (8) ∀p [p ⇒ ♦(p ∈ α) ] Here the quantiﬁer ranges without restriction over an indeﬁnitely extensible totality. By the earlier arguments it appears that the truth-condition for (8) rests upon the transmissibility of the property of knowability across the principles of extendibility that collectively yield the indeﬁnitely extensible totality of all propositions. We can certainly take as our ‘base’ in spelling out how this might work the restricted principle of knowability for simple propositions. But the real

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work in spelling out the truth-condition for (8) comes with what we might term the transmissibility principles—those principles that ensure that the totality of knowable propositions extends and expands at exactly the same rate as the totality of propositions. Since the extendibility principles for the totality of propositions are precisely those governing the truth-functional connectives and the quantiﬁers, we can expect the transmissibility principles to track the principles laying down the truth-conditions for compound propositions. In fact, it is possible to argue that the transmissibility principles just are the principles specifying truth-conditions for compound propositions. If we concede that intuitionistic logic is the appropriate logic for indeﬁnitely extensible totalities then the truth-conditions for compound propositions will have to be given in intuitionistic terms. They will, in fact, take something like the following form, where in each clause the connective/quantiﬁer/operator on the right is to be understood intuitionistically: (14) (15) (16) (17) (18) (19) (20) (21)

‘not ϕ’ is true ⇔ ¬ ϕ ‘ϕ or ψ’ is true ⇔ ϕ ∨ ψ ‘if ϕ then ψ’ is true ⇔ (ϕ ⇒ ψ) ‘ϕ and ψ’ is true ⇔ ϕ ∧ ψ ‘For some x ϕ’ is true ⇔ ∃x ϕ ‘For all x ϕ’ is true ⇔ ∀x ϕ ‘Possibly ϕ’ is true ⇔ ♦ϕ ‘Necessarily ϕ’ is true ⇔ ϕ

The key point here is that, precisely because the connectives/quantiﬁers/operators are being understood intuitionistically, principles (14) through (21) secure the transmissibility of the property of knowability across the indeﬁnitely extensible totality of all (true) propositions. Suppose we take an implication of the form ‘if ϕ then ψ’ where ϕ but not ψ is a simple proposition. If we understand ‘if . . . then . . . ’ intuitionistically then ‘if ϕ then ψ’ is true iff there is some form of procedure that will transform a proof of ϕ into a proof of ψ (we assume that we can formulate an analog of the proof-conditional interpretation of intuitionistic logic when we ﬁnd ourselves outside the domain of mathematical proof ). If we have such a procedure then the conditional is knowable and, by assumption, ϕ is knowable. Hence ψ is knowable. Pari passu for the other logical constants—and, of course, for quantiﬁcation (since ‘∀x’ and ‘∃x’ make ineliminable reference to effective operations when understood intuitionistically). In fact, Dummett himself proposes (Dummett 2001) that the correct response to Fitch’s paradox is an inductively speciﬁed theory of truth of the type given by principles (14) through (19), coupled with a basic principle akin to our (13) to the effect that simple propositions are knowable. Although Dummett (in what can only be described as a rather elliptical paper) does not put the matter in quite these terms, the inductive speciﬁcation secures the knowability of the totality

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of all true propositions without a global quantiﬁcation such as (8)—and hence without allowing the type of substitution that yields Fitch’s paradox. We can see how this works as follows. Suppose that p is a true simple proposition that has never been and never will be known. From (13) we still have that [p ∧ ♦(p ∈ α)]. But this, after all, simply tells us what we already knew, which is that p is a true simple proposition that could be known at some time. Since [p ∧ ♦(p ∈ α)] is a compound proposition we cannot substitute it back into (13) to yield the paradox. The basic point, then, is that (8) is not the correct way to express the basic principle that all truths are knowable. Because we are dealing with a domain given by an indeﬁnitely extensible concept our expression of the basic principle needs to reﬂect the principles of extendibility in terms of which we grasp that domain. In this case these are the principles governing the logical constants and quantiﬁers. By spelling out those principles in an intuitionistic metalanguage, as in principles (14) through (19) we effectively give expression to the basic principle that all truths are knowable without stating it explicitly—and so without either compromising the indeﬁnite extensibility of the concept proposition or opening the door to Fitch’s paradox. The ﬁnal step in the argument is plainly in view in Dummett 2001 where he sets out the inductive speciﬁcation as a way of blocking Fitch’s paradox. Few commentators, however, have seen the rationale for the inductive speciﬁcation (and Dummett himself has nothing to say about it). This paper has tried to make that rationale explicit. As I have argued, the motivation for the inductive speciﬁcation is to be found in the indeﬁnite extensibility of the concept proposition. The inductively speciﬁed theory of truth gives the principles of extendibility that we must grasp if we are to have a grasp of the indeﬁnitely extensible totality of all propositions. If Dummett is right, moreover, that the correct logic to use over a domain given by an indeﬁnitely extensible concept is intuitionistic, then the inductive speciﬁcation must itself be intuitionistic, which is sufﬁcient to ensure that the principles of extendibility are also principles of transmissibility in the sense we have discussed.

Pa r t I I I Pa r a c o n s i s t e n c y a n d Pa r a c o m p l e t e n e s s

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7 Beyond the Limits of Knowledge Graham Priest

1 . In t ro d u c t i o n Are there limits to knowledge? Well, there are certainly many things that we do not, as a matter of fact, know. We do not know (at the moment) whether Iraq will continue its downward spiral into anarchy. We will know in due course. We do not know how to make the Theory of Relativity and Quantum mechanics consistent with each other. Maybe we will in due course. More interesting is the question of whether there are things that it is not possible to know. Perhaps there are things that are so difﬁcult, remote, or recondite, that they transcend anything we could ﬁnd out. If this is the case, there are even limits to what it is possible to know. Whether or not this is so is the main topic of this paper. ‘Possible’ is a highly ambiguous word in philosophy. It can mean ‘logically possible’, ‘physically possible’, ‘epistemically possible’, and doubtless many other things. It may therefore reasonably be asked what sense of possibility is at issue here. The answer is that it doesn’t really matter. For most of the purposes of this paper, it can mean any sense of possibility one likes. One group of people who assert that all truths are knowable (in some appropriate sense) comprises veriﬁcationists, including mathematical intuitionists. For them, this is a constraint on truth itself (or maybe on meaning): everything that is true is such that it is possible (at least in principle) to know it. What sense of ‘possible’ veriﬁcationists have in mind here, I leave them to explain. But at least in their honour, I call the principle that all truths are possibly known the Veriﬁcation Principle. This Principle settles the matter at issue in one way. Ancestors of this paper were given under the title ‘The Limits of Knowledge’ at Mt Holyoke College, the Graduate Center, City University of New York, the University of California at San Diego, the University of Melbourne; and at the conferences The Philosophy of Uncertainty: Epistemic Limits, Probability, and Decision, held at the University of Tokyo, and Logica 2005, held in the Czech Republic. I am grateful to the audiences present for their helpful discussions, and to Masake Ichinose, Tim Childers and Vladimir Svoboda, for organizing the conferences.

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On the hand, there is a well known argument, usually attributed to Fitch, to the effect that the Veriﬁcation Principle is false. If this is the case, then there are some truths that it is impossible to know. This resolves the issue in the opposite direction. We may therefore approach the matter by considering the tenability of the Veriﬁcation Principle in the light of the Fitch argument.

2 . Se t t i n g u p t h e Is s u e Let us start by getting the geography of the issue straight. Some notation: I will use lower case Greek letters for sentences of whatever language is at issue. ♦ and are the usual modal operators of possibility and necessity. Kx will be the predicate ‘is known’, and . is a name-forming device. Now, let T be the set of truths. The question is how what we know relates to this. There are two relevant subsets. The ﬁrst comprises the truths that are (actually) known, K = {x : Kx}. The second comprises the truths that it is possible to know, P = {x : x ∈ T ∧ ♦Kx}. (Note that K α entails that α is true; but ♦K α does not—only that α is possibly true.) Since what is known is possibly known (in any normal sense of possibility), the general relationship between the three sets is as shown in Figure 7.1. T

P

K

Figure 7.1.

K is certainly non-empty. Melbourne, for example, is known to be in Australia. P − K is also non-empty. As I have already observed, there are things about the future that we do not know, but will; so that knowledge is certainly possible. Similarly, the Ancient Greeks did not know that there was a planet beyond Uranus; but it is possible to know this: we do. The status of T − P is less clear. The Veriﬁcation Principle says that α → ♦K α. If this is true, T − P is empty; if there is a counter-example to the Principle then there are truths that it is not possible to know.

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Now to the Fitch argument.¹ This is to the effect that if it is possible to know whatever is true, then everything true is not just possibly known, but actually known. That would appear to be a reductio ad absurdum of the view. It is clear that not everything is actually known—even if one is a veriﬁcationist. A priori, there is something highly suspect about the argument, however. Surely one cannot get from the mere fact that it is possible to know something to the fact that it is known? Informally, Fitch’s argument goes as follows. Suppose that everything true is knowable, and suppose for reductio that there is something, α, which is true but not known, α ∧ ¬K α. Then it must be possible to know this, ♦K α ∧ ¬K α. By a few straightforward inferences concerning knowledge, it follows that it is possible to both know α and not know it, ♦(K α ∧ ¬K α), which it isn’t. 3 . T h e Fi t c h A r g u m e n t

3.1. Stage 1: Knowledge Let me spell out the argument in detail (in natural deduction form), so that we may look at the moves in it more carefully. For the purpose of discussing the argument, and in the cause of simplicity, I will write K α as K α, effectively turning the predicate K into the more usual operator. (As long as we are not quantifying-in, there is no real difference.) The part of the proof concerning knowledge goes as follows. Call it 1 . K ( ∧¬K ) K

[ ∧¬K ]

K ( ∧ ¬K ) ∧ ¬K ¬K

K ∧ ¬K

1 uses four inferences: [ ] ∧ ∧

K

K K

In the fourth of these, the column from β to γ represents an argument with premise β (and only β), and conclusion γ. The square brackets represent the fact that the inference discharges β, so that the ﬁnal argument no longer depends on it. ¹ Fitch (1963). Fitch himself attributes the argument to an anonymous source. It lay dormant for some time, but was published again by Hart and McGinn (1976), whose attention was drawn to it, again, anonymously.

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The ﬁrst two inferences require no comment; nor does the third: what is known is true. The fourth says that knowledge is closed under entailment. This is certainly not correct. What is actually known is not closed under entailment. For example, medieval monks knew that Aristotle was Greek. They did not know that (Aristotle was Greek or the formalism of quantum mechanics deploys Hilbert spaces), even though this entails it. Or consider the Peano postulates. I know all these. But I do not know all their consequences (amongst which are probably the solutions to some famous unsolved problems in number theory). But the Fitch argument cannot be dismantled by simply rejecting this principle of inference. This is because the only use made of the principle in the argument is to infer a special case: that the knowledge of a conjunct follows from the knowledge of a conjunction. Hence, the rule could be replaced by the much simpler: K (β ∧ γ) Kγ This seems much harder to contest.² In particular, the sorts of counter-example just mentioned relevant to the failure of the closure of knowledge under entailment (in general) seem to get very little grip on it. The knowledge of a conjunct seems implicit in the knowledge of a conjunction. There is therefore little scope for faulting this part of the argument.

3.2. Stage 2: Possibility The second part of the argument embeds 1 in an argument concerning possibility. This is as follows, where the right-hand column represents 1 . Call this part 2 . [K (α ∧ ¬K α) ] α ∧ ¬K α ∇ K α ∧ ¬K α ♦K (α ∧ ¬K α) ♦(K α ∧ ¬K α) 2 applies two new rules, which are as follows: [β] .. . β ♦K β

♦β γ ♦γ

² Harder, but not impossible. Connexivist logicians (including some medievals) held that β ∧ γ does not entail γ—for example, if β is ¬γ, this simply cancels out the γ. Such a logician could know β ∧ γ, but not believe, and a fortiori know, γ. To avoid this kind of problem we can just restrict the class of knowers in question to those who have the normal beliefs about the validity of inferences concerning conjunction—which includes us.

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The ﬁrst of these is simply the Veriﬁcation Principle, which is what the argument assumes (for the sake of reductio). The second says that possibility is closed under entailment. This seems to hold for any notion of possibility. If α is true in a possible world (of any appropriate kind), and α entails β, then β is true in that world, and so possible (in the same sense). There is little in this stage of the inference that one can balk at, then.

3.3. Stage 3: Contraposition The third part of the argument embeds 2 in an argument deploying negation. This is as follows, where the left-hand column represents 2 . Call this 3 . [α ∧ ¬K α] ∇ ♦(K α ∧ ¬K α) ¬♦(K α ∧ ¬K α) ¬(α ∧ ¬K α) 3 employs one premise and one further rule of inference. The premise is ¬♦(β ∧ ¬β), or equivalently, given the usual connections between and ♦: ¬(β ∧ ¬β) The inference is contraposition: [β] .. . γ

¬γ ¬β

The only plausible way to contest these steps is to suppose that contradictions may be true. The rationale for contraposition is that if β delivers something that is not true, γ, it must be false. This rationale collapses if γ can be true despite the truth of ¬γ. Unsurprisingly, then, the inference fails in many paraconsistent logics (including the one whose semantics I will describe below). Suppose, for example, that the logic contains the Law of Excluded Middle (LEM), β ∨ ¬β. Then we have γ β ∨ ¬β. Contraposing, ¬(β ∨ ¬β) ¬γ, that is (assuming De Morgan Laws), β ∨ ¬β ¬γ—which fails, since γ was arbitrary. This stage of the argument may therefore be broken by appealing to dialetheism. It might be thought that dialetheism would invalidate the new premise of the argument as well: if contradictions may be true, one might expect ¬(β ∧ ¬β), and so its necessitation, to fail. Surprising as it might be to those meeting paraconsistency for the ﬁrst time, it does not. There are many paraconsistent logics where the law holds (including the one whose semantics I will describe below). Of course, any contradiction, β ∧ ¬β, will then generate a secondary

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contradiction, (β ∧ ¬β) ∧ ¬(β ∧ ¬β), but there is nothing in a paraconsistent logic to rule this out. Actually, the simplest way of avoiding ¬(β ∧ ¬β) (and so its necessitation) is to appeal, not to truth-value gluts, but to truth-value gaps. If β is neither true nor false, so (given the natural semantics for the connectives) is ¬(β ∧ ¬β). Appealing to truth-value gaps also invalidates contraposition unless the logic is paraconsistent. If the logic is not paraconsistent, we have β ∧ ¬β γ, and so ¬γ ¬(β ∧ ¬β), i.e., ¬γ β ∨ ¬β, which is not the case if we do not have the LEM. It might therefore be thought that appealing to truth-value gaps is a way of avoiding the argument without an appeal to gluts. Unfortunately (for the friends of consistency) it is not. As 2 shows, given the Veriﬁcation Principle, α ∧ ¬K α already leads to ♦(K α ∧ ¬K α), and thus to the possibility of true contradictions. Moreover, if the logic is not paraconsistent, we have, for an arbitrary β, K α ∧ ¬K α β. By the closure of possibility under entailment, we have ♦(K α ∧ ¬K α) ♦β. Given that ♦(K α ∧ ¬K α), everything is possible—not an enticing conclusion (for the friends of consistency). One way or another, then, true contradictions are required to break this step of the argument.

3.4. Stage 4: Double negation There is one ﬁnal part of the argument. This embeds 3 in the argument which actually takes us from α to K α. This goes as follows, where the right-hand column represents 3 . ¬♦(K α ∧ ¬K α) ∇ ¬(α ∧ ¬K α) ¬¬K α Kα This stage of the argument uses contraposition again, discharging ¬K α. (And in this application, there is also another assumption in the sub-proof. As is to be expected, this does nothing to restore validity in a paraconsistent logic. It just makes matters worse.) It uses one further rule, double negation: α [¬K α] α ∧ ¬K α

¬¬β β Double negation fails in intuitionist logic, which is intimately connected with veriﬁcationism. Hence, breaking the argument by denying this step is a very plausible move. If we do, we can get from α only to ¬¬K α, which is not so bad. Well, not really. Given α → ¬¬K α, we obtain ¬¬¬K α → ¬α by a

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form of contraposition that is intuitionistically valid. And in intuitionist logic, ¬β ↔ ¬¬¬β. So by transitivity, ¬K α → ¬α. Even intuitionists cannot accept this in general. Let α be ‘Alpha Centauri has a planetary system’. I do not know that α; I do not know that ¬α. (Nor does anybody else—maybe for ever.) It cannot follow that ¬α and ¬¬α.³ 4 . A Si m p l e Mo d e l We have seen that appealing to dialetheism breaks the Fitch argument against veriﬁcationism. We can do more than this, however. It can be shown that once contraposition (and only contraposition) is removed from the principles employed, the inference from α to K α is not forthcoming. I demonstrate this with a counter-model based on the semantics for a simple paraconsistent modal/epistemic logic.⁴ Interpretations are of the form W , ∞, R, S , ν. W is a set of worlds. ∞ is a distinguished member of W. R is the modal binary accessibility relation, and we require that for every w ∈ W , wR∞. S is the epistemic binary accessibility relation, which is at least reﬂexive. ν maps every world and propositional parameter to {1}, {0} or {1 , 0} (true, false, both). I write the value of α at w as νw (α). Truth-conditions at worlds, w, other than ∞ are as follows: 1 0 1 0 1 0 1 0

∈ νw (α ∧ β) iff 1 ∈ νw (α) and 1 ∈ νw (β) ∈ νw (α ∧ β) iff 0 ∈ νw (α) or 0 ∈ νw (β) ∈ νw (¬α) iff 0 ∈ νw (α) ∈ νw (¬α) iff 1 ∈ νw (α) ∈ νw (♦α) iff for some w such that wRw , 1 ∈ νw (α) ∈ νw (♦α) iff for all w such that wRw , 0 ∈ νw (α) ∈ νw (K α) iff for all w such that wSw , 1 ∈ νw (α) ∈ νw (K α) iff for some w such that wSw , 0 ∈ νw (α)

∞ is the trivial world. That is, for every α: ν∞ (α) = {1 , 0} Validity is deﬁned in terms of truth-preservation at all worlds. Leaving aside the Veriﬁcation Principle for the moment, it is not difﬁcult to check that the semantics verify all the inferences involved in the Fitch argument (including the closure of knowledge under entailment, and the premise ¬♦(β ∧ ¬β) ) except contraposition. ³ On this and related objections, see Percival (1990). ⁴ This is an extension of the propositional paraconsistent logic LP (see Priest (1987), ch. 5). The existence of the trivial world, ∞, does not affect the logic of the extensional connectives.

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For the veriﬁcationist inference: for any α, 1 ∈ ν∞ (K α), and so for every w (including ∞), 1 ∈ νw (♦K α). The inference α ♦K α is therefore (vacuously) valid. To ﬁnish the job, we just need an interpretation where there are worlds, w 0 and w 1 , such that 1 ∈ νw0 (p) , w 0 Sw 1 , but 1 ∈ / νw1 (p). Then 1 ∈ / νw0 (Kp). We can depict the simplest interpretation of this kind as shown in Figure 7.2 (+ indicates that a formula holds; − indicates that it fails; square brackets indicate things that hold at worlds, other than what is part of the speciﬁcation). Notice that R and S can be made as strong as one likes without ruining the argument. In other words, the modal logic of K and ♦() can be beefed up to S5 without affecting the result. ∞

S w0

p+

[◊Ka+, Kp −]

R R S

[Ka+, ◊Ka+] S w1

p−

[◊Ka+]

Figure 7.2.

Note, also, that we may take the language to contain a conditional operator, →, with strict truth/falsity conditions as follows.⁵ At any world, w, other than ∞: 1 ∈ νw (α → β) iff for all w such that wRw , if 1 ∈ νw (α) then 1 ∈ νw (β) 0 ∈ νw (α → β) iff for some w such that wRw , 1 ∈ νw (α) and 0 ∈ νw (β) Assuming that R is reﬂexive, these semantics verify (at least) the inferences: [α] .. . α

α→β β β α→β (where α is the only undischarged assumption in the second inference). In the above model (with the additional proviso that R is reﬂexive), α → ♦K α holds for all α at w 0 , but p → Kp fails. 5 . E n t e r t h e K n owe r We have seen that the Fitch argument may be blocked by an appeal to dialetheism. Moreover, it is the only way that we have found in which the argument may ⁵ See Priest (1987), ch. 6.

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be blocked.⁶ But—it might well be argued—an appeal to dialetheism in this context is extreme and unmotivated. Better to take the argument to be a simple reductio of the Veriﬁcation Principle. Matters are not that simple, though. First, there are situations in which the Veriﬁcation Principle appears to hold (at least in some sense of possibility) and where the agent in question does not know everything true. It is coherent, I take it, to suppose the existence of an omniscient (and omnipotent) being. Let us call them ‘God’. Everything true it is possible for God to know; indeed, everything true God actually does know. But God has a friend; call him ‘Gabriel’. Gabriel is not omniscient. There are many things that Gabriel doesn’t know, and doesn’t care about—such as who won the 4.30 at Flemington. But Gabriel knows at least that God is omniscient. Moreover, he knows that he can always ask God if he wants to know something; God, being a decent and trustworthy fellow, will tell him. Hence, anything that is true, it is possible for Gabriel to know—just by asking. Yet Gabriel does not know everything true. The Fitch argument must therefore fail. The Fitch argument itself suggests an objection to this. Let us suppose that Red King Hit won the 4.30 at Flemington—call this κ—and that, as a matter of fact, Gabriel does not know this, since he never bothers to ask. Then: (∗ )

κ and Gabriel does not know (at any time) that κ

is true. God knows it. It might be argued that it is, none the less, not possible for Gabriel to know it. To do so, he would have to know κ and know that he does not know κ (at any time), which is impossible. But could he not ask God whether (∗ ) is true, and get an answer? Of course he could. If, as we suppose, (∗ ) is true, God will tell him so. Hence, Gabriel will know κ, and (∗ ) is false. Suppose, on the other hand, that (∗ ) is false. Then God will tell him so. At this point Gabriel still does not know whether κ is true or false. Suppose we then shoot him; he never will. So (∗ ) is true. None of this shows that Gabriel cannot know (∗ ); all it shows is that, if he does ask the question, the situation is a paradoxical one. In fact, the paradox is a version of a well known one—the Bridge. A person has to cross a bridge; on the other side there is a bridge-keeper who asks a question. If the person answers truly, they are allowed to pass; if not, the bridge-keeper hangs them. The bridge-keeper asks ‘what will you do when you get to the other side of the bridge?’ The person answers ‘I will be hanged by you’.⁷ Again, the question forces a paradoxical situation. ⁶ Human ingenuity being what it is, there may, of course, be other suggestions. A number of these are discussed (and rejected) in Williamson (2000a), Ch. 12. The chapter also contains references to other discussions of the argument in the literature. ⁷ The paradox is one of Buridan’s sophismata but, according to Sorensen, it probably goes back to Chrysippus. A version of it is told by Cervantes in Don Quixote. See Sorensen (2003), pp. 207–9.

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A much simpler version of the paradox is forthcoming by just letting κ be the sentence ‘Gabriel (or even God) does not know κ’. Let us make this more precise. By applying techniques of self-reference, we can construct a sentence, κ, that says of itself that it is not known. That is, κ is of the form ¬K κ. (I now revert to writing K as a predicate. Self-referential constructions require this.⁸) Suppose that K κ; then κ is true, so ¬K κ. Hence, ¬K κ. That is, κ, but we have just demonstrated this, so it is known to be true, K κ. (This is the Knower paradox.) We have demonstrated K κ ∧ ¬K κ. This is therefore necessarily true (in whatever sense of necessity one cares for); a fortiori, ♦(K κ ∧ ¬K κ). And the Veriﬁcation Principle ﬁgures nowhere in the argument for this. We see, in particular, that quite independently of the Fitch argument there are sentences of the form required to invalidate the contraposition in 3 . Appealing to dialetheism to break the Fitch argument is, therefore, not at all ad hoc or unmotivated. In the context, it is very natural.⁹

6 . C o n t r a d i c t i o n a n d t h e L i m i t s o f K n ow l e d g e We can bring this to bear explicitly on the question of the limits of knowledge as follows. Let X ⊆ K. Provided that X has a name, and given appropriate techniques of self-reference, we can form a sentence that says of itself that it is not in X ; that is, a sentence, αX , of the form αX ∈ / X . We can show that αX ∈ /X but that αX ∈ K as follows: αX ∈ X ⇒ αX ∈ K ⇒ αX ⇒ αX ∈ /X Hence, αX ∈ / X . But this is αX , and we have just established this, so it is known to be true; that is, αX ∈ K. The situation may be depicted as shown in Figure 7.3. When X is the empty set, αX can be located anywhere in K − X (= K). As X gets bigger and bigger,¹⁰ there is less and less space in which αX can be consistently located; until, at the limit, when X coincides with K there is nowhere consistent for αX to go. αK ∈ K ∧ αK ∈ / K. (This is the ⁸ In fact, we can maintain K as an operator provided that we have a truth-predicate, T , in the language. We can then deﬁne an appropriate predicate, K x, as KTx. (Thanks to Jc Beall for this observation.) ⁹ The ﬁrst person to moot the possibility of a connection between the Fitch argument and the Knower was Routley (1981) (see esp. p. 112, n. 26). The connection was made more robustly by Beall (2000). ¹⁰ Of course, X does not literally grow. In particular, we are not considering the case where more and more is known. (That would be a case of K growing.) This is just a picturesque way of saying that for larger and larger X . . .

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Knower paradox. κ is just αK .) The limit of what is known is dialetheic. That is, there are certain truths that are both within the known and without it. K X × aX

Figure 7.3.

Exactly the same is true of P. Let X ⊆ P. As before, we can construct a sentence, αX , of the form αX ∈ / X. αX ∈ X ⇒ αX ∈ P ⇒ αX ∈ T ∧ ♦K αX ⇒ αX ⇒ αX ∈ /X Hence, αX ∈ / X . But this is αX , and we have just established this, so it is true and known to be so, K αX . A fortiori, it is possible to know it, ♦K αX . Thus, αX ∈ T ∧ ♦K αX . That is, αX ∈ P. Just as with K, when X is small, there is plenty of room for αX to reside, consistently, outside it but inside P. As X gets bigger and bigger, there is less and less room, until when X is P, a contradiction arises: αP ∈ P ∧ αP ∈ / P. The boundary of possible knowledge is inconsistent too. An Inclosure involving a set, , a predicate, ψ, and a function, δ, is a structure satisfying the following conditions: 1. ψ() 2. if X ⊆ and ψ(X ) (a) δ(X ) ∈ / X (Transcendence) (b) δ(X ) ∈ (Closure) Whenever we have an Inclosure, a contradiction arises at the limit, when X = . For we then have δ() ∈ / ∧ δ() ∈ . All the standard paradoxes of self-reference are limit-paradoxes of this kind.¹¹ The two contradictions we have just looked at are of this form. In the ﬁrst, is K; in the second, is P. In both, ψ(X ) is ‘X is deﬁnable (has a name)’, and δ(X ) is αX . Hence, both are inclosure contradictions. ¹¹ See Priest (1995), part 3. For the Knower paradox, see 10.2. There, is deﬁned as {x : ϕ(x)}, where ϕ is the appropriate predicate.

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7. Conclusion Let us recall our original diagram, and take stock (Figure 7.4). K, we know, is non-empty, as is P − K. And this may be so even if the Veriﬁcation Principle is correct, and so T − P is empty, since the Fitch argument fails. We have also learned that the boundaries between K and T − K, and P and T − P are dialetheic. That is, there is a true sentence, αK , such that αK ∈ K and αK ∈ / K, and a true sentence, αP , such that αP ∈ P and αP ∈ / P. (This is what the ‘×’s on the new version of the diagram indicate.) And since αP is true, αP ∈ T ∧ αP ∈ / P, so T − P is also non-empty. For all I have said, this might be its only denizen. It cannot, therefore, be ruled out that T − P is empty as well. Whether or not this is so might well depend on the sense of possibility at issue. It is, at any rate, a matter for another occasion. T

P

K

Figure 7.4.

×

×

8 Knowability and Possible Epistemic Oddities Jc Beall

1 . No n - o m n i s c i e n ce a n d t h e K n ow a b i l i t y Ru l e Our world is non-omniscient. Nobody knows all truths, and nobody ever will. Does it follow that there are unknowable truths? Frederic Fitch (1963) ‘proved’ the afﬁrmative. In short, if some truth is unknown, then that it is unknown is itself unknowable; hence, given non-omniscience, there is some unknowable truth. Veriﬁcationists, who tie truth to veriﬁability, are committed to the so-called knowability rule (henceforth, KP).¹ Let K be the epistemic operator it is known by someone at some time that . . . , and ♦ the aletheic it is possible that. . . . KP is the following rule. α ♦K α Non-omniscience gives us α ∧ ¬K α, for some α. KP, in turn, gives us ♦K (α ∧ ¬K α). A few related rules governing ♦ and K quickly yield ♦(K α ∧ ¬K α), the possibility of ‘true contradictions’. (For the relevant rules, see Section 2) Fitch’s proof qua reductio makes the ﬁnal step: KP is unsound. In this paper, my concern is not so much with Fitch’s ‘proof’ against KP (or the conditional version). The proof is blocked in familiar ‘paraconsistent’ and ‘paracomplete’ logics (see Sections 2 and 3), both of which are independently motivated and, hence, available to veriﬁcationists. Rather, my concern is with the apparent commitment to ‘possibly true contradictions’. For discussion I am grateful to Colin Caret, Carrie Jenkins, Graham Priest, Greg Restall, and various members of the AHRC Arch´e Centre for Logic, Language, Mathematics, and Mind. Thanks also to Joe Salerno for editing the volume. ¹ The key idea is often given as a (universally quantiﬁed) conditional principle, but, to simplify current discussion, I focus on the rule form. ( The conditional version is often called ‘KP’, short for the knowability principle, but little confusion should result.)

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Regardless of its effect on veriﬁcationism, Fitch’s ‘proof’ highlights the oddity of epistemic optimism in a non-omniscient world. Given non-omniscience, KP involves something out of the ordinary. My main aim is to brieﬂy explore a few options for cashing out the given oddity. The paper runs as follows. In Section 2, I brieﬂy set out the relevant rules (and one corresponding premise) on which the Fitch argument relies. Section 3, in turn, brieﬂy reviews the main point of Beall, which suggests a paraconsistent response to Fitch’s ‘proof’ qua reductio of KP; however, the same considerations also motivate a similar paracomplete, non-paraconsistent, response, on which I will focus.² In Section 4 I set out the main focus: what to make of Fitch’s (initial) argument for the ‘possibility of gluts’. While the paracomplete response undercuts Fitch’s ‘proof ’ qua reductio, the issue of ‘possible, true contradictions’ remains open—especially in a nonparaconsistent framework, which is the target. I explore two options. In Section 5 I suggest but reject a ﬂat-footed option: living with possible—but merely possible—inconsistency. Section 6, in turn, explores the other option: avoiding even the ‘mere possibility’ of ‘true inconsistencies’. As a sort of synthesis, Section 7 brieﬂy sketches another option: a paracomplete and paraconsistent framework. Section 8 closes with some general comments and (brief ) responses to objections. 2 . Fi t c h’s Pro o f, i n Sh o r t For present purposes, the basic rules, involved in Fitch’s proof, may be divided into four categories: epistemic, (aletheic) modal, modal–epistemic (viz., KP), and ‘background’. The rules run as follows.³

1. Epistemic rules Veridicality (KV). The idea is that ‘knowledge implies truth’. Kα α Distribution (KC). That a conjunction is known implies that its conjuncts are known. K (α ∧ β) Kα∧Kβ ² See Priest’s (Ch. 7, this volume) for a development of the LP-based paraconsistent (indeed, dialetheic) position, and Section 7 for an alternative paraconsistent framework. ³ To facilitate comparison with Priest’s (Ch. 7, this volume), which discusses the related paraconsistent—indeed, dialetheic—response, I use a natural deduction version of the rules. As in Priest’s paper, [α], in the context of rule (or proof ), indicates a discharged assumption.

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2. Aletheic modal rules Non-contradiction (LNC). It is false that it’s possible that α ∧ ¬α is true, for any α.⁴ ¬♦(α ∧ ¬α) Closure (CP). That α is possible and that α implies β implies that β is possible. [α] .. . β

♦α ♦β

3. Modal–Epistemic rule Knowability (KP). The idea is that ‘truth implies knowability’, the key veriﬁcationist position. α ♦K α

4. Background Rules Adjunction and Simpliﬁcation. Conjunction behaves normally. α β α∧β

α∧β α β

Contraposition. That β is not true and that α implies β implies that α is not true. [α] .. . β

¬β ¬α

Fitch’s Proof, in short: Suppose, for reductio, α ∧ ¬K α, for some α. KP yields ♦K (α ∧ ¬K α). Given KC, we have that K (α ∧ ¬K α) K α ∧ K ¬K α. Simpliﬁcation, VK, and Adjunction (and transitivity of implication), yield that K (α ∧ ¬K α) K α ∧ ¬K α. But, then, CP gives ♦(K α ∧ ¬K α). LNC, in turn, delivers ¬♦(K α ∧ ¬K α). Contradiction. ⁴ Given the inter-deﬁnability of α and ¬♦¬α, which is assumed, LNC (as here put) amounts to the validity of ‘inferring’ ¬(α ∧ ¬α), for all α, from no premises—i.e., as theorem.

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3 . Ve r i ﬁ c a t i o n i sm , t h e K n owe r, a n d Fi t c h’s ‘ Pro o f ’ While I am not a veriﬁcationist, I agree with those who think that veriﬁcationism is not undermined by the ‘proof’. The result shows only that, given nonomniscience, veriﬁcationists cannot consistently endorse all of the rules involved in Fitch’s ‘proof ’. Since the rules in question are largely ‘classical’ (e.g., LNC, Contraposition), veriﬁcationism is best understood in a non-classical framework, one in which some of the given rules are invalid. One might worry that going non-classical is ad hoc. Were there no independent reason to reject some of the given rules, the worry would be warranted. But there are independent reasons to reject some of the given rules. Familiar semantic paradoxes, cases of vagueness, or other commonly ‘deviant’ phenomena, motivate familiar logics in which some of the given rules fail—notably, the LNC or Contraposition. Consider, for example, the Knower paradox, which involves a sentence κ that says of itself (only) that it is not known.⁵ Given LEM, κ is either known or not. In the latter case, κ is not known, and hence true. In the former case, KV gives us that κ is true, in which case κ is not known. Either way, κ is not known, and, so, κ is true. But, now, we have a proof that κ is true, and hence—on the basis of our proof—we know that κ is true. The upshot: there is a sentence, namely, κ, such that we know that κ is true but, as κ says, do not know that κ is true. In response to the Knower (or many such paradoxes), one might, as in Beall (2000), take the Knower to independently motivate ‘dialetheism’ with respect to knowledge—that K α ∧ ¬K α is true, for some α. On such a line, a paraconsistent logic, in which such inconsistency is ‘harnessed’, is motivated. But, then, at least Contraposition is invalid—and, hence, Fitch’s ‘proof ’ fails. Graham Priest (Ch. 7, this volume) advocates just such a line in the context of LP, a paraconsistent logic in which both LEM and LNC are valid. On the other hand, a paracomplete (and non-paraconsistent) response to the Knower is equally natural. A paracomplete logic is one in which LEM is invalid.⁶ Unlike the dialetheic response, a paracomplete theorist rejects the Knower instances of LEM. One familiar paracomplete framework is K 3 , the Strong Kleene framework.⁷ In such a logic, not only is Contraposition invalid, ⁵ Here, I simply use ‘is known’ rather than the operator. In a suitably non-classical framework, we can enjoy a genuine (intersubstitutable) truth predicate that, in turn, diminishes the importance of distinguishing between operators and predicates. ⁶ Accordingly, Intuitionistic logic counts as paracomplete. I will not discuss the Intuitionistic options, as these are well known. (Besides, Intuitionistic logic does not afford viable options for the broader class of paradoxes—e.g., Liars, etc.) ⁷ I set issues of a suitable conditional aside, and concentrate mostly on the given ‘rules’. A suitable conditional is an important issue, but it is one that would take the discussion too far aﬁeld.

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but LNC is also invalid.⁸ Accordingly, a veriﬁcationist who, for independent reasons, endorses K 3 (or some suitable extension), need not worry about Fitch’s proof qua reductio of KP. While I am (very) sympathetic with the paraconsistent—indeed, dialetheic—framework, I will focus on non-dialetheic and, except, for Section 7, non-paraconsistent but paracomplete responses. Either way, veriﬁcationists have independent reason—e.g., Knower or the like—to endorse a non-classical logic in which Fitch’s ‘proof’, qua reductio of KP, fails.

4 . T h e Re a l Is s u e : Po s s i b l y Tr u e Gl u t s ? Though veriﬁcationists needn’t worry about Fitch’s proof qua reductio of KP, there is more to Fitch’s argument than the ﬁnal few (reductio) steps. As in Section 1, Fitch’s argument highlights the oddity of epistemic optimism in a non-omniscient world. Contraposition and LNC aside, the remaining rules (see Section 2) still leave a curiosity: the apparent commitment to ‘possibly true contradictions’. To see the issue, we assume an extension of K 3 in which the relevant rules remain, except, of course, for Contraposition and LNC. (See Section 5 for the natural semantics.) The initial steps of Fitch’s proof still go through: nonomniscience gives us α ∧ ¬K α, for some α. KP gives us ♦K (α ∧ ¬K α). But K (α ∧ ¬K α) K α ∧ K ¬K α from KC. Simpliﬁcation, VK, and Adjunction (and transitivity of implication), yield that K (α ∧ ¬K α) K α ∧ ¬K α. But, then, CP gives ♦(K α ∧ ¬K α). So, losing Contraposition or LNC still leaves the noted oddity. In a non-paraconsistent setting, ‘possibly true contradictions’ are at least curious. The real issue, then, is what to make of the given oddity in a non-paraconsistent, paracomplete setting. How should a paracomplete, nonparaconsistent veriﬁcationist—or KP theorist, in general—respond to the apparent ‘possibly true contradictions’? As in Section 3, I will explore two salient options. The ﬁrst option is to simply live with the given ‘oddity’. The second option, rejecting even the ‘mere possibility’ of ‘true inconsistency’, involves expanding one’s space of possibilities while restricting one’s account of validity. After discussing such (non-paraconsistent) options, I turn to a brief sketch of a ‘compromise’, a non-dialetheic but nonetheless paraconsistent and paracomplete framework. I will then close (in Section 8) with general comments, brieﬂy answering two objections. ⁸ According to K 3 , ‘Explosion’ is valid: α ∧ ¬α β. But LEM is invalid: ¬(α ∧ ¬α). Were Contraposition valid, we’d immediately have ¬β ¬α ∨ α, for any β and α. But we don’t have that in K 3 , since, as said, we do not have LEM.

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5 . L i v i n g w i t h Me re l y Po s s i b l e Gl u t s Given non-omniscience, veriﬁcationists—and KP theorists, in general—are apparently committed to the possibility of ‘true contradictions’. In a nonparaconsistent context, which is the chief concern in this paper, such a commitment is curious. The question is: what to make of it? The ﬂat-footed option is to just live with it. On the surface, the ‘possibility of true contradictions’ is startling. Upon inspection, though, the situation is in many respects mundane, especially if there’s exactly one—non-actual, merely possible—such ‘possibility’. The ﬂat-footed response acknowledges a (unique) trivial world, and she learns to live with it. To make the idea clearer, I brieﬂy sketch a basic—paracomplete but nonparaconsistent—semantics. I then return to the ﬂat-footed response.

5.1. Paracomplete semantics with the trivial world We are considering a paracomplete and non-paraconsistent framework for veriﬁcationism (or KP theorist, in general), one in which there’s a unique possibility of ‘true contradictions’. By way of contrast with the natural LP-based paraconsistent framework,⁹ I focus on an extension of K 3 , the Strong Kleene framework. Our set of semantic values, namely, V = {1 , .5 , 0}, is ordered in the standard way. D, our designated values, comprises exactly 1. In addition to our usual extensional connectives, we add two unary connectives, the epistemic K and the aletheic ♦. (We deﬁne as ¬♦¬.) K and ♦ are intended to be modal connectives. Accordingly, we pursue a modal extension of K 3 . Interpretations are structures W , R, E , v, w⊥ , where W ∩ {w⊥ } comprises ‘worlds’, with w⊥ ∈ / W the trivial world. R and E are binary relations on W ∪ {w⊥ } (each at least reﬂexive), and v : A × W → V is a valuation from atomics and worlds into {1 , .5 , 0}. For convenience, we let vw (α) = v(α, w), this being the value of α at w. The value of any sentence at any w ∈ W is achieved via the following clauses.¹⁰ vw (¬α) vw (α ∧ β) vw (α ∨ β) vw (♦α) vw (K α)

= = = = =

1 − vw (α) m i n{vw (α) , vw (β)} m a x{vw (α) , vw (β)} m a x{vw (α) : wRw for any w ∈ W ∪ {w⊥ }} m i n{vw (α) : wEw for any w ∈ W ∪ {w⊥ }}

⁹ See Priest (Ch. 7, this volume). ¹⁰ Except for the clause concerning w⊥ , the following are the standard Kleene clauses for modal connectives.

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With respect to w⊥ , the trivial world, the clause for any interpretation is the obvious (trivial!) one: v⊥ (α) = 1 Finally, we deﬁne validity as ‘truth-preservation’ over all worlds of all interpretations. A few features of the framework are notable. To begin, the semantics is clearly paracomplete in that α ∨ ¬α is invalid. Just consider a model in which vw (α) = 0.5, for some w ∈ W. Since, as one may verify, α ∨ ¬α and ¬(α ∧ ¬α) are equivalent in the semantics, the same (counter-) model serves to invalidate LNC (see Section 3). Similarly for Contraposition.¹¹ On the other hand, it is clear that Adjunction and Simpliﬁcation are valid. Moreover, and more to the current point, the remaining epistemic and aletheicmodal rules are all validated. (See Section 2.) KV. Suppose that vw (K α) = 1, for some w ∈ W and α. Since R is reﬂexive, we have it that vw (α) = 1. KC. Suppose that vw (K (α ∧ β) ) = 1. Then vw (α ∧ β) = 1 = vw (α) = vw (β) for all w ∈ W such that wRw . But, then, vw (K α) = 1 = vw (K β). CP. Suppose that α β and, for some interpretation, vw (♦α) = 1. Then vw (α) = 1, for some w such that wRw . But, by supposition, there’s no world, in any interpretation, at which α is true and β not true. Hence, vw (β) = 1, and so vw (♦β) = 1. The question, of course, turns to our essential modal–epistemic rule KP, which is not valid on the current semantics. Can the semantics be tweaked to ensure the validity of KP? Yes. Indeed, the whole point of invoking w⊥ , which has thus far played no role, is to ensure the validity of KP. To achieve KP we stipulate that wRw⊥ , for all worlds w (including w⊥ ). That KP is now valid is obvious; it is vacuously so.¹² So, except for Contraposition and LNC—which, as in Section 3, are suspect for independent reasons—the current framework preserves all of the key rules, including KP. Because KP is preserved, (actual) non-omniscience forces an oddity: the trivial world. Indeed, so long as validity is deﬁned as ‘all points validity’ (e.g., truth-preservation at all worlds), then, unless one goes with a paraconsistent framework, I see no way to avoid the trivial world without giving up KP.¹³ The issue, to which I now return, is whether such oddity is too odd. ¹¹ The corresponding LP-based paraconsistent framework validates LNC but, as here, not Contraposition. See Priest’s (Ch. 7, this volume). ¹² Compare the LP-based dialetheic model (Ibid.), which likewise invokes w⊥ for the same job. (One difference, of course, is that the trivial world naturally falls out of the LP framework, whereas here it is at least curious.) ¹³ If one endorses a paraconsistent logic, a more natural paracomplete framework might be had. See Section 7 for a sketch.

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5.2. The ﬂat-footed response KP, the reﬂection of high epistemic optimism, produces an oddity in a non-omniscient world. The apparent oddity, given the going rules (except Contraposition and LNC), is the possibility of true contradictions. The ﬂatfooted response to such oddity is to accept it, but accept it as merely possible and, importantly, a unique case. Given non-omniscience, KP is preserved in virtue of the unique trivial world—the possibility in which ‘true contradictions’ occur. Is the trivial world too high a price to pay for KP? The answer is not obvious. Admittedly, it may be very difﬁcult to fully understand the trivial world. While one can easily understand that the trivial world is the world at which every sentence is true, it is not easy to understand what such a world is like. Still, there are a few things that can be said on the trivial world’s behalf. 1. KP! As in Section 5.1, if validity is to be understood as all points validity (e.g., truth-preservation at all worlds), it is difﬁcult to retain KP without the trivial world—unless one goes with a paraconsistent logic, which is set aside at this stage. (See Section 7.) So, one virtue of the trivial world is that it affords the chief desideratum for a non-classical veriﬁcationism: it preserves KP. 2. Concrete Explosion! In the current semantics we have ‘explosion’, that is, α, ¬α β. In many (most) non-paraconsistent logics, explosion itself is vacuously achieved: it is valid in virtue of no interpretation in which the premises are true. Here, we have ‘concrete evidence’ of explosion: any world in which α ∧ ¬α is true is the explosive one in which everything is true. There is something to say for such ‘concrete evidence’ (although I wouldn’t put too much weight on this). 3. Merely possible! Similarly, while the possibility of ‘true contradictions’ sounds startling at ﬁrst, the current proposal is rather mundane. After all, in discussing the possibility of ‘true contradictions’, one may quickly point out that we’re talking about a unique and limit case—the merely possible trivial world. In addition to (1)–(3), there is another—perhaps the strongest—point to consider. As throughout, KP’s validity in a non-omniscient world is indeed odd. One reason we might think it odd is that it clashes against the ‘normal’ behaviour of our connectives—which behaviour, perhaps, is by and large classical. The trivial world, which, on the current proposal, is the result of KP’s validity (and a non-omniscient world), might best be seen as a world in which our connectives ‘go on holiday’. Clearly, the connectives are not behaving normally at w⊥ . Perhaps such abnormal behaviour is the price of KP’s validity, given non-omniscience.

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5.3. Trouble with ﬂat-footedness Despite its virtues (if virtues they be), the trivial world is nonetheless disappointing in the current context. While I do not think the trivial world itself is terribly objectionable, its role in the current context is prima facie problematic. The heart of veriﬁcationism is KP, a rule that, at least traditionally, has served to distinguish veriﬁcationists from non-veriﬁcationists. On the current proposal, KP is preserved—indeed, its validity achieved—solely in virtue of the trivial world. But, now, the traditional role of KP cannot be served. After all, it is obvious that anyone—even a classical logician—could acknowledge the trivial world, at least in the fashion in Section 5.1. But if anyone can have the trivial world, anyone can have KP. Surely veriﬁcationism is more demanding than that.¹⁴ The ﬂat-footed response, then, is ultimately unsatisfactory. Unfortunately, without going paraconsistent (though not necessarily dialetheic), there is no obvious way to preserve KP without the trivial world, at least if validity remains ‘all points validity’. Giving up such a notion of validity provides an alternative paracomplete approach, to which I now brieﬂy turn. 6 . Ab n o r m a l Ep i s t e m i c Po s s i b i l i t i e s In Section 5, I suggested—but found wanting—the ‘ﬂat-footed’ paracomplete response to the veriﬁcationist’s apparent commitment to possibly true contradictions. Might an alternative paracomplete response do away with the ‘possibly true contradictions’ altogether? In this section, I brieﬂy explore one route towards doing as much.¹⁵ On this approach, the oddity of KP in a non-omniscient world is not ‘possibly true contradictions’, but rather the sheer oddity of possibly knowing an unknown truth. I will ﬁrst give a philosophical sketch of the idea, followed by a slightly more formal sketch, and then offer a few comments on the overall framework.

6.1. The philosophical story Veriﬁcationists tie truth to veriﬁcation. An essential ingredient of the connection is reﬂected in (at least) KP. What Fitch seemed to show is that, given nonomniscience, KP leads to possibly true contradictions. But perhaps another ¹⁴ I should say that this point might affect Priest’s LP-based proposal (Ch. 7, this volume). A more natural (paraconsistent) approach, not subject to the same problem, is brieﬂy sketched in Section 7. ¹⁵ Other options are available, of course, if the extensional connectives (negation, conjunction, disjunction) behave non-standardly, but I am chieﬂy interested in ‘normal’ behaviour for such (extensional) connectives—i.e., classical input, classical output. (One could go weaker than Strong Kleene, but independent motivation for such logics is more difﬁcult to ﬁnd than for the K 3 case.)

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lesson may be drawn. In particular, what veriﬁcationists are committed to is not some possibly true contradiction; rather, they’re committed to epistemically abnormal—but none the less entirely (aletheically) possible—worlds, worlds in which, for example, knowing an unknown truth happens. Veriﬁcationists are committed to ♦(α ∧ ¬K α), for some α, but the possibility in question is epistemically abnormal, a world, perhaps, in which ‘epistemic ﬁctions’ transpire.¹⁶ At such worlds, the normal behaviour of K breaks down in various respects. In particular, given that possibilities are one and all consistent (though not necessarily complete), the normal distributive behaviour of K breaks down. Such abnormal worlds are precisely where the oddity—but not inconsistency—of KP’s clash with non-omniscience emerges.¹⁷ The proposal, then, is to avoid ‘possibly true gluts’ via expanding one’s range of possibilities. Speciﬁcally, the veriﬁcationist acknowledges epistemically abnormal possibilities in which K is deviant. At the same time, the veriﬁcationist is committed, on the whole, to the validity of standard K -rules. While knowledge might deviate from its normal behaviour at odd points, the validity of standard K -rules ought to remain intact. Accordingly, in addition to expanding her range of possibilities, the veriﬁcationist narrows her account of validity—or, what comes to much the same, keeps her account of validity focused on the non-deviant, normal possibilities. A formal—and, in some respects, familiar—picture will be helpful. I will return to philosophical discussion in Section 6.3.

6.2. A formal picture The basic idea can be modelled along ‘non-normal lines’.¹⁸ We make a distinction among worlds—the normal and non-normal (or abnormal, as I will say). In turn, we deﬁne validity as ‘truth-preservation’ over only one sort of world, not as ‘all points (worlds) validity’. The behaviour of target operators at the abnormal worlds is recognized, but such behaviour is (in effect) ignored for purposes of deﬁning validity. A simple account is as follows. Let V and D be as in Section 6 (Strong Kleene base). Our interpretations are structures W , N , N ∗ , R, E, v, ε, where W = N ∪ N ∗ , with N (normal worlds) and N ∗ (abnormal) non-empty, and N ∩ N ∗ = ∅. R and E are as before, each being at least reﬂexive on W. ε, to which I’ll return, has the job ¹⁶ Compare Priest (1992). ¹⁷ Admittedly, if one acknowledges ‘abnormal epistemic worlds’ in which, e.g., K ’s normal distributive behaviour breaks down, there may be no strong reason to reject other such abnormal worlds in which more radical deviance occurs (such as knowing a contradiction!). Even so, the current proposal aims at avoiding ‘possibly true inconsistency’ altogether. ¹⁸ The idea behind ‘non-normal semantics’ comes from Kripke (1965), wherein the aim was to model Lewis systems weaker than S4. Arguably more signiﬁcant philosophical use of non-normal semantics has emerged in literature on ‘relevant logics’. See Dunn and Restall (2002) and references therein.

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of evaluating K -claims at abnormal worlds. v : A × W −→ V assigns values to all atomics at all worlds, normal and not. Interpretations are extended to all sentences at all worlds via the following clauses. 1. Extensional. For any w ∈ W, vw (¬α) = 1 − vw (α) vw (α ∧ β) = m i n{vw (α) , vw (β)} vw (α ∨ β) = m a x{vw (α) , vw (β)} 2. Possibility. For any w ∈ W, vw (♦α) = m a x{vw (α) : wRw for any w ∈ W} 3. Knowledge. (a) Normal worlds. For any w ∈ N vw (K α) = m i n{vw (α) : wEw for any w ∈ W} (b) Abnormal worlds. For any w ∈ N ∗ vw (K α) = εw (K α) The job of ε, as above, is to give values to K -claims at our abnormal worlds. ε may be viewed as an ‘arbitrary evaluator’ of K -claims at abnormal worlds, though the arbitrariness, to avoid inconsistency at abnormal worlds, is subject to the following constraint. εw (K α) = 1 ⇒ vw (α) = 1 Finally, validity is deﬁned as ‘truth-preservation’ over all normal worlds of all interpretations. So given, the semantics delivers some, but not all, of the target principles. Importantly, we do not get KP. For example, consider an interpretation in which N = {w}, N ∗ = {w∗ }, and, in addition to reﬂexivity, we have only wEw∗ . Now let vw (α) = 1 and vw∗ (α) = 0 = εw∗ (K α). Figure 8.1 shows diagram of the counter-example. R-accessibility R

w

w*

E-accessibility E

w

Values at worlds

w*

α

Kα

◊Kα

w

w

w

1

0

0

w*

w*

w*

0

0

0

Figure 8.1.

The trouble, of course, is that our class of interpretations is too big.

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Towards narrowing our class of interpretations, let a narcissistic world—an n-world, for short—be any world w (normal or abnormal) such that, for any w ∈ W, wRw or wEw ⇒ w = w N-worlds see only themselves, in either relevant sense of ‘see’. Now, deﬁne a V∗ -model to be any interpretation (as above) such that the following holds. V∗ . For any normal w, if vw (α) = 1, then there is some abnormal n-world w∗ such that εw∗ (K α) = 1 and wRw∗ . In turn, validity is deﬁned as ‘truth-preservation’ over all normal worlds of all V∗ -models. That there are V∗ -models may be seen by tweaking the previous counter-example to get the results shown in Figure 8.2 (where ‘starred’ worlds are abnormal).¹⁹ R-accessibility R

w

E-accessibility

w1* w2* w3*

E

Values at worlds α

w1* w2* w3*

w

Kα K(α

Kα)

◊Kα

w

w

w

1

0

0

1

w1*

w1*

w1* 1

0

1

0

w2*

w2*

w2* 0

.5

.5

.5

w3*

w3*

w3* 1

1

.5

.5

Figure 8.2.

The corresponding picture is shown in Figure 8.3.²⁰ R R,E

R,E

R,E

w3*

w2*

α

α

Kα

E

w α Kα

R,E R

w1* α K(α

Kα Kα)

Figure 8.3. ¹⁹ .5 is not forced at w∗2 . One could also give K α the value 0. ²⁰ In general, a doubly squared world is abnormal.

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Notice that each of the abnormal worlds except for w∗2 serves as an n-world for w. The V∗ -model above also serves to invalidate Fitch’s chief inference—from knowability to known. In the model, w is a non-omniscient (normal) world with respect to α, but—thanks to the abnormal worlds—it is possible to know α.

6.3. Comments I turn to a few comments about the ‘abnormal’ approach. I begin with a few salient virtues of the semantics, and then brieﬂy turn to the broader, philosophical picture (returning to the topic in Section 8). As expected, Contraposition and LNC are invalidated, and the regular extensional connective remain normal (as in Strong Kleene). More importantly, the semantics validate each of the standard K -rules, including the essential KP. KV. Let vw (K α) = 1 for some w ∈ N . Then vw (α) = 1 for all w ∈ W such that wEw . Since E is reﬂexive, vw (α) = 1. KC. Let vw (K (α ∧ β) ) = 1 for some w ∈ N . Then vw (α ∧ β) = 1 = vw (α) = 1 = vw (β) for all w ∈ W such that wEw . Hence, as E is reﬂexive, vw (K α) = 1 = vw (K β).²¹ KP. Let vw (α) = 1. Then, by V∗ , there’s some abnormal n-world w∗ such that wRw∗ and vw∗ (K α) = εw∗ (K α) = 1. On the other hand, not everything is retained. Not surprisingly, the deviation from ‘all points validity’ to ‘all normal points’ invalidates certain inferences, notably, CP (see Section 2). For example, as above, KC is valid, and so K (α ∧ ¬K α) implies K α ∧ K ¬K α. Moreover, KP is valid (as above). Yet, ♦K (α ∧ ¬K α) ♦(K α ∧ K ¬K α), since the relevant world—the world at which K (α ∧ ¬K α) is true—might be abnormal.²² In abnormal worlds, K can deviate from its normal behaviour. One might think of such worlds not only as ‘odd epistemic possibilities’ but, further, as worlds at which valid K -behaviour breaks down. Turning to the broader philosophical picture, a few virtues of the current account may be noted. 1. Consistency. A main motivation behind the ‘abnormal’ approach was to avoid even the possibility of ‘true contradictions’. While achieving as much requires constraints on ε, the aim seems to be realized, for what that is worth. ²¹ But see further discussion below! ²² This is not surprising given non-normal semantics. Indeed, as mentioned, Kripke’s original motivation beyond non-normal worlds semantics was to model Lewis systems weaker than S4, systems in which Necessitation fails. Moreover, in subsequent non-normal approaches to conditionals, the aim is often to model conditionals for which there is no (standard) deduction theorem.

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2. Fitch’s Lesson. Fitch’s ‘proof’, as in Section 3, points to a genuine oddity in combining KP and non-omniscience. In the current ‘abnormal’ case, the oddity is fully acknowledged; it shows up as ‘abnormal possibilities’, possibilities that seem inconsistent but, in the end, avoid outright inconsistency via deviant K behaviour. 3. Failure of CP. While CP’s failure is odd, the current story comes with an explanation: distribution of K fails inside (aletheic) modal contexts because such contexts are pointing to epistemically deviant worlds. By my lights, there is a coherent story along the ‘abnormal’ lines—odd, but coherent. Some oddness, as Fitch highlighted, is inevitable, at least given KP and non-omniscience. The question, of course, is whether the ‘abnormal’ approach to the inevitable oddness is overly odd. Ultimately, that is an issue for veriﬁcationists. As far as I can see, there is nothing in veriﬁcationism that either rules out or implausibly conﬂicts with (something like) the foregoing ‘abnormal’ approach. Whether, in the end, the abnormal approach is ultimately viable is something that I leave for debate. Doing away with even the possibility of ‘true contradictions’ is difﬁcult. Perhaps, ultimately, veriﬁcationists are better off accepting Fitch’s argument for apparently possible ‘true contradictions’ in a broader paraconsistent (but non-dialetheic) framework. I turn now to a brief sketch of such an approach. In Section 8 I (very brieﬂy) return to the overall philosophical viability of the canvassed approaches.

7 . Sy n t h e s i s : Ga p s a n d Me re l y Po s s i b l e Gl u t s If, as in Section 5, one goes with ‘all points validity’ in a (normal) paracomplete but non-paraconsistent framework, the veriﬁcationist seems to be stuck with the trivial world—and a vacuous KP. Dropping ‘all points validity’, as in Section 6, affords more options, but one is forced to give up a few more rules (e.g., KC in the context of aletheic modalities). While each option may hold promise (especially the second), a further option is worth noting. In this section, I brieﬂy sketch—without arguing for—another option: an ‘all points validity’ approach that is both paracomplete and paraconsistent but nonetheless non-dialetheic. The paraconsistent veriﬁcationist blocks Fitch’s ‘proof ’ at the same place(s) that K 3 does—either LNC or Contraposition. With respect to the ‘oddity’ of KP in a non-omniscient world, the paraconsistent response is straightforward: non-omniscience and KP generate an inconsistent possibility—knowing an unknown truth. But such possible inconsistency needn’t generate actual inconsistency, at least in a paracomplete paraconsistent framework. In a paracomplete framework, the

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paraconsistent veriﬁcationist may acknowledge ‘possible gluts’ without thereby accepting dialetheism—the view according to which there is actual inconsistency.²³ Here, I sketch a basic four-valued framework for veriﬁcationists, an extension of the familiar Anderson–Belnap framework (1992).

7.1. The basic model Interpretations are structures W , R, E , v, where W , R, and E are as before (with R and E at least reﬂexive). Here, it is convenient to let V, our semantic values, be P ({1 , 0}).²⁴ Then v is any function from S × W into V subject to the following constraints. 1. Negation (a) 1 ∈ vw (¬α) iff 0 ∈ vw (α) (b) 0 ∈ vw (¬α) iff 1 ∈ vw (α) 2. Conjunction (a) 1 ∈ vw (α ∧ β) iff 1 ∈ vw (α) and 1 ∈ vw (β) (b) 0 ∈ vw (α ∧ β) iff 0 ∈ vw (α) or 0 ∈ vw (β) 3. Disjunction (a) 1 ∈ vw (α ∨ β) iff 1 ∈ vw (α) or 1 ∈ vw (β) (b) 0 ∈ vw (α ∨ β) iff 0 ∈ vw (α) and 0 ∈ vw (β) 4. Possibility (a) 1 ∈ vw (♦α) iff 1 ∈ vw (α) for some w such that wRw . (b) 0 ∈ vw (♦α) iff 0 ∈ vw (α) for all w such that wRw . 5. Knowledge (a) 1 ∈ vw (K α) iff 1 ∈ vw (α) for all w such that wEw . (b) 0 ∈ vw (K α) iff 0 ∈ vw (α) for some w such that wEw . Validity is deﬁned as ‘truth preservation’ over all worlds of all interpretations. With the expected exception of Contraposition and LNC, the semantics, with validity so deﬁned, preserve most of the target principles: KV, KC, CP, Adjunction, etc. The question, of course, concerns KP. ²³ In Priest’s alternative LP (dialetheic) setting, which is not paracomplete, the ‘mere possibility’ of ‘true contradictions’ immediately generates actual inconsistency. In LP (or the target extension), we have ¬♦(α ∧ ¬α). Hence, given any β such that ♦(β ∧ ¬β) is actually true, we immediately have actual inconsistency. For further discussion, see Restall (1997), wherein Restall ﬁrst discussed the point regarding the LP situation, and Beall and Restall (2006) for broader discussion [3]. ²⁴ This idea is due to Dunn (1966, 1976).

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Alas, KP is invalid. Just consider an interpretation in which W = {w, w } and, in addition to the required reﬂexivity of R and E, we have wRw and wEw , but also vw (α) = {1} and vw (α) = {0}.²⁵ This serves as a counter-example to KP. The diagram shown in Figure 8.4 may be useful. R-accessibility R

w w′

E-accessibility E

Values at worlds

w w′

α

Kα

◊Kα

◊K(α

w

w

w

{1}

{0}

{0}

{0}

w′

w′

w′

{0}

{0}

{0}

{0}

Kα)

Figure 8.4.

A picture of the counter-example is shown in Figure 8.5. I give only the value of α. R,E

R,E R,E w

w′

α

¬α

Figure 8.5.

So, KP fails. The trouble, of course, is that our class of interpretations is too big. To get the target interpretations, we need to pare down our class of interpretations.

7.2. The target: V-models The natural remedy is to invoke n-worlds, as in Section 6 (but now without abnormal worlds). Let an epistemically narcissistic world—an n-world, for short—be a world w such that, for any w ∈ W, wEw ⇒ w = w Since E is reﬂexive, every world epistemically sees itself; n-worlds (epistemically) see only themselves. In turn, we deﬁne a V-model (for Veriﬁcationism model) to be any interpretation (as above) that conforms to the following. ²⁵ For that matter, you could let vw (α) = ∅.

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V. If 1 ∈ vw (α) then there’s some n-world w such that wRw and 1 ∈ vw (α ∧ ¬α).²⁶ Validity is deﬁned as before, but now only over V-models. And with that we get KP. That there are V-models is clear. In particular, we have V-models that invalidate Fitch’s chief inference—from knowability to knowledge. A simple V-model—perhaps the simplest—in which the Fitch inference fails (viz., from knowable to known) is shown in Figure 8.6. R-accessibility R

w0 w1 w2

E-accessibility

Values at worlds

w0 w1 w2

E

α

Kα

◊Kα

◊K(α

w0

w0

w0

{1}

{0}

{1,0}

{1,0}

w1

w1

w1

{1,0}

{1,0}

{1,0}

{1,0}

w2

w2

w2

{0}

{0}

{1,0}

{1,0}

Kα)

Figure 8.6.

A picture of the model is shown in Figure 8.7. R R,E

R,E w2

α

E

w0

α

R,E w1

R

α

α

Figure 8.7.

This is a model in which the given Fitch inference fails, since α is knowable at w 0 but not thereby known. The trouble, of course, is that the paracomplete (but paraconsistent) V-models were supposed to afford an entirely consistent actual world while allowing for ‘merely possible inconsistency’. In the simple model above, such a promise does not show up. After all, w 1 serves as an n-world for ²⁶ A simpler, perhaps more natural, route would be to add a distinguished actual world and impose V only on that, but I will go with the ‘all worlds of all models’ approach.

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both w 0 and w 2 (and itself). Since there are only three worlds, the above model, in effect, is basically an LP-based model. (If we demanded LEM for all atomics, it would be an LP-based model.) The upshot is that, in the above model, the possibility of α-inconsistency—and, in particular, K α-inconsistency—trickles back into actual inconsistency: ♦K α is true and false at all worlds, and hence the actual. To get a consistent but non-omniscient ‘actual world’ (say, w 0 ), we simply add more worlds. The simplest addition is the null world w∅ , shown in Figure 8.8.²⁷ Values at worlds

R-accessibility

E-accessibility

R w0 w1 w2 w0

E w0 w1 w2 w0

w0

w0

α

Kα

◊Kα

◊K(α

Kα)

w0

{1}

{0}

{1}

{1}

{1,0}

{1,0}

{1,0}

w1

w1

w1

{1,0}

w2

w2

w2

{0}

{0}

{1}

{1}

w0

0

0

0

0

w0

w0

Figure 8.8.

The corresponding picture can be seen in Figure 8.9. R R,E

R

R,E w0

R

R,E w2 α

E

w0 α

R,E w1

R α

α

Figure 8.9.

This model is more attractive than the former, simpler model, as it leaves ‘the actual world’ consistent while nonetheless refuting the Fitch inference. I move to a few general comments.

7.3. Comments There are various virtues of the V-models over the LP-based paraconsistent approach. For present purposes, I list the salient ones. ²⁷ Note that the null world is not essential; one merely needs the appropriate ‘incompleteness’.

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1. Merely Possible Inconsistency. The foregoing approach shares the basic response common to any paraconsistent veriﬁcationism: namely, that KP (and the other set of K-rules) forces inconsistency in the face of non-omniscience. But since the current approach is also paracomplete, there’s no threat that ‘merely possible inconsistency’ implies actual inconsistency—as is the case in an LP-based approach.²⁸ The upshot is that a veriﬁcationist can admit that the possibility of knowing unknown truths forces inconsistency; however, it need only force inconsistency ‘elsewhere’ and only elsewhere—some merely possible world. 2. Trivial world. While the trivial world, without further constraints, certainly shows up in V-models, it isn’t required to ensure KP (or, as discussed below, the countermodel to Fitch’s basic inference). There are V-models, of course, in which LEM holds among all worlds of the given models (viz., LP-models!); however, being based on a broader four-valued framework, LEM certainly isn’t valid. In short, V-models allow for ‘incomplete worlds’, worlds in which neither α nor ¬α show up (as it were). 3. Not entirely inconsistent. Moreover, while KP is fully ensured, as above, by inconsistency ‘elsewhere’, V-models allow for ‘local inconsistency’ to do the work. In particular, the n-worlds, into which knowing ‘non-omniscience truths’ (e.g., α ∧ ¬K α) forces inconsistency, need not themselves be entirely inconsistent. Because of incompleteness, there can be many n-worlds throughout which the given inconsistency is distributed, and many of them can be perfectly consistent in proper quarters. There are probably other notable virtues vis-`a-vis the LP-based (paraconsistent) approach, but I turn to one ﬁnal matter. One might think that V, which invokes suitable en-worlds to ensure KP, is ad hoc. Such charges are notoriously difﬁcult to adjudicate, and I won’t pursue the issue in any depth here. By my lights, V is not at all ad hoc. After all, V reﬂects the (paraconsistent and paracomplete) veriﬁcationist’s chief tenet: that all truths are knowable—even those that reﬂect non-omniscience, and hence generate inconsistency elsewhere. Rather than being some ad hoc posit, the relevant en-worlds that V invokes might best be seen as an implicit feature of veriﬁcationism. One thing is uncontroversial about Fitch’s argument: veriﬁcationism’s commitment to KP makes for some oddity in its confrontation with our actual non-omniscience. The paraconsistent-cum-paracomplete framework accepts that the given oddity is indeed as it appears: possibly true inconsistency. But veriﬁcationists are not thereby dialetheists; such inconsistency, in virtue of incompleteness, is harnessed at the merely possible. ²⁸ Of course, enriching the language might raise further problems, but the aim here is merely to sketch a beginning option.

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8 . C l o s i n g Re m a r k s I would like to end this paper by arguing for the supremacy of one of the canvassed options, but I cannot. As above, I am not a veriﬁcationist, and so not committed to KP via a prior theory of truth (or meaning, or etc.). Moreover, I know of no good arguments for KP.²⁹ Still, I ﬁnd KP plausible and think that each of the canvassed approaches has merit. Instead of trying to settle which, if any, of the given approaches is best, I will close by answering the most salient worries that confront each of the two chief options—setting aside the ‘ﬂat-footed response’ (see Section 6).

8.1. Abnormal epistemic possibilities? The suggestion, here, is that KP holds in virtue of abnormal epistemic possibilities, where these are possibilities in which normal K behaviour breaks down. The chief worry about such a picture is that we are no longer talking about knowledge when we are talking about ‘abnormal K behaviour’. Put differently (with echoes of Quine), the charge is that necessarily, K behaves like such and so—in particular, distributes over conjunction (and is such that, e.g., CP is valid). Hence, the ‘abnormal epistemic possibility’ framework is really one in which we are introducing two distinct epistemic operators, one reﬂecting our ‘real knowledge operator/predicate’, the other some ‘deviant’ (but distinct) operator/predicate. As such, veriﬁcationists—and KP theorists, in general—are still stuck with the original problems confronting our ‘real’ item.³⁰ By way of reply, the way I look at the situation is (brieﬂy) as follows. Veriﬁcationists are committed to some sort of oddity. If veriﬁcationists are likewise committed to the bulk of the given rules (see Section 3) and ‘no possibly true inconsistency’, then a natural suggestion, as in Section 6, is that K behaves differently at different sorts of worlds. Now, the charge, as above, maintains that we have two different K s, rather than a single K that, as said, behaves differently at different points. I’m not sure how to adjudicate this. If the project is to give the veriﬁcationist—or KP theorist, in general—entirely consistent worlds across the board, while also retaining the bulk of the given rules, then it’s unlikely that there’s a distinct K of the sort presupposed in the charge (as opposed to a single K that behaves differently at different points, as per the proposal). After all, if ‘the real K ’ is like that (e.g., supports distribution inside the diamond), then the veriﬁcationist is stuck with inconsistent worlds. Again, I don’t know how to ultimately adjudicate the matter. I’m not a veriﬁcationist, but I think it worthwhile to see how the veriﬁcationist might ²⁹ I do not consider appeals to ‘intuition’ good arguments. ³⁰ I am grateful to Carrie Jenkins for pushing this point.

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enjoy entirely consistent worlds and the bulk of the rules. Of course, she has to give up something, and the Section 6 proposal gives up ‘all points validity’ (and, in turn, CP). In the end, perhaps the resulting picture is too implausible to suit veriﬁcationists. I don’t know. But I do not see why they can’t have a single K that behaves differently at different points. Indeed, veriﬁcationists—or, again, KP theorists, in general—can take the lesson of Fitch’s ‘proof’ to be that we were ignoring various possibilities, namely, ones in which our unique K behaves in very abnormal ways.³¹

8.2. Paraconsistent but non-dialetheic V-models There may well be various worries about this approach, many of which might spring from general worries about ‘possibly true contradictions’. This paper is not the place to address such broad worries.³² Instead, I will assume a general openness to the idea of (merely) ‘possibly true gluts’. There remains a salient worry for the V-model approach.³³ The worry, in short, is that the proposal calls for too much. In particular, the proposal commits us to the possibility of α ∧ ¬α for every true α. Even if one is prepared to acknowledge merely possible gluts, it is hard to accept that for every truth α, it is possible that α is true and false! I think that, by way of reply, one needn’t quite accept as much as the V-model approach yields. The V-models were so given as a simple example, but one should be able to restrict matters further so as to avoid the going worry. For example, one approach might be to restrict condition V to any ‘non-omniscience truth’, any truth of the form α ∧ ¬K α.³⁴ Whether this would immediately yield KP is not obvious, but it would at least deal with the main worry over KP—namely, the sort of ‘non-omniscience’ claims involved in Fitch’s argument. ³¹ One might also argue that the veriﬁcationist—or KP theorist, in general—ought to acknowledge possibilities in which the constraints on K (on knowledge, in general) vary. In the case of veriﬁcationism, it is not implausible to think that knowledge might be achieved in some (admittedly, abnormal or remote) possibilities in which veriﬁcation criteria are weaker than normal. I think that this line is worth exploring, but for space reasons I omit further discussion. ³² For discussion of such broader issues, see Priest, Beall, and Armour-Garb (2004). ³³ I am grateful to Greg Restall for pushing this concern. ³⁴ In this case, it might be easier to add a distinguished ‘actual world’ to the models, and deﬁne validity as ‘truth-preservation’ over actual worlds (of all such models), but I will leave details for another occasion.

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Pa r t I V Ep i s t e m i c a n d Te m p o r a l Op e r a t o r s : Ac t i o n s , Ti m e s a n d Ty p e s

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9 Actions That Make Us Know Johan van Benthem

1 . T h e Pr o b l e m : Ve r i ﬁ c a t i o n i sm In c u r s t h e Fi t c h Pa r a d o x Veriﬁcationism is an account of meaning and truth whose origins lie in logical proof theory, especially, in its constructivist versions. The idea is that ‘truth’ can only be assigned to propositions for which we have evidence. This view can be found with logical authors like Dummett and Martin-L¨of from the 1970s onwards, but it has also penetrated since into general philosophy. Stated as a sweeping claim, this take on truth implies the general veriﬁcationist thesis that what is true can be known: ϕ → ♦K ϕ

VT

Here the K can be taken as a relatively unproblematic knowledge modality, while the ♦ is an as yet unspeciﬁed modality ‘‘can’’ of ‘feasibility’ in some relevant sense. Now, a surprising argument by Fitch trivializes this principle. It uses just a weak modal epistemic logic to show that VT collapses the notions of truth and knowledge, by taking the following clever substitution instance for the schematic formula ϕ, like elsewhere: q ∧ ¬Kq → ♦K (q ∧ ¬Kq) Then we have the following chain of three conditionals—which works in quite weak and apparently unproblematic modal logics: ♦K (q ∧ ¬Kq) → ♦(Kq ∧ K ¬Kq) → ♦(Kq ∧ ¬Kq) → ♦ ⊥ → ⊥ Thus, a contradiction follows from the assumption q ∧ ¬Kq, and we have shown overall that q implies Kq, making truth and knowledge equivalent. Is there a real problem here? How plausible was Veriﬁcationism anyway? There can be legitimate doubt on this score—but all the same, looking at ‘paradoxes’ like Fitch’s can be worthwhile. Of course, not every

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paradoxical argument points at a genuine problem. Some are just spats on the Apple of Knowledge, which can be removed with a damp cloth. But others are the telltale brown spots of worm rot inside, and deep surgery is needed—and the Apple may not even remain in one piece when restoring consistency. Professional paradox hunters and puzzle-driven researchers always claim the ‘deep trouble’ diagnosis—and sometimes they are right. Proposed remedies for the Paradox fall mainly into two kinds (cf. Brogaard and Salerno 2002; van Benthem 2004). Some solutions weaken the logic in the argument still further. This is like tuning down the volume on your radio so as not to hear the bad news. You will not hear much good news either. Other remedies leave the logic untouched, but weaken the veriﬁcationist principle itself. This is like censoring the news: you hear things loud and clear, but they may not be so interesting. Some choice between these strategies is inevitable. But what one really wants is a new systematic viewpoint beyond plugging holes, and opening up a new line of thinking with beneﬁts elsewhere. In our view, the locus for this is not Fitch’s proof as such, but rather our understanding of the two key modalities involved, either the modal K or the epistemic , or both.

2 . A Fi r s t Q u i c k A n a l y s i s : Ep i s t e m i c L o g i c a n d Ev i d e n c e Let us ﬁrst get to the essence of Fitch’s argument. The above substitution instance exempliﬁes a much older problem called Moore’s Paradox. Originally stated about belief, it consists in the observation that the statement ‘‘P, but I don’t believe it’’ can be true, whereas it cannot be consistently believed. Transposed to knowledge, this same problem was discussed by Hintikka in the 1960s, using the inconsistency of the formula K (q & ¬Kq) in epistemic logic. So, it is easy to understand the issue. Some truths are ‘fragile’ whereas knowledge is ‘robust’: and hence the former need not support the burden of the latter. Thus, one sensible and straightforward approach to the paradox weakens the scope of applicability of VT as follows (Tennant 2002): Claim VT only for propositions ϕ such that K ϕ is consistent CK CK has clear merits, but it fails our more general desideratum: it provides no exciting new account of either knowledge K or feasibility . We have put our ﬁnger in the dike, but no larger polder management system has emerged. Indeed, there seems even an obvious missing link in CK , reﬂecting one’s intuitive semantic understanding of the setting for VT . We have the truth of ϕ in some epistemic model M with actual world s, representing our current information

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state. But consistency of K ϕ per se gives us only the truth of K ϕ in some possibly quite different epistemic model (N, t). The real issue is rather: What natural step of ‘coming to know’ would take us from (M, s) to (N, t)? One could see this as asking for a principled account of the above operator , while the K can retain its standard meaning from epistemic logic. One way in which the has been unpacked in the literature goes back to the proof-theoretic origins of VT . In well-established type-theoretic approaches to provability, the evidence for a conclusion is displayed and manipulated in binary assertions of the form p: ϕ, where p is a proof for ϕ, or a piece of evidence in a more general sense. Type theory seems the most sophisticated underpinning of Veriﬁcationism to date. Van Benthem (1993) took this idea to standard epistemic logic, and proposed an explicit calculus of evidence for its K -assertions. One striking modern realization of this is the ‘logic of proofs’ of Artemov (1994, 2005), which replaces the box ϕ of the usual modal provability logic by operators [p] ϕ ‘p is a proof for ϕ’. Indeed, labels p of many sorts appear in the ‘labeled deductive systems’ of Gabbay (1996). This ‘evidence parameter’ for logical investigation seems a deep response to any paradox—but I am not aware of an inspiring solution to Fitch-style problems in this proof-theoretic setting. Thus, I take a different tack in this essay, in terms of dynamic semantic actions that produce knowledge. 3 . D y n a m i c s o f In f o r m a t i o n a n d C o m i n g t o K n ow Broadening our view of what a feasibility modality might stand for, van Benthem (2004, 2006a) looks at general mechanisms producing knowledge. Mathematical proof, no matter how liberally construed, is not the best paradigm for understanding how we come to know things, since it does not add new truths beyond our premises. Genuine actions by which we come to know new things seem much more domestic: we observe, or we ask some expert who knows! The latter actions involve a notion of change beyond proof steps: new information changes the current epistemic model—and in the process our knowledge changes, too. The simplest mechanism achieving this reﬂects the folklore sense in which ‘new information shrinks the current range of possibilities’: An announcement of some proposition P changes the current range of possible worlds, leaving those where P holds, while removing all others. More precisely, consider an epistemic model (M, s), with designated actual world s. What can be known in this setting seems restricted to what might be known correctly about that actual situation s. We know already that it is one of the worlds in M . What we might learn is that this model can be shrunk further, zooming

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in on the location of s. In this dynamic epistemic setting, we can recast the Veriﬁcationist Thesis as follows. Saying that every true statement may be known amounts to stating that: What is true in the current setting may come to be known there

VT-dyn

What this means in a simplest scenario is that some authoritative true statement could be made which changes the current model (M, s) to some submodel (M |ϕ, s) where the relevant proposition ϕ is known. Indeed, announcing ϕ itself seems an obvious and infallible candidate for this purpose, but more on this in a moment. The dynamic turn toward knowledge-producing actions involves some delicate issues. A ﬁrst thing to note is that making announcements is not just a matter of accumulating knowledge. This is true for atomic facts—but truth-values of more complex epistemic assertions can change in the process. When I tell you that p, which you did not know, the statement Kyou p changes its truth-value from false to true. But at the same time, the iterated knowledge statement Kyou ¬Kyou p goes from true to false—and so on upward, with changes in iterated statements of epistemic reﬂection. Thus, one single action !ϕ of publicly announcing ϕ can have repercussions for truth-values across the epistemic language. In particular, the Moore sentence shows that some propositions ϕ have the ‘self-afﬂicting’ property of changing their own truth-value when they are announced: A true public announcement !(q & ¬Kq) of q & ¬Kq makes the fact q into common knowledge, thereby invalidating the conjunct ¬Kq. Thus, announcing a truth is not an infallible way of turning it into knowledge. We will investigate the subtleties of epistemic update in the next section. For now, we contrast our new dynamic view with the earlier consistency requirement on CK. Here is the connection between our new proposal VT-dyn and the earlier CT : Fact

(a) VT-dyn implies CK (b) CK does not imply VT-dyn for all propositions ϕ

Proof Implication (a) is obvious. Its converse (b) is not, as we need truth of K ϕ not in just in any model (which would sufﬁce for consistency), but in some submodel of the current one. Here is a counter-example. Not surprisingly by now, it works with a relative of the Moore-type assertion q & ¬Kq: ϕ = (q & ¬Kq) ∨ K ¬q,

where q is a proposition letter.

This is knowable in the sense of CK , since K ( (q & ¬Kq) ∨ K ¬q) is consistent. For instance, this formula holds in a model consisting of just one world with ¬q. Indeed, in S5, the statement K ϕ is equivalent to K ¬q. But now consider the following two-world epistemic S5-model M with an actual world s and an

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epistemically indistinguishable world t, where the atomic formula q holds at s but not at t. In this situation, no truthful announcement would ever make us learn the above ϕ:

s

t

q

q

Figure 9.1.

In the actual world, (q & ¬Kq) ∨ K ¬q holds, but it fails in the other one. Hence, K ( (q & ¬Kq) ∨ K ¬q) fails in the actual world. Now, there is only one truthful proper update of this epistemic model M , which just retains its actual world with q: the actual world q

Figure 9.2.

But in this one-world model, the formula K ( (q & ¬Kq) ∨ K ¬q) fails. The preceding example suggests that CT, though correct in spirit, is still too weak in a dynamic setting. This point is somewhat technical, but telling all the same. It shows how, in a natural semantic scenario of coming to know things, the Veriﬁcationist Thesis places stronger requirements on propositions than those found in the literature so far. How can this happen? Why does not a true assertion (q & ¬Kq) ∨ K ¬q stay true when we ‘learn more’? Once again, the learning intuition behind world elimination is only valid for factual propositions. But epistemic propositions involving K -modalities may change their truth-value when a model contracts, as ignorance has now turned into knowledge. To understand this better, let us now look in more detail at logical mechanisms for epistemic change and learning (van Benthem 2002, 2006a, 2006b). 4 . Ep i s t e m i c L o g i c D y n a m i ﬁ e d

4.1. Static epistemic logic The basic language of epistemic logic and its semantics are well-known, with the individual knowledge modality Ki ϕ interpreted as follows: Ki ϕ is true at a world s iff ϕ is true in all worlds t with s ∼i t, where ∼i is the epistemic accessibility relation for agent i.

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In what follows, for convenience of exposition, we use an S5 version, where world accessibility is an equivalence relation. This simple semantics of knowledge has inspired much philosophical discussion, partly by the logical precision that it offers, but also, it has to be said, by its perceived deﬁciencies. Hotly debated until today are ‘logical omniscience’ (closure of knowledge under valid implications), and ‘introspection’ (automatically knowing that one knows or does not know a proposition): cf. van Benthem (2006a). Moving beyond single agents, epistemic logic can also analyze new forms of ‘social’ knowledge in groups. In particular, common knowledge. CG ϕ for a group G says intuitively that everyone knows that ϕ, they also know that the others know, and so on to any ﬁnite depth of iteration of mutual knowledge operators. Semantically, the corresponding epistemic modality CG ϕ is true at a world s whenever ϕ is true in the whole ‘component’ of the model consisting of all worlds accessible from s by some ﬁnite sequence of agent accessibility steps. In scenarios with just a single agent 1, common knowledge C{1} ϕ is just the same as knowledge K1 ϕ. (This would not work in weaker epistemic semantics than that for S5.) One can read the following discussion up to Section 7 either way, as being about knowledge of a single agent, or about common knowledge in a group. As for epistemic inference, well-known complete axiom systems exist for the valid laws in this language over standard model classes, such as multi-agent S5 (plus common knowledge) for models where the accessibilities are equivalence relations. Finally, as to computational complexity, most current versions of epistemic logic are decidable.

4.2. Dynamic epistemic logic To deal with the dynamics of Section 4, we need to add epistemic actions to this framework. Here, the driving engine for update of agents’ information is model change. The simplest case described earlier is that of a truthful public announcement !ϕ of an assertion ϕ. This does not just evaluate ϕ truth-conditionally in the current model (M, s). It rather updates that model to a new model M |ϕ , s, a submodel of (M , s)—and it does so by eliminating all those worlds from it which fail to satisfy ϕ: from

s

to

(M, s)

f

(M|f, s)

f

Figure 9.3.

s

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This update scenario can analyze questions and answers producing new information, and it even works for much more intricate puzzles involving knowledge and ignorance (van Benthem 2002). Thus, we get a dynamic-epistemic logic, as a general semantic setting for information ﬂow and learning. There is a family of epistemic models: the relevant information states, and a repertoire of announcement actions, which increase information by moving from one model to another. Full-ﬂedged dynamic-epistemic logics arise from standard epistemic ones by adding an action modality from dynamic logics of computation. It expresses what holds after an action was performed: M , s |= [ !ϕ ]ψ

iff

if M , s |= ϕ , then M |ϕ , s |= ψ

Thus, the dynamic modality [!ϕ]ψ says that ‘‘after ϕ has been truthfully announced, ψ. holds at the current world.’’ With this language, one can express systematic effects of communication, using combined knowledge-action statements such as [!ϕ]Kj ψ: after a public announcement of ϕ, agent j knows that ψ There are complete and decidable logical calculi for this richer language, too. Their key axioms systematically analyze the result of an epistemic action in terms of things that were true before. Dynamic epistemic logic does not magically solve the problems of static epistemic logic, as perfect reasoning with logical omniscience, and perfect reﬂection with epistemic introspection are still assumed. But our new logics do help analyze and even high-light further issues of potential philosophical interest. Sometimes, it is just liberating to move to new problems instead of remaining stuck with old ones. In particular, one additional idealization of the dynamic setting seems worth pondering. The central valid law of the logical calculus of public announcement reduces knowledge resulting from communication to relativized knowledge that was true before: [!A] Ki ϕ ↔ (A → Ki (A → [ !A] ϕ)) The semantic soundness of this principle has its own further presuppositions, including perfect memory of agents (Liu 2006). This idealization has been called into question in game theory and cognitive psychology under the heading of ‘bounded rationality’. Moving beyond single knowers, however, the most exciting applications of epistemic logics today emphasize the multi-agent character of speakers, hearers, and audiences. In particular, even Hintikka’s original language can iterate knowledge assertions, as in K1 ¬K2 P

‘‘1 knows that 2 does not know that P.’’

Also, common knowledge was a group phenomenon par excellence. ‘Social’ epistemic notions are crucial to information ﬂow and communication. Some

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philosophers think such issues are not profound, having to do with gossip, ICT, and other shallow necessities of living with a lot of people on one small planet. But the pursuit of knowledge and rational behavior consists to a large extent of intelligent interaction with others—and we need to understand that success. This point will return below, as so-called paradoxes afﬂicting lonesome knowers may look brighter in groups.

4.3. Interaction, partial observation, and event update Public announcement is a basic mode of transmitting information. But information can ﬂow in many more subtle ways. For example, we observe informative events without overt linguistic aspects. And, crucially, observation can then be different for different observers. I see which card I am drawing from the current stack; you only see that I am drawing one. By now, sophisticated event update mechanisms exist for such phenomena, far beyond simple world elimination (Baltag, Moss, and Solecki 1998). These can model complex multi-agent forms of communication mixing public actions and information hiding. Think of whispering to your colleagues during a seminar, or sending an email using the button bcc. In cases like these, the current epistemic model need not shrink: it may even grow in size. Example: Reading a Letter You have taken an exam, but neither you nor your friend knows the outcome yet. Here is a simple epistemic model, where in fact (viz. the bold-face actual world to the left), you passed: me pass

fail you

Figure 9.4.

Now you receive a letter in the presence of your friend, and read that you have passed. If this were a case of public announcement, the model would just shrink to the left-hand world as before. But this time, you cannot tell whether your friend has seen the content of the letter, though she does know it is an ofﬁcial notiﬁcation. She might, and she might not have seen what you read—and so, as far as she is concerned, you might also have been reading a letter which says that you failed. In this case, taking both your situations into account, there are three relevant possible pairs of simultaneous events: (you read Pass, she reads Pass) (you read Pass, she sees nothing) (you read Fail, she sees nothing)

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The ﬁrst two joint events can only occur if you have passed, the third if you failed. Note also that these events themselves have epistemic relations. For example, you cannot distinguish the ﬁrst from the second, and she cannot distinguish the second from the third. Next, as for update, the new epistemic model resulting from incorporating the new information in the reading/observing event into the preceding two-world model has three instead of two worlds now, with the relevant pairs (old world, new event) as depicted here: pass, (you read Pass, she sees nothing) you

she

pass, (you read Pass, she reads Pass) fail, (you read Fail, she sees nothing)

Figure 9.5.

Here the new epistemic relations arise as follows. Agents cannot distinguish two pairs (s, e) and (t, f ) if they can distinguish neither the old worlds s, t nor the new events e, ϕ. Suppose that in fact your friend read what was in the letter. Then the actual world is pass, (you read Pass, she reads Pass) In that world, by standard evaluation in terms of epistemic logic, you know that you passed, she knows it, too, but you do not know that she knows. These are typical asymmetries of information that may arise between players in the course of a card game. General event update takes a model M for the current information of a group of agents, plus some event model A modeling all relevant events, and then compute a new ‘product model’ MxA. This construction covers much of the information ﬂow in communication, games and other more realistic activities. Again, there are complete and decidable dynamic-epistemic logics dealing with what agents know stage-by-stage as general actions of this sort take place (van Ditmarsch, van der Hoek, and Kooi 2006). 5 . L e a r n i n g by Up d a t e The self-refuting nature of true Moore-type assertions noted in Section 4 shows that Fitch-style issues about Veriﬁcationism reﬂect core phenomena in information update. Indeed, this analogy is the main point of this paper. But

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it is of interest to see how these issues play in dynamic epistemic logic. They do not have the same doom-laden atmosphere. Informal studies of speech acts sometimes state that the generic effect of a public announcement !ϕ is simply that ϕ becomes common knowledge. The neat corresponding axiom in the earlier calculus would read as follows: [!ϕ]CG ϕ But this principle is not valid in general, witness the earlier-mentioned Moore sentence ϕ = q & ¬Kq. The latter assertion, once announced, cannot be true any more—and it even makes its own negation common knowledge! The reason is that announcing ϕ makes q common knowledge, and hence also Kq, but Kq implies ¬(q & ¬Kq). This is not an isolated curiosity. Gerbrandy (2007) gives a new analysis of the well-known Paradox of the Surprise Examination, which revolves around a teacher’s problematic assertion that some upcoming exam in the following week will take place on a day ‘when the student does not expect it’. Gerbrandy shows how the usual perplexity dissolves once we see that the teacher’s assertion can be of the above true-but-self-refuting type. For example, with a two-day time span, the formula for the teacher’s statement in our dynamic-epistemic logic is this (writing Ei for ‘the exam is on day i’): (E1& ¬Kyou E1) ∨ (E2 & [ !¬E1]¬Kyou E2) This says that the exam is on Day 1, and you do not know that now, or it will be on Day 2, and even learning that it is not on Day 1, you will not know that it is on Day 2. For details and a further defense of this analysis, we refer to the cited publication. Simple epistemic models of the above sort then clarify various surprise exam scenarios.

5.1. From paradox to typology These observations do not suggest at all that one must ban self-refuting assertions—as has been proposed in some remedies to the Fitch Paradox. To the contrary, they rather bring to light a rich diversity of types of behavior which calls for a dynamic typology of epistemic assertions. For example, we can investigate which precise forms of assertion are ‘self-fulﬁlling’, in that they do become common knowledge upon announcement. For instance, all universal modal formulas are self-fulﬁlling in this sense. These are the ones constructed using atoms and their negations, conjunction, disjunction, Ki and CG . But there are other self-fulﬁlling types of statement, and a complete syntactic characterization has been an interesting open model-theoretic problem since the late 1990s (Gerbrandy 1999; van Benthem 2002; van Ditmarsch and Kooi 2006).

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Logical studies in this vein have brought to light further delicate phenomena. In particular, some epistemic assertions ϕ are only self-fulﬁlling or ‘self-refuting’ in the long run. When announced truly for as long as possible, they either result in common knowledge CG ϕ, or the opposite: CG ¬ϕ. Van Benthem (2002) applies this insight to game theory, and shows how well-known game solution procedures may be analyzed in terms of repeated announcement of formulas ϕ expressing the ‘rationality’ of all players. Such statements are informative in general, and remove possible strategic equilibria, but at the ﬁrst stage where they no longer shrink the model, common knowledge of rationality sets in. Thus, instead of exorcizing paradox, we chart the diversity of epistemic behavior. This turn may be compared to that in Kripke’s theory of truth, where self-reference of propositions became an object of study, rather than a taboo. Another interesting typology goes back to the ‘coming to know’ of Section 4, our dynamic setting for learning true propositions. Indeed, van Benthem (2004) deﬁnes three possible types of learnability for propositions ϕ, using an existential action modality ψ: one can truly announce A and then get ψ true. |= ϕ → ∃AK ϕ Local Learnability ∃A :|= ϕ → K ϕ Uniform Learnability |= ϕ → K ϕ Autodidactics He shows that each successive type is more demanding than the preceding. Moreover, at least on epistemic S5-models, all three notions of learnability are decidable. Further notions of learning arise with iteration of true assertions, perhaps even the same one. Baltag, van Ditmarsch, Herzig, Hoshi, and de Lima (2006) present sophisticated update calculi of this sort, and they prove in particular that, when added to our basic logic of public announcement, the logic of ‘truth after some announcement’ stays axiomatizable. Thus, once again, the ‘paradox of knowability’ turns from a nuisance into an interesting phenomenon, and a source of intriguing new logical questions.

5.2. Digression: reachability with event updates The event updates of Section 5 took an epistemic model M and an event model A, and computed a new product model MxA. Many more statements may be made true by such drastic changes. Call a model N ‘reachable’ from M , if, for some event model A, N is equal (or better: ‘epistemically bisimilar’) to MxA. Could this approach rescue CT ? We gave a model (M , w) and a statement ϕ true in it—but, even though K ϕ was consistent, having a model N with ϕ true throughout, no eliminative update took M to such a model. But could some more general event update do that job? For the single-agent case, this is not so.

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Every model MxA is then bisimilar to some submodel of M . But, there might be another way of saving CT. To link with the current (M , w), we might require that K ϕ be consistent with a description of the current world in (M , w). But, if we make K ϕ consistent with the state description of w (its true and false atomic propositions), update may still be impossible. If we make K ϕ consistent with the complete modal theory of w, we do get a model bisimilar to (M , w) (van Benthem 2002), but this seems a trivial victory.

5.3. Explicit temporal perspectives on knowledge We conclude with a common criticism of the ‘learning problem’ in the dynamic epistemic setting. Self-refuting Moore-type assertions evoke strong responses. One either loves this sort of subtlety, or one thinks it fundamentally misguided, disregarding the role of time. And, indeed, there is a sense in which announcing any true proposition should always lead to common knowledge. When I say that ϕ is true right now, at time t0 , immediately afterwards, it becomes common knowledge that ϕ was true then at time t0 ! This insight is not in conﬂict with the type of logic we have used. We can add explicit temporal operators to the dynamic epistemic framework, say a Y for yesterday in the time of our epistemic process. Then we get a complete update logic again, including the following attractive validity: ϕ → [ !ϕ ]CG Y ϕ This says that, if ϕ is true now, announcing it makes it common knowledge that it was true at the preceding stage. Some conversational moves work in just this way—like when people say in response to some assertion that ‘‘I knew already what you told me.’’ One might see this as one plausible sense in which the Veriﬁcationist thesis does hold: Every local truth right now can come to be known as being true now at some later stage of investigation. Indeed, analyzing the Paradox of Knowability in an explicit temporal epistemic logic has been proposed before, e.g., in Edgington (1985). In such a formalism, all the above issues still make sense. In particular, we now want to know precisely which assertions will persist over time, from Y ϕ to ϕ. For some further explorations at the border-line with dynamic-epistemic logic, cf. Sack (2006), Yap (2006), van Benthem and Pacuit (2006). It has to be said that this greater expressive power also has its price. In particular, statements of valid ‘learning

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principles’, and complexity of epistemic-temporal logics, depend in subtle ways on which precise strength we give to the temporal operators. This section has presented a number of technicalities that may seem nongermane to our general discussion. But the way we see it, these demonstrate that any ‘banning’ response to the Fitch paradox would be a bad idea, as it would deprive us of a rich area of investigation offering a lot of genuine insight into how we come to know things.

6 . Ma n y A g e n t s , C o m m u n i c a t i o n , a n d In t e r a c t i o n O ve r Ti m e Modern epistemic logic is no longer about lonely knowers in rickety armchairs in leaking attics. It unfolds its true attractions in multi-agent settings, analyzing what agents know about each other, and how they interact: in communication, games, or any other social activities where information ﬂows. The earlier-mentioned notion of common knowledge is crucial then. Here, self-refuting assertions come up naturally without any Moore-like paradoxical ﬂavor, witness the following evergreen from the literature.

6.1. Puzzles of repeated announcement Like other areas of logic, dynamic epistemic logic has its ‘icons’. In the well-known puzzle of the Muddy Children, whose epistemic importance was recognized in Fagin, Halpern, Moses, and Vardi (1995), it is successive public announcements of ignorance which drive the solution process toward common knowledge of the true state of affairs. In a simple version, the story runs as follows: After playing outside, two of three children have mud on their foreheads. They all see the others, but not themselves, so they do not know their own status. Now their father comes and says: ‘‘At least one of you is dirty.’’ He then asks: ‘‘Does anyone know if he is dirty?’’ The children answer truthfully. As questions and answers repeat, what will happen?

Nobody knows in the ﬁrst round. But in the second round, each muddy child can ﬁgure out her status, by explicit reasoning, or by updates. To display these, draw an epistemic model whose worlds assign D or C to each child. The actual world is DDC : that is, child 1 and 2 are dirty, while child 3 is clean. Initially, a child knows only the status of the others’ faces, but not her own. The corresponding epistemic uncertainty relations are indicated by the labeled lines

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in the following diagrams. Epistemic updates start with the father’s elimination of the world CCC: from DDD 1

3 DDC∗

2

CDD

DCD

3

1 2

2 1

CDC

3

2

CCD

DCC

3

1

CCC

Figure 9.6.

to DDD 3

1

DDC∗

2

CDD

DCD

3

1 2

2 1 CCD

CDC

3 DCC

Figure 9.7.

One can see this as a simple ‘symmetry breaking’ of the original pattern which will have startling consequences—like the way, say, a professional starts a snooker game. Next, when it turns out that no one knows his status, the bottom worlds disappear as shown in Figure 9.8:

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DDD 1 CDD

2

3 DDC ∗

DCD

Figure 9.8.

Finally, when the muddy children 1 and 2 say simultaneously that they know their status, all worlds where at least one of them still has an uncertainty line left disappear. Thus, this statement, too, was highly informative-and the ﬁnal update is to: DDC ∗ With k muddy children, k rounds of public ignorance assertions achieve common knowledge about who is dirty, while the announcement that the muddy children know their status achieves common knowledge of the whole situation. Thus, public assertions of ignorance can drive a positive process of gathering information, and their ability to eventually invalidate themselves (the earlier-mentioned phenomenon of ‘self-defeating’ assertions) may even be the crowning event. The last announcement of ignorance for the muddy children led to their knowing the actual world. This puzzle highlights the interplay of many agents, and also the passage of time. We consider both in turn.

6.2. Multi-agent learning Our scenario suggests that learning becomes more interesting, and less ‘paradoxical’ in a multi-agent setting. Indeed, we do not need Muddy Children to make this point. Much simpler epistemic models represent interesting scenarios of communication which might be hard to keep straight just in words. Consider the example in Figure 9.9, with three worlds and two agents: p

2

p

1 p

Figure 9.9.

In the actual world to the top left, indicated in black, p is in fact the case, but neither agent knows if p. Now what can the agents learn by internal

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communication? First, neither can tell the other something factual about p. And yet the agents can discover where they are by communicating their epistemic state. First 1 says ‘‘I don’t know if p.’’ This rules out the right-most world, where K1 ¬p holds. After the update, only the left-hand worlds remain, and so 2 now knows that p. Saying that will then also inform 1, and the agents have achieved common knowledge of p. This is just one case: three-world models support a range of communication scenarios. Thus, epistemic models with different accessibility structure for agents encode useful information exchange, either of ground facts, or of epistemic attitudes. The general issue here is what agents can learn if they communicate what they know, and keep doing so until the model no longer shrinks. Van Benthem (2002) describes such scenarios completely, showing they lead to a unique submodel (with the technical proviso of ‘modulo bisimulation’). Essentially, internal communication turns the ‘implicit knowledge’ of a group into common knowledge. Similar scenarios have been studied in game theory when making hidden correlations in information explicit between players. But there may be other, more complex sorts of speciﬁcation for a communication process. For example, we may want only some group members to learn that ϕ, keeping the others in the dark. This can also happen with Moore-type statements. Here is one more scenario showing such multi-agent phenomena. In Figure 9.10, consider the following model M with actual world p, q: p, q 1

p, q

2 p, q

Figure 9.10.

Announcing q will make 2 know the Moore statement that ‘‘p and 1 does not know it.’’ But this can never become common knowledge in the group {1, 2}. What can become common knowledge, however, is p & q, when 1 announces that q, and 2 then says p.

6.3. Fitch paradoxes for plural knowers? Next, consider the Paradox of Knowability once more. When q is true and you don’t know it, there is nothing problematic with others knowing both those facts.

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Indeed, general communicative actions—though not full public announcement to the whole group—can ensure the truth of: K2 (q & ¬K1 q) ! Thus we might amend the Veriﬁcationist Thesis VT once more, and recast it as: If ϕ is true, then someone could come to know it.

VTmulti-agent

This principle is true, at least in some construals! In any model (M , s) where ϕ holds, adding a perfectly informed agent whose epistemic accessibility relation is identity between worlds is consistent, and in the expanded model that new agent knows that ϕ. More interesting is the issue of coming to know facts about the whole group. Here are two new possibilities. First, let ϕ be true but not common knowledge: ϕ & ¬CG ϕ This cannot be common knowledge in the group G, as the old Fitch argument still applies. But ϕ & ¬CG ϕ can be known by individual agents, and even whole subgroups. Next, consider a stronger case: ϕ is true, but there is a false common belief that it is not: ϕ & CBG ¬ϕ This time, no agent in the group can come to know this—at least in a very plausible epistemic-doxastic logic. For if agent i were to know ϕ & CBG ¬ϕ, we would have (a) Ki ϕ → Ki Ki ϕ → Ki Bi ϕ, (b) Ki CBG ¬ϕ → Ki Bi ¬ϕ, and so (c) Ki (Bi ¬ϕ & Bi ¬ϕ ), and Ki Bi ⊥, and hence a contradiction, at least, if our logic does not allow belief in contradictions. Veriﬁcationism becomes a more varied issue in communities of epistemic agents.

6.4. Temporal perspective once more: game theory and learning theory Dynamic epistemic logics describe single steps in larger processes where information ﬂows. There seems to be a growing consensus that such long-term procedures are crucial to ‘coming to know’. Our concerns so far then merge into larger issues about interactive agents with goals and strategies for achieving them. Thus, dynamic-epistemic logic meets game theory (Osborne and Rubinstein 1994) and learning theory (Kelly 1996), including strategic equilibria and convergent learning procedures in both ﬁnite and inﬁnite settings. These links go beyond the present paper, but their import is clear. In the ﬁnal analysis, what one can come to know is intimately intertwined with the how!

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7. Conclusion We have looked at the Paradox of the Knower in a dynamic-epistemic perspective where learning means changing the current epistemic model. The problematic Moore sentence driving the paradox turns out to be the typical ‘probe’ for investigating the sometimes surprising, but always useful, effects of successive assertions. Moreover, the multi-agent setting of epistemic logic places Veriﬁcationism in a richer interactive setting. This change in perspective trades the atmosphere of paradox and disaster for one of free exploration of dynamic typology of epistemic assertions, learning and reachability, and many further surprising twists in the logic of communication. Even so, we do not claim the last word on Veriﬁcationism, the origin of the Fitch puzzle. The proof-theoretic paradigm of evidence for what we know also has a ring of truth. And, indeed, the dynamic approach so far has no insightful take on the ‘information’ that comes to us via deduction (cf. Egr´e 2004; Jago 2006). Updating with logical consequences of what we know does not change any of the models used here. A uniﬁed account of learning from deduction and from observation is a long way off. For one recent attempt at merging the relevant kinds of information: observational ad inferential, in a dynamic logic setting, cf. van Benthem 2008. And even our own semantic perspective has told only half of the story. In a truly multi-agent setting, learning is not a single-agent matter, and the basic paradigm should have at least two roles: the Learner and the Teacher. And then, the issue with learning is not just what information we get when updated by some given assertion—say, an answer—or some more general observation of an event. It is just as much the other side of the coin: what we ask of others, and how we enquire. Veriﬁcation and veriﬁcationism seems really about both seeking and ﬁnding intertwined: a point made long ago in Hintikka (1973). In that light, our story so far has only addressed half of the real topic.

10 Can Truth Out? John Burgess

. . . truth will come to light; murder cannot be hid long; a man’s son may, but at the length truth will out. The Merchant of Venice, II.ii.73

1 It is rather discouraging that forty years have passed since Frederic Fitch ﬁrst propounded his paradox of knowability without philosophers having achieved agreement on a solution (1963: 135–42).¹ As a general rule, when modal phenomena prove puzzling, it is a good idea to look at the corresponding temporal phenomena, and accordingly I propose to examine here not the knowability principle that whatever is true can be known, but rather the discovery principle that whatever is true will be known. As Fitch’s modal paradox attacks the knowability principle, so an analogous temporal paradox threatens the discovery principle. The formulation of the paradox is as follows. Start with the minimal tense logic with G and H for ‘‘it is always going to be . . .’’ and ‘‘it always has been . . .’’ as primitive, and F and P for ‘‘it sometime will be . . .’’ and ‘‘it once was . . .’’ deﬁned as ∼ G ∼ and ∼ H ∼ (see Burgess 1984).² Add a one-place epistemic operator K for ‘‘it is known that,’’ and add as axioms minimal assumptions for this new operator, expressing that anything known is true, and that if a conjunction is known, so are both conjuncts: (1) Kp → p (2) K(p & q) → Kp & Kq Acknowledgment: Thanks to Michael Fara, Helge Rückert, and Timothy Williamson for perceptive comments on earlier drafts of this paper. ¹ For a summary of recent debates, see Brogaard and Salerno (2004). ² The various theorems of tense logic cited below can all be found in this source.

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In an attempt to formalize the discovery principle, add one further axiom: (3) p → FKp The paradox is that one can then derive the following: (4) p → Kp The derivation of (4) using (3) is, apart from replacing and ♦ by G and F, the same as Fitch’s derivation, which is too well known to bear repeating here. The operator K is intended to indicate human knowledge, not divine omniscience. The grounds for belief in the discovery principle have indeed traditionally involved a belief in divine omniscience, but it is not this belief alone that supports the principle, but rather this belief plus a further belief that on some future day God will bring it about that whatever is hidden is made manifest (quidquid latet apparebit). Obviously that day has not yet come, and the conclusion (4), that everything true is already humanly known, is an absurdity, and so we have a reductio of the principle (3). The ‘‘dialethists’’ and other proponents of radical revisions of classical logic can be counted on to tout their proposed revisions as solutions to this paradox, as they have touted them as the solutions to so many others. But a priori it is overwhelmingly more likely that the problem lies not in the underlying classical logic, but in the least familiar element, the axiom (3), the only axiom in which temporal and epistemic operators interact. And, indeed, that is where the problem lies. One has to be careful in going back and forth between symbolism and English prose, and Fitch, or rather, his hypothetical temporal analogue, wasn’t careful enough. In tense logic p, q, r , . . . are supposed to stand for tensed sentences, whose truth-value may change with time (or if one wants to speak of ‘‘propositions,’’ then they must be propositions in a traditional rather than a contemporary sense, propositions that are themselves tensed, and whose truth-value may change with time). FA is supposed to be true at a given time if A is true at some later time. What (3) actually expresses thus amounts to this: (5) If p is true now, then at some later time it will be known that p is true then. The proposed formalization as (3) has in effect turned the principle that any truth will become known into the principle that any sentence that expresses a truth will come to be known to express a truth. But this last formulation invites the immediate objection that the sentence in question may cease to express a truth before the knowledge of the truth it once expressed is acquired. And so (5) surely does not express what Shakespeare meant in saying ‘‘Truth will out.’’ He meant to imply that if Smith murders Jones secretly, so that no one knows, then it will become known that Smith murdered Jones secretly, so that no one knew. He did not mean to imply that if what the form of words ‘‘Unknown to all, Smith has murdered Jones’’ now expresses is true, then there

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will come a time when what that same form of words then expresses will be known to be true. Thus the temporal analogue of Fitch’s argument does not discredit the discovery principle, because the target of that argument is not a correct expression of that principle.

2 That one particular objection to a principle fails is no proof of the principle itself, and indeed no proof that it may not be open to simple, straightforward objection along other lines. And in fact the discovery principle is open to two kinds of objection, each of which requires us either to impose a restriction on the principle, or to assume charitably that a restriction on the principle was already intended by its advocates. As background to a ﬁrst objection consider the timing of the collision of two ordinary extended material objects. The boundaries of such objects generally are sufﬁciently ill-deﬁned on a scale of nanometers as to make dating their collision on a scale ﬁner than nanoseconds meaningless. If murders, say, are all the events we want to talk about, we do not need to conceive of ‘‘times’’ as durationless ‘‘instants,’’ but may conceive of them as very brief ‘‘moments,’’ of no more than, say, a nanosecond’s duration. In this case, chronometry—by which I here mean no more than our usual ways of dating events by year, month, day, hour, minute, second, and on to milli- or micro- or nanosecond and beyond if one wishes, all tacitly understood relative to some ﬁxed time zone—supplies a term for every time. But it may be otherwise if we wish to speak of point-particles and their collisions. The worry is that there will be truths that can never be known because they can never be stated. Suppose, for instance, that x = 0.1 8 2 5 6 4 7 9 3 . . . is an irrational number, and that exactly x seconds before 12:00 p.m., particle i collided with particle j. Can it ever become known that particles i and j collided at exactly x seconds before 12:00 p.m. on June 1, 2003? According to the discovery principle, all the following will become known: (1) Particles i and j collided at 182 ± milliseconds before 12:00 p.m. on June 1, 2003. (2) Particles i and j collided at 182564 ± microseconds before 12:00 p.m. on June 1, 2003. (3) Particles i and j collided at 182564793 ± nanoseconds before 12:00 p.m. on June 1, 2003. Here ‘‘±’’ abbreviates ‘‘the nearest unit’’, to the nearest milli- or micro- or nanosecond, as the case may be. (For the sake of argument, set aside any quantum-mechanical doubts about whether the series (1)–(3) really could be continued indeﬁnitely.)

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But for it to be knowable that i and j collided at exactly x seconds before 12:00 p.m., would it not have to be sayable that i and j collided at exactly x seconds before 12:00 p.m.? And for this to be sayable, there would have to be some means in language or thought of referring to the irrational number x—I mean, of course, some means other than referring to it as the number of seconds before 12:00 p.m. when i and j collided. √ Mathematics supplies such means for relatively few irrational numbers, such as 2 , π, e, and so forth. Coincidence may supply a few others: the time when i and j collided may be describable also as the time when k and l collide, if the two collisions happen to be simultaneous. But by cardinality considerations we inevitably lack means of reference to most irrational numbers. The discovery principle must be understood to exclude ineffable truths. It must be understood as restricted to truths expressible in our language. Such a restriction will be built into any tense-logical formalization of the principle, if the letters p, q, r , . . . are understood as standing for sentences of our language. Such a restriction seems in one sense not too serious, because the principle still tells us that the true answer to any question we have the language to ask will become known.

3 A second objection to the discovery principle is more subtle. Suppose that as I write it is 12:00 p.m., June 1, 2003. Then the following is true: (1) Now, this moment, it is 12:00 p.m., June 1, 2003. Obviously (1) itself will never be true in the future. And it seems that no sentence of our language will ever express in the future exactly what (1) expresses now. Thus the truth that (1) now expresses seems to be one that will be unknowable in the future because it is unsayable in the future. Moreover, the demonstrative ‘‘this moment’’ and the indexical ‘‘now’’ are both pleonastic, what they indicate being already sufﬁciently indicated by the fact that the verb ‘‘is’’ is in the present tense. Thus what has just been said about (1) is equally true of the following: (2) It is 12:00 p.m., June 1, 2003. And indeed, if now, this minute, Smith is murdering Jones, then the following is another example subject to the same difﬁculty as (1) and (2). (3) Smith is murdering Jones. The truth that Smith is (now, this moment) murdering Jones seems one that will be unsayable and therefore unknowable in the future, even though it is sayable now and even knowable now. The discovery principle must be understood to exclude not only ineffable truths, which are never expressible in our language, but also ephemeral truths,

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which are expressible for a moment, and then never again. Such a restriction seems in one sense not too serious, because it does not leave us with any question that can always be asked and never be answered. The ephemeral will be equally inexpressible interrogatively as assertorically. Such a restriction seems not too serious for another reason, because the truths it excludes from human knowledge in the future are excluded even from divine knowledge in eternity, if one follows those theologians who make the latter knowledge timeless. For (1)–(3) are no more true in a timeless eternity than they will be true in the seconds and minutes and hours and days and months and years to come. The old riddle that suggests an exception to the principle that God can see anything I can see is a joke.³ But the counterexamples (1)–(3) to the principle that God knows anything I can know are not. This point seemed worth digressing to mention, if only because a desire to have a formal apparatus in which such issues could be discussed was an important part of the motivation of the creation of tense logic by Arthur Prior.

4 We have seen that (1.3)—displayed item (3) of §1—is not the right formalization of the discovery principle. What is? It cannot be claimed that a complete solution to the paradox has been obtained until this question is answered. One answer suggests itself at once. Now that we have restricted the principle to truths that will remain expressible in our language in the future, it is tempting to formulate the principle as the principle that any sentence that will continue to express a truth in the future will come to be known to express a truth. This goes over into symbols as follows: (1) Gp → FKp And (1) is, unlike (1.3), immune to Fitch-style paradox, even if one considerably strengthens the background tense logic. For deﬁniteness, let us consider the tense logic, call it Llinear , that is appropriate for linearly ordered time without a last time. Then the immunity of (1) from Fitch-style paradox is the content of the following proposition. Proposition. Let T be Llinear plus (1.1), (1.2), and (1). Then (1.4) is not a theorem of T . ³ I mean the riddle: Q. What is it that God never sees, that the king seldom sees, but that you and I see every day? A. An equal. This seems less a problem for theologians than for partisans of ‘‘substitutional quantiﬁcation.’’

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Proof. Consider an auxiliary theory T ∗ , obtained from Llinear by adding a constant π and the following axiom: (2) Fπ Then π is not a theorem of T ∗ . For if we take any model of Llinear , and let π be true at and only at the times later than the present, then (2) will be true at all times, but π will not, being false at all past times and at the present time. Next assign each formula A of the language of T a translation A∗ into the language of T ∗ , by taking Kp to abbreviate p & π. Thus (1.1), (1.2), (1), and (1.4), respectively, are translated as follows: (3) (4) (5) (6)

p&π → p (p & q) & π → (p & π) & (q & π) Gp → F(p & π) p → p&π

Note that the translation (6) of (1.4) is not a theorem of T ∗ . For, if it were, substituting ∼ π for p and applying truth-functional logic, π would be a theorem, and as we have seen it is not. To show that (1.4) is not a theorem of T , it will sufﬁce to show that the translation of any theorem of T is a theorem of T ∗ . And, to show this, it will sufﬁce to show that the translations (3)–(5) of the three axioms of T are theorems of T ∗ . For the ﬁrst two axioms this is trivial, since (3) and (4) are truth-functional tautologies. For the third axiom, this follows using the following theorem of Llinear : (7) Gp & Fq → F(p & q) And (5) follows by truth-functional logic from (2) and (7), to complete the proof.

5 The formalization (4.1) has several corollaries worth noting. Proposition. Let T be as in §4. Then the following are theorems of T : (1) (2) (3) (4)

Pp → FKPp p → FKPp Fp → FKPp Gp → FKGp

Proof. First note that each of the following is either an axiom or a theorem of Llinear : (5) Pp → GPp (6) p → GPp

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(7) Fp → FGPp (8) FFp → Fp (9) Gp → GGp Also, the following is a derived rule of Llinear : (10) If A → B is a theorem, then FA → FB is a theorem. (For the cognoscenti, the assumption here is that the rule of temporal generalization, on which (10) depends, continues to imply after the formal language has been enriched by the addition of the epistemic operator K.) (1), (2), and (4) are immediate from (5), (6), and (9), respectively. As for (3), it can be derived as follows: (11) FPp → FFKPp (12) FPp → FKPp

from (1) by (10) from (11) and (8) 6

To illustrate these corollaries just derived, if Smith has murdered Jones, or is murdering Jones, or will murder Jones, then according to whichever of (5.1)–(5.3) is applicable, it will become known that Smith has murdered Jones. Let us write brackets around present tense verbs to indicate omnitemporality, so that, for instance (1) Smith [murders] Jones. is to be understood as meaning (2) Smith has murdered, is murdering, or will murder Jones. Then we may say that if Smith [murders] Jones, then it will become known that Smith murdered Jones. And similarly in any other case. Murder cannot be hid—though (4.1) does not go so far as to join the Bard in claiming (unfortunately, erroneously) that murder cannot be hid long. And if the memory of Smith’s victim will never cease to be honored, then according to (5.4) this fact will become known—though there is (again, unfortunately) no guarantee it will become known soon enough to comfort the victim’s grieving friends and relations. And if the universe will be forever expanding, according to (5.4) this fact, too, will eventually become known—though there is (yet again, unfortunately) no guarantee it will become known soon enough to satisfy the curiosity of present-day cosmologists. Still, despite its corollaries, (4.1) may look unsatisfactory for the following reason. Consider what the corollary (5.2) tells us about a present truth: (3) If p is true now, then at some later time it will be known that p was true once.

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The ‘‘once’’ here invites the question, ‘‘When?’’ And (5.2) provides no answer. Or so it may seem. But in a sense (5.2), taken together with chronometry, does provide an answer. If (3.2) and (3.3) are true, then their conjunction is true: (4) It is 12:00 p.m., June 1, 2003, and Smith is murdering Jones. Applying (5.2) not to (3.3) alone, but to this conjunction, we obtain It will become known that it was once 12:00 p.m., June 1, 2003, and Smith was murdering Jones. Or more idiomatically: (5) It will become known that Smith murdered Jones at 12:00 p.m., June 1, 2003. What more could one want by way of answer to a when-question? Quite generally, an event occurs at a given time, one can conjoin to a sentence p asserting the event’s occurrence a sentence q giving the standard chronometric speciﬁcation of the time, and then apply (5.2), not to p alone, but to the conjunction.

7 Nonetheless, it may seem that the most obvious correction of (1.5) would be the following: (1) If p is true now, then at some later time it will be known that p was true now. And (1) seems to tell us more than (4.1) (by way of (6.3)) tells us. It is known that (1) cannot be expressed using just the temporal operators G and H and F and P. But tense logicians have considered other operators. Most to the point in the present context, they have considered a ‘‘now’’ operator J, so interpreted that even within the scope of a past or future operator Jp still expresses the present, not the past or future, truth of p. And with this operator (1) can be symbolized, as follows: (2) p → FKJp One may be tempted to think that (2) would do better as a formalization of the discovery principle than does (4.1). But this is a misleading way of putting the issue. For if the operator J is admitted, subject to its usual laws, then (4.1) implies (2). For one of the usual laws is precisely (3) p → GJp and (2) is immediate from (3) and (4.1). So the temptation here is simply the temptation to add J to the language.⁴ ⁴ I owe this observation to Williamson.

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I think the temptation should be resisted for a double reason. My ﬁrst reason is that introducing the J-operator is unnecessary in order to answer a when-question. For I have just ﬁnished arguing that (4.1) does, after all, provide answers to such questions. Against this it may be said that (2) appears to have the advantage of doing so without depending on chronometry. But my second reason for avoiding the J-operator is that this apparent advantage comes at the cost of involving us with the problematic notion of a de re attitude towards a time. This truth is perhaps most easily brought to light by switching temporarily from regimentations using tense operators to regimentations using explicit quantiﬁcation over times. So let t , u, v, . . . range over times. And let t < u mean that time t is earlier than time u, or equivalently, time u is later than time t. Let each tensed p be replaced by a one-place p∗ (t ) for ‘‘p [is] the case at time t.’’ Every formula A built up from the letters p, q, r , . . . will similarly be replaced by an open formula A∗ (t ). PA and FA, respectively, will be replaced by: (4a) ∃u(u < t & A∗ (u) )

(4b) ∃u(t < u & A∗ (u) )

In a formula A(t ) the parameter t may be thought of as standing for that time which is now present. Leaving open how to symbolize the epistemic operator, (5.2) and (2) above go halfway into symbols as follows: (5) p(t ) → ∃u(t < u & it is known at time u that ∃v(v < u & p(v) )) (6) p(t ) → ∃u(t < u & it is known at time u that p(t )) There is this difference between the two semi-formalizations, that what occurs towards the end of (5) can be understood in a de dicto way, thus: (7) At time u, ‘‘p was true once’’ [is] known to be true. By contrast, what occurs towards the end of (6) must be understood in a de re way, thus: (8) At time u, ‘‘p was true then’’ [is] known to be true of time t. The symbol-complex KJp in (2) above may be pronounced ‘‘it is known that p was true now,’’ but what it really amounts to is more like this: (9) It is known of t that p was true then, where t is that time which is now present. 8 There are (at least) three major difﬁculties in making sense of the notion of a de re knowledge about an object a. Or to put the matter another way, there is only one obvious strategy for making sense of the notion of a de re attitude, namely, reduction to a de dicto attitude, and there are (at least) three major obstacles to this strategy. The strategy is to understand a subject as knowing of an object a

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that F (x) holds of it if and only if the subject knows that F (a) where a is a term denoting a. The three obstacles or problems relate to the choice of term a. A ﬁrst general problem with de re knowledge is that of anonymity. There may simply be no term a denoting a. This problem has been encountered in the case of times in §2, and given the restriction on the discovery principle imposed there, it may be set aside here. A second general problem with de re knowledge, and one relevant to the question whether J should be admitted is the problem of aliases. The problem is that there may be two terms a and b denoting an object a, and it may be that the subject knows that F (a) but does not know that F (b), or the reverse. The star whose common name is ‘‘Aldebaran’’ has also the ofﬁcial name ‘‘Alpha Tauri.’’ It seems that a subject may have been told by different authoritative sources, and hence may know that: (1) Aldebaran is orangish. (2) Alpha Tauri is the thirteenth brightest star. and yet, being in ignorance that the two names are names for one and the same heavenly body, the subject may not know that: (3) Alpha Tauri is orangish. (4) Aldebaran is the thirteenth brightest star. And this makes it hard to answer the question whether the subject knows of the star itself, independently of how it is named, that it is orangish, or the thirteenth brightest. The existence of aliases is a problem insofar as privileging one of them over the other seems arbitrary. The same problem can arise for times. Robinson may know that one rainy day Smith committed murder, and may know that Jones was murdered, and not know that the murder Smith committed was that of Jones. In this case Robinson will know that: (5) At the time when Smith committed murder, it was rainy. But not that: (6) At the time when Jones was murdered, it was rainy. And this makes it hard to answer the question whether Robinson knows of the time itself, independently of how it is described, that it was rainy then. Where there exists some standard term for each object of a given kind, one can always stipulate that a subject is to be credited with de re knowledge about the object a that F (x) holds of it, if and only if the subject has de dicto knowledge that F (a) where a is the standard term for a. Admittedly, such a stipulation may be more a matter of giving a sense to a kind of locution (ascriptions of de re knowledge) that previously had none, than of ﬁnding out what sense this kind of locution had all along. Pretty clearly, it would be a case of giving rather

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than ﬁnding if one took as canonical terms for heavenly bodies the ofﬁcial names adopted by international scientiﬁc bodies, preferring ‘‘Alpha Tauri’’ over ‘‘Aldebaran.’’ For times, the obvious candidates for standard terms are those provided by chronometry. If one is content with (4.1), there is no need to enter into the problem of de re knowledge about times at all, and so no need to ﬁx on any standard terms for times. If one adopts (7.2), reliance on chronometry is the only obvious way to impose a solution on the problem of aliases. But in that case the one advantage (7.2) appeared to have over (4.1), that of not depending on chronometry, must be recognized to have been illusory. This consideration argues, I claim, in favor of the J-free formalization (4.1) and against the J-laden formalization (7.2). A third general problem with de re belief is the problem of demonstratives (and with them indexicals). When the star Alpha Tauri, alias Aldebaran, is visible in the night sky, one can point to it and say ‘‘that star,’’ and so achieve reference to it. Now it seems someone looking at the star may well know (5) That star is orangish. And yet not knowing the name of the star, she may well not know either (1) or (2). This is, so far, just a special case of the problem of aliases. But demonstratives are especially troublesome because, on the one hand, when available, they seem to provide so direct a way of referring that it is hard to insist that nonetheless it is some other way of referring that provides the canonical terms for reduction of de re to de dicto; but on the other hand, demonstratives themselves are not viable candidates for canonical terms, simply because they are usually not available: if we took demonstratives as canonical terms, most objects would suffer from anonymity most of the time. Demonstratives act, so to speak, as spoilers, making any other candidates for the ofﬁce of canonical term look unworthy, while themselves not being eligible for that ofﬁce. But this problem has been encountered in the case of times in §3, and given the restriction on the discovery principle imposed there, it may be set aside here, as the problem of anonymity was set aside. The problem of aliases, I claim, is enough to make the admission of J undesirable.

9 I have done with the topic of the discovery principle. But what of the knowability principle, and the original, modal version of Fitch’s paradox? I began this essay by recalling that there is a close parallel between temporal and modal. I should now note that while there are many analogies, in connection with Fitch’s paradox there is also one glaring disanalogy, that makes the original, modal problem more

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refractory than its temporal analogue. Perhaps the best way of proceeding would be to begin by simply listing pairs of analogous notions in parallel columns, as shown in Table 10.1.

(1.3) (4.1) (5.2) (7.2)

Temporal

Modal

Discovery Principle present tense, future tense G, F Llinear p → FKp Gp → FKp p → FKPp now J p → FKJp times, instants, moments chronometry

Knowability Principle indicative mood, subjunctive mood , ♦ S5 p → ♦Kp p → ♦Kp p → ♦K♦p actually @ p → ♦[email protected] possibilities, worlds, situations ???

(1) (2) (3) (4)

But, returning to what is formally representable, I have recalled in the left margin in the table the numbers of temporal formulas we have met earlier, and assigned in the right margin numbers to the analogous modal formulas. Fitch’s (1) is dismissable for reason analogous to those that led to the dismissable of its analogue (1.3).⁵ The difﬁculty comes when one seeks a replacement. The absence of any obvious analogue for possibilities of standard chronometric speciﬁcations for times makes (2) and (3) much less satisfactory than (4.1) and (5.2)—and (4) correspondingly much more tempting than (7.2). But the same absence makes the problem of de re knowledge of possible situations connected with (4) at once more critical and more difﬁcult to solve or evade than was the problem of de re knowledge of temporal moments connected with (7.2). I will not enlarge further here, partly because it would be a good exercise for readers to work out the analogy for themselves, but mainly because I would be largely repeating points that have been made by Dorothy Edgington in her proposed solution to the paradox, and by Timothy Williamson in his criticisms thereof.⁶ A further disanalogy emerges in discussion of Edgington and Williamson that is not formally representable, and is therefore not indicated in the above table. It is just this: that, generally speaking, the fact that something is only actually true and not necessarily true tends to matter less to us than the fact ⁵ Though there is a lot more than can be said. For a full exposition of the essentially grammatical fallacy in the paradox, see Rückert (2004). Rückert draws on Wehmeir (2005). ⁶ See Edgington (1985); Williamson (1987a). For further relevant publications, see Brogaard and Salerno (2004).

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that something is only at the present moment true and not permanently true. Or, to put the matter another way, what will be true when the world is older matters more to us than what could have been true if the world had been otherwise, since we hope to live on into ‘‘future worlds’’ but do not expect to transmigrate into ‘‘possible worlds.’’ So far as the present investigation is concerned, it seems that the analogy between mood and tense takes us only so far, and in the end provides us not with a solution, but only with a better understanding of just what makes the problem difﬁcult.

10 Before giving up, however, perhaps we should try the combination of temporal and modal. That is to say, perhaps instead of considering the knowability principle as the principle that anything true could have been known, we should consider it as the principle that anything true could become known. The natural setting for such a principle would be a system like Prior’s logic of ‘‘historical necessity’’ (see Prior 1967: ch. VIII). In the most elaborate version, which he calls ‘‘Ockhamist,’’ Prior uses both tense operators G, H, F, P, subject to the axioms for Llinear , and modal operators , ♦, subject to the axioms of S5. But the modal operators are themselves understood in a tensed way, as meaning necessity and possibility given the course of history up to the present. Prior uses special letters a, b, c, . . . for sentences with the special property that their truth is independent of the future course of history in addition to the usual letters p, q, r , . . . for arbitrary sentences. Not all formulas, but only certain special ones, with the same special property as the special letters, may be substituted for those special letters. These include the special letters themselves, any formula beginning with a modal operator or ♦, and any formula obtainable from formulas of these two kinds using the truth-functions and past-tense operators H and P. A single axiom links the temporal and modal operators: (1) a → a One can obtain by substitution (2) Pa → Pa One cannot derive (3) Fa → Fa Taking for a in (1)–(3) ‘‘A sea ﬁght is occurring,’’ in Prior’s system one can conclude that if a sea ﬁght is occurring or has occurred, then the occurrence of a sea ﬁght is (historically) necessary; but even if a sea ﬁght is only going to occur,

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then its occurrence is (historically) contingent, though once it does occur, it will become (historically) necessary. A version of the knowability principle can be expressed in this context by the formula (4) Ga → ♦FKa And from (4) one can derive, using various tense-logical and modal theorems, the following rough analogues of the corollaries in the proposition of §3: (5) (Pa ∨ a ∨ Fa) → ♦FKPa (6) Ga → ♦FK♦Ga The details will not be given here, because the system is ultimately unsatisfactory. Let me explain how. From (4), by way of its corollaries, one can conclude the following, wherein I contract ‘‘possibly will’’ to ‘‘may’’: (7) If Smith is murdering Jones, then it may become known that Smith has murdered Jones. (8) If the memory of Smith’s victim will always be honored, then it may become known that the memory of Smith’s victim may always be honored. (9) If the universe is always going to be expanding, then it may become known that the universe may be always going to be expanding. What one cannot conclude is: (10) If the memory of Smith’s victim will always be honored, then it may become known that the memory of Smith’s victim will always be honored. (11) If the universe is always going to be expanding, then it may become known that the universe is always going to be expanding. So (4) seems too weak. The strengthening of (4) to (12) Ga → ♦FKGa would provide assurance of (10) and (11), but unfortunately (12) is too strong. For it would also provide assurance of the absurd: (13) If Smith murdered Jones but will forever escape detection, then it may become known that Smith murdered Jones but will forever escape detection. This is Fitch’s paradox, adapted to the present context.

11 In sum, Prior’s Ockhamist framework fails to provide a formula that is not too weak and not too strong, but just right. A glance back at the examples above will

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help us localize the difﬁculty: it is with truths about the actual future (and more particularly about what will always hold throughout that actual future). There are philosophers, however, who question the very meaningfulness of assertions about the actual future (for a recent expression of this view see Belnap and Green 1994, 365–88). And Prior (1967: ch. VII) has developed a logic he calls ‘‘Peircean’’ for them. In this logic one has only the special letters a, b, c, . . ., and only the formulas built up from them using truth-functions, past-tense operators, and four operators amounting to the combinations G, ♦G, F, ♦F. Substitution for the letters a, b, c, . . . is allowed for all formulas so built up. The operators , ♦, G, F do not appear separately, apart from the four combinations just mentioned. The pertinent feature of this logic in the present context is that it bans as meaningless the examples that caused trouble in the preceding section, and (10.4) seems adequate as an expression of the knowability principle for all such sentences as are still accepted as meaningful. Banning statements about the actual future is a radical step. Presumably the friends and relations of Jones know that his memory possibly will always be honored, and possibly will not always be honored. They know ♦Ga and ∼ Ga, where a is (1) The memory of Smith’s victim is honored. The Peircean, however, rejects as meaningless (2) The memory of Smith’s victim will always be honored. unless ‘‘will’’ is either strengthened to ‘‘necessarily will’’ or weakened to ‘‘possibly will.’’ The Peircean it seems, can’t allow the friends and relations to hope that (2) is true, or to fear that it isn’t (this observation is repeated from Burgess 1978). Likewise, cosmologists presumably already know that it is possible the universe will expand forever, and possible that it won’t. The Peircean can’t allow them to wonder if it in actual fact will. So Peirceanism is a radical doctrine. But then, so is the knowability principle. The question is, do the two forms of radicalism cohere? If an adherent of the knowability principle were to embrace Peirceanism, would the resulting position have any coherent motivation? Or would embracing Peirceanism be mere ad hoc epicycling, avoiding counterexamples by declaring them meaningless? This is too large, and too non-logical, an issue to go into here, but at least a word may be said about the historical sources of epistemological views like the discovery and knowability principles on the one hand, and of Peirceanism on the other. Belief in the discovery principle, I said at the outset, has traditionally rested on theological grounds. Belief in the knowability principle has, by contrast, been mainly an expression of a commitment to a certain philosophical theory of meaning, veriﬁcationism. The radical epistemological view that there are no unknowable truths has usually been a consequence of the even more radical

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semantical view that understanding a sentence consists in grasping under what conditions it would be known to be true. Belief in Peirceanism has had several sources. Prior cites late-mediæval logicians who have held a similar view on theological grounds, but the more recent proponents of the view seem to base their adherence on grounds that ultimately are veriﬁcationist. Thus combining the knowability principle with Peirceanism could be viewed as combining two manifestations of an underlying veriﬁcationism. Of course, there are many varieties of veriﬁcationism, and it remains to be seen whether a single variety can cogently motivate both these manifestations simultaneously. A key issue will be the veriﬁcationist’s attitude towards the reality of the past.

11 Logical Types in Some Arguments about Knowability and Belief Bernard Linsky

Of course the foregoing refutation of Fitch’s deﬁnition of value is strongly suggestive of the paradox of the liar and other epistemological paradoxes. It may therefore be that Fitch can meet this particular objection by incorporating into the system of his paper one of the standard devices for avoiding the epistemological paradoxes. If this is possible it will involve a drastic rewriting of the paper, not just a footnote here and there.¹

Over the years a number of arguments have been formulated in elementary modal logic purporting to show that there are limits to what can be known or believed. These include the ‘‘Fitch’’ style arguments that will be the main interest of this paper, versions of the paradoxes of the ‘‘Surprise examination’’ and the ‘‘Preface,’’ and several arguments against analyses of truth in terms of veriﬁability under ideal conditions. A use of iteration of operators and even apparent self-reference seems to reappear in various of these arguments and so one might wonder what exactly is common to these arguments and if that common element reveals something about their validity.² In recent years it has also been claimed that veriﬁcationism is subject to logical difﬁculties revealed by these arguments. It would be a challenge to veriﬁcationism to have a proof that some true sentences simply cannot be known, or believed, even by an idealized agent. The understanding of these arguments is thus a pressing issue for the veriﬁcationist program. Proposals for analyses of truth in terms of veriﬁcation in ideal conditions also confront difﬁculties when ¹ From Alonzo Church’s anonymous referee reports on Frederic Fitch’s ‘‘A deﬁnition of Value,’’ in Chapter 1 of this volume. Church here considers the liar paradox as ‘‘epistemological’’ in its formulation involving propositions that are asserted, following Whitehead and Russell ((1910): p. 62), rather than in the later formulation in terms of linguistic expressions and truth, and so as ‘‘semantic’’. See Church (1976) for his classic treatment of these semantic paradoxes using the theory of types. I am grateful to Joseph Salerno for sharing the reports, which he discovered as this volume was in preparation. ² As I wondered at the end of my (1986).

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one worries about the truth conditions for statements asserting that such ideal conditions do or do not obtain. There appears to be at least self-application of the theory of truth to its own preconditions. In accord with the suggestion from Alonzo Church in the epigram that precedes this paper, I propose below to identify the elements of ‘‘self-reference’’ in these various arguments, distinguish self-reference proper from the use of iteration of operators expressing epistemic conditions, and then provide a uniform account of them making use of the idea of logical types of propositions. The argument with which I begin is due to Frederic Fitch (1963). Fitch gives credit for this argument to an anonymous referee of an earlier version of the paper from 1945.³ As described in the introduction to this volume that referee has recently been identiﬁed as Alonzo Church himself, and two referee reports have been found, although both drafts of the paper are missing. The quotation that begins this paper is from the second report, responding to a revised version from Fitch. Fitch’s original argument is put in terms of ‘‘truth classes’’ of propositions, classes all of whose members are true, but current discussions consider its application to the case of the factive operator ‘‘Knows that’’ (K) which only holds of true propositions. (Hence the ‘‘T’’ axiom in standard logics of knowledge: K φ ⊃ φ): The Fitch Argument 1 1) K (p & ∼Kp) Assumption 2) K (p & ∼Kp) ⊃ Kp & K ∼ Kp Distribution Axiom 1 3) Kp & K ∼ Kp 1,2, Prop Logic 4) K ∼ Kp ⊃ ∼Kp T Axiom 1 5) Kp & ∼Kp 3,4, Prop Logic 6) ∼K (p & ∼Kp) 1–5 reductio But since p & ∼Kp can surely be true in some situations, we would have then something true that couldn’t be known. The argument is brought to point as follows: ∼ K (p & ∼Kp) follows from the last line by necessitation. Suppose that all truths are knowable, including that a given truth isn’t known, that is to say, p & ∼Kp ⊃ ♦K (p & ∼Kp). Then we derive that the antecedent is false, so ∼(p & ∼Kp), in other words it is provable that p ⊃ Kp, a manifest absurdity.⁴ In his ﬁrst referee report, Church offers the following suggestion to Fitch: In spite of the preceding argument I think Fitch has a good defense (but only one). This defense is that there is no law of psychology according to which one who believes ³ See p. 138 n. 5 in Fitch (1963). ⁴ Some discussion of the argument has hinged on the fact that in intuitionistic logic only p ⊃ ∼ ∼Kp can be proved, something that ought not to bother the veriﬁcationist who would use such a logic. This raises another approach to the Fitch argument that I won’t consider.

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a proposition must believe all its logical consequences; on the contrary, historical counter-examples are well known. To be sure, one who believes a proposition without believing its more obvious consequences is a fool; but it is an empirical fact that there are fools. It is even possible that there might be so great a fool as to believe the conjunction of two propositions without believing either of the two propositions; at least, an empirical law to the contrary would seem to be open to doubt. (pp. 2–3)

A ‘‘fool’’ can believe a conjunction but not one of the conjuncts, and so the Distribution Axiom is not in general true. Church’s suggestion is that the above version ‘‘Fitch Argument,’’ in standard epistemic logic that treats ‘K ’ as a sentential operator, can be blocked as invalid at step 2. In what follows this objection will be considered again, but after reformulating the argument within the theory of types, one of the ‘‘standard devices for avoiding the epistemological paradoxes’’ to which Church refers above. Even a version of the Distribution Axiom limited to conjuncts of the same logical type will not make the argument valid. Consider next a proof of a related result, involving in this case a proposition that cannot be believed. ‘‘Moore’s Problem’’ (also known as ‘‘Moore’s Paradox’’) is the seeming oddity in the expression ‘‘p but I don’t believe it.’’⁵ Jaakko Hintikka proposed to analyze the oddity by proving that the sentence might well be true but cannot be believed consistently with the principles of epistemic logic. This might be seen as a challenge to veriﬁcationism in so far as it shows that some truths can’t even be believed, much less veriﬁed or known. Here is a reformulation that uses proposed axioms of the logic of belief to replace Hintikka’s ‘‘reductive,’’ semantic, argument:⁶ The Hintikka Argument 1 1) B(p & ∼Bp) Assumption 2) B(p & ∼Bp) ⊃ Bp & B ∼ Bp Distribution Axiom 1 3) Bp & B ∼ Bp 1,2, Prop Logic 4) Bp ⊃ BBp ‘‘KK’’ principle for B 1 5) BBp & B ∼ Bp 3,4, Prop Logic 1 6) B(Bp & ∼Bp) 5, Conjunction 7) ∼ B(Bp & ∼Bp) Logical Omniscience 8) ∼ B(p & ∼Bp) 1–7 reductio In this argument Hintikka invokes a variant of his notorious KK-thesis (Kp ⊃ KKp) only in this case for belief (Bp ⊃ BBp), as well as requiring a certain amount of consistency on the part of believers, here described as the stronger requirement of ‘‘Logical Omniscience’’. Both assumptions have been challenged.

⁵ See the discussion in Sorensen (1988): chapter 1. ⁶ Presented at p. 69 of his (1962).

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Now consider a third argument in this family, the result of adapting the Fitch argument to belief, so that ‘K’ becomes a ‘B’. Call what we get ‘Fitch-B.’ The Fitch-B Argument 1 1) B(p & ∼Bp) Assumption 2) B(p & ∼Bp) ⊃ Bp & B ∼ Bp Distribution Axiom 1 3) Bp & B ∼ Bp 1,2, Prop Logic 4) B ∼ Bp ⊃ ∼Bp (Disbelief Principle) 1 5) Bp & ∼Bp 3,4, Prop Logic 6) ∼ B(p & ∼Bp) 1–5 reductio Step (4) in Fitch-B cannot be justiﬁed as in the original Fitch argument, as an instance of the T schema, as belief surely does not obey that principle in general. Not everything believed is true. However, it is important to note that while this is a particular instance of a T schema, it is also an instance of a weaker principle, that might be true in general, namely B ∼ Bφ ⊃ ∼Bφ. That the argument can be run for B in this way has been noted, for example by Binkley (1968). Binkley sees it as a version of the KK thesis for belief like the one that Hintikka uses, namely that if one believes φ then one doesn’t believe that one doesn’t believe φ. It is better to view this as an instance of a principle that expresses the ‘‘transparency’’ of (dis-)belief, namely that when one believes that one doesn’t believe φ, one can’t be wrong. While explicitly contrary to Freud’s fundamental insights in psychology, such a principle is in keeping with the generally Cartesian and rationalist tone of much work in the epistemic logic of belief. To indicate that it is a principle on its own. Above I call it the ‘‘Disbelief Principle.’’ This is indeed a principle that one may well want to deny if one wants to block the Fitch argument.⁷ Surely some of our beliefs are opaque to us. However, in addition to indicating another point at which one might object, I hope to present a better logical picture of the nature of such error about our own beliefs. The arguments presented so far all make use of iterated operators, believing that one believes or knowing that one doesn’t know. Some of these principles make strong claims, even about some sort of idealized belief, and so the arguments might be faulted simply for using false premises or ‘‘axioms’’ of epistemic logic. The use of iteration as such does not seem to be at fault. These arguments do not involve self-referential beliefs, or some similar vicious circularity. Still, it may be argued, each involves some strong principle relating beliefs of different logical types, principles that may also easily be denied. The next argument to be considered does not show that a truth cannot be believed, but rather that certain beliefs will guarantee a truth. The correct ⁷ Frederik Stjernberg (1997) shows how a hierarchy of knowledge operators would block the Fitch argument, in a paper that may be one of the targets of Williamson’s (2000a) brief dismissal of this proposal.

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response to this argument will introduce the use of logical types in propositions about belief and knowledge, and so lead us back to a more careful examination of our three arguments so far. The ‘‘Paradox of the Preface,’’ while originally presented to different effect, can be formulated so as to involve a proposition that asserts something about the truth value of a totality of propositions including itself.⁸ Consider a modest author who is convinced, from study of his own earlier work, and the work of others, that his new book is likely to contain a mistake, a false statement. The author then announces this in the preface: ‘‘There is a false assertion in this book.’’ The mere act of making this assertion in the preface has remarkable logical impact. First, it becomes thereby true that there is a false assertion in the book, for if the author had so far avoided making one, the statement in the preface is guaranteed to be false. But in fact we guarantee that it is some other statement that must be false. For if only that one is false, then it must be true, so on pain of contradiction, there must be some other statement in the book that is false. What a powerful way to introduce falsehoods into the body of potentially error free texts! Rather than a case of a truth that cannot be believed we have a case of a belief that cannot but be true. We can symbolize the argument presenting the paradox as about belief, and having the belief that one has a false belief, using an abbreviation ‘F φ’ for ‘one falsely believes that φ’, i.e. ∼φ & Bφ: The Paradox of the Preface (for Belief) 1 1) B ∃p Fp Assumption 2 2) ∼∃p Fp Assumption 1,2 3) B ∃p Fp & ∼ ∃p Fp 1,2, Prop Logic 1,2 4) ∃q (Bq & ∼q) 3, E.G. 1,2 5) ∃q Fq Def F 1 6) ∃p Fp 2–5 reductio? 7) B ∃p Fp ⊃ ∃p Fp 1–6 ⊃ Intro 1 8) ∃p (Fp & ∼(p = ∃p Fp) ) 1,7, Various steps The connection with self-reference is introduced when one considers whether the very belief in the preface is the false one. If one observes the types of propositions, as suggested in a ramiﬁed theory of types, one will say that steps (2) and (5) involve beliefs of different types and so are not in contradiction. A mistake occurs then at step (6); there is no reductio ad absurdum because the p in step (2) is of a different logical type from the q of step (5). Now there is no explicit self-reference here, simply a violation of restrictions on types produced by allowing a proposition to include a quantiﬁer over propositions that ranges over itself. This is, however, how a system of type theory will analyze paradoxes of self-reference. ⁸ Originally from Makinson (1965), I follow the formulation of A. N. Prior (1971): p. 87.

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A simpliﬁed version of Alonzo Church’s (1976) r-types will present enough of the ramiﬁed theory of types to represent the points to be made here.⁹ If premise (1) in the Paradox of the Preface has these type assignments: B (2) (∃p 1 F (1) p 1 ) then when we get to line (4) the assignment of r-type to the generalized variable q will yield: ∃q 2 (B (2) q 2 & ∼q 2 ) hence the q in line (5) will have the r-type 2, while the p in (2) has r-type 1: ∃p 1 (B (1) p 1 & ∼p 1 ) so there is no contradiction and no reductio argument. The Fitch argument and its relatives do not involve illegitimate self-reference, but rather this distinct sort of failure to observe principles of logical types. There is more than only one sort of paradox or fallacy involved in these sorts of ‘‘paradoxes,’’ and they do not all call for the same sort of resolution. Given the range of ‘‘type free’’ solutions to the liar paradox, it has become common wisdom in philosophical logic that type oriented solutions to paradoxes are unintuitive, or violate principles of ordinary language. That doesn’t seem right for this propositional version of the Preface Paradox. My suggestion is that the same is the case for Fitch’s argument. Put another way, one can proﬁtably investigate whether the Fitch argument is valid in ‘‘Russellian’’ intensional logic, i.e. the ramiﬁed theory of types with propositions, independently of whether this is the right way to approach all such paradoxes. Provably self-referential sentences will be banned by such a logic, but that is not the only source of potential violations of principles of types. The Fitch argument does not rely on any sort of self-reference. Determining whether certain arguments are valid in intensional logics with logical types is distinct from the general issue of the appropriateness of using type theory to resolve paradoxes in philosophical logic. Something may be learned from applying different logical tools to different paradoxes. Indeed it is an important step to be able to separate this larger family of results into those that do and those that do not involve self-reference. For example, Roy Sorensen (1988) has gone to great pains to identify those ‘‘assimilators’’ who identify some of what he calls ‘‘blindspot’’ arguments with self-referential paradoxes. An important example is the Surprise Examination Paradox: the case ⁹ The notation includes expressions of the form (ι)/1 which would be the r-type of a predicative, or lowest level, functions of individuals, which are of r-type ι. A proposition of the lowest level will be (a 0-place function) of r-type ()/1. Propositions deﬁned in terms of the totality of propositions of r-type ()/1, i.e. those whose deﬁnitions involve quantiﬁers over that totality, will be of r-type ()/2. When dealing with propositions it is convenient to abbreviate the r-type ()/n simply with the integer n. Thus propositions will be of levels 1,2,3, etc. Finally, when dealing with only predicative functions one can ignore the (potentially crucial) level index, thus, for example, writing (n)/1 as simply (n). The upshot of these abbreviations is that numbers indicate the order of propositions, as Church deﬁnes that term.

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of the teacher who informs the students that there will be an exam during the next week, but that the students won’t know until the day of the exam when it is to be given. Sorensen identiﬁes yet another of his blindspots in the Surprise Exam, namely a proposition that can be true but not known, in this case the truth that there will be an exam, say, on Thursday, but that it will not be known that there will be an exam on Thursday (until Thursday). This is then a case of an assertion p & ∼Kp of the sort we have already encountered. Following a long line of commentators, including Quine, this ‘‘solution’’ to this paradox is simply to point out that the students can’t know that there will be a surprise exam, even though the teacher may announce it in class. A distinct and almost equally long line of commentators has tried to assimilate this puzzle to various paradoxes of self-reference by analyzing the teacher’s assertion that the test will be a surprise as saying something like this: ‘‘There will be a test, but you won’t be able to deduce that there will be from this assertion.’’ While this does produce a puzzle, it seems to miss the straightforward analysis of this problem as one involving blindspots, more propositions that can be true but not known.¹⁰ Indeed some formulations of the ‘‘one day’’ version of the paradox simply involve the teacher saying that there will be an exam tomorrow but it will be a surprise, and so you don’t know it that there will be an exam tomorrow. This is a straightforward use of a Fitch style ‘‘blindspot.’’ The Fitch argument doesn’t involve self-reference, any more than the Surprise exam does on my preferred formulation, even though some paradoxes in this vicinity may well do so. Thus, when formulated in Russellian intensional logic, the ﬁrst part of the Paradox of the Preface involves a violation of the theory of types distinct from an explicit self-reference.¹¹ I now turn to an investigation of whether the Fitch argument survives the imposition of types on assertions about belief. This is a radical proposal in terms of logical form by contemporary standards, if not when Church wrote, but it has a less radical result, namely that of challenging ¹⁰ Even commentators such as Williamson (2000a) who argue that there is something special about the number of days in which the exam could be given, so that at the beginning one can know that there will be an exam, agree that by the last day that is impossible. ¹¹ It appears that in his revision to the ﬁrst draft of his paper in response to Church’s ﬁrst comments, Fitch proposes introducing a principle that if one desires something then one knows that one desires it. In the ‘‘Second Referee Report,’’ page 5, Church responds: ‘‘To further my objection—that there is no law of psychology according to which it can be inferred from the fact that a knows something that therefore a desires something—Fitch replies by pointing out that a might know that a desires p. If, however, Fitch consents to adopt one of the standard devices for avoiding the epistemological paradoxes, this reply will no longer be open to him. For example, on the basis of Russell’s original theory of types, ‘a desires p’ is of higher order than p, whereas the two ‘something’ ’s in my assertion must of course be understood as of the same order.’’ Church here considers the analysis of Fitch’s suggestion in terms of type theory and points out that ‘‘knows that’’ applied to a proposition ‘‘a desires p’’ will be of a different type from ‘‘knows that’’ applied to p. The same would hold for an iterated proposition ‘‘a knows that a knows that p’’ and the proposition ‘‘a knows that p.’’

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certain principles that involve iteration of epistemic operators.¹² Proposing the type theoretic analysis challenges the standard practice, in epistemic logic, and indeed all modal logic, of innocently allowing iteration of operators with regard to syntax. From the point of view of type theory an operator such as B becomes a propositional function applied to propositions. If p is of order 1 then the proposition Bp might have type assignments as following: B (1) p 1 . The whole proposition Bp will then be of order 2. As a result the iteration BBp will involve the following types: B (2) (B (1) p 1 ) Changing the analysis of ‘B’ from an operator that can be indeﬁnitely iterated to a function that differs according to its arguments does not mean that there is any error in the use of modal logic for such notions. Rather what is presented is a different analysis of the syntax of the language used. This allows one to reconsider the validity of various principles, but does not automatically rule any of them out as ill-formed. It should be noted that analyzing some apparent operators, such as the epistemic operators ‘K’ and ‘B’ as second order functions does not require that all operators, including sentential connectives, be so treated. It is possible to analyze a negation ∼p as being the result of applying a one place function to a propositional variable: N (1) p 1 , and so see it as literally a function of propositions as I have proposed for knowledge and belief. In a formulation of type theory, there indeed will be such a function. But it need not be required as the only way of formulating a negation. The language which introduces B and K can keep ∼ whether or not it introduces N . If it does there will be provable an equivalence: N (1) p 1 ≡ ∼p, but ∼ itself can continue to be read as a connective, rather than an expression of N (1) p 1. (Indeed one might consider leaving modal operators as operators in a language where epistemic operators are read as ‘‘propositional attitudes’’ or functions.) The proposal to replace operators with functions can thus be taken selectively, on the basis of considerations that suggest that it does not innocently iterate without raising the type of the proposition to which it applies.¹³ On the proposed account, what appears as one schema, for example the principle B ∼ Bφ ⊃ ∼Bφ, will in fact have to be read as a ‘‘typically ambiguous’’ ¹² Thus Williamson (2000a) argues that the Surprise examination, and several other arguments in epistemic logic incorrectly assume either the KK principle, or some other principle that involves iteration of knowledge claims. But this he does treating K and B as an operator that can be iterated freely, at least syntactically. ¹³ C. A. Anderson has developed a Russellian intensional logic in his (1989) which would allow for negation, for example, to be treated as a connective, while propositional functions can also be represented and the equivalences such as that between N and ∼ above also proved. Despite some unclarity about just how ﬁne the classiﬁcation of types is to be, Whitehead and Russell say at ∗9 · 1 3 1 of Principia Mathematica that a negation is of the same type as the negated proposition. Other passages, however, go against this, suggesting that connectives are functions differing from ‘believes that’ only in being extensional (pp. 6, 8) and that they are ‘‘typically ambiguous,’’ because they apply to different types (p. 43).

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schema ranging over various types for the proposition φ and itself involving functions B of two different types. One instance of this will be for the case where φ is a proposition of the lowest order 1, and so the instance will be what might be called the ‘Disbelief ’ principle: B (2) ∼ B (1) p 1 ⊃ ∼ B (1) p 1 What reason might one have to reject such a principle? If the antecedent B (2) ∼ B (1) p 1 is true, but the consequent ∼ B (1) p 1 is false, then we have a false second level belief about a ﬁrst level belief. How is this coherent? The picture of belief that is suggested by using the theory of types for propositions is of a hierarchy. At the bottom are certain basic propositions. A logic of belief, like a logic of knowledge, is intended to represent not just the particular, potentially arbitrarily collected psychological states of an agent that may be identiﬁed as beliefs. Rather the logic is based on some notion of epistemic consistency (and so of a consequence relation) which is constrained by the logical properties of propositions believed. If propositions believed are distinguished by logical type, then the sort of belief appropriate to each will be distinguished by type. Belief in a basic proposition p 1 of the ﬁrst type will be reported with a second level proposition B (1) p 1 . Belief in that second order proposition, and any other such propositions about ﬁrst level propositions will be reported with a proposition of the type of B (2) p 2 . Thus above the ﬁrst order propositions are propositions that depend on that ﬁrst totality, including propositions that some of those ﬁrst order propositons are believed, then yet another third sort of propositions depending on that second totality, and so on. The failure of the ‘‘BB’’ principle is easy to understand on this model. The proposition asserting belief in a proposition of the ﬁrst level, say p 1, will be a second level proposition, B (1) p 1, and the assertion of belief in that B (2) B (1) p 1 a distinct third level proposition. No logical connection requires that one imply the others. The same holds with the ‘‘Disbelief ’’ principle. The higher level belief may be in error, but it is in no way logically suspect. Given that beliefs must be stratiﬁed into types, there will be no beliefs about all beliefs whatsoever, and any principle such as ‘‘BB’’ or the ‘‘Disbelief’’ principle, which apparently ranges over all beliefs must be seen as ‘‘typically ambiguous.’’ Each speciﬁcation for a distinct level will be a distinct proposition, any or all of which might fail. Admittedly, as an account of natural language expressions about knowledge and belief, and of the semantic paradoxes concerning the predicate ‘true,’ the theory of types may not be the most natural account. I propose that since the

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ramiﬁed theory of types is a natural way to handle the Paradox of the Preface, it may also be a promising way of getting a uniﬁed account of the more selfreferential seeming epistemic paradoxes and the ‘‘blindspot’’ or Moorean cases which involve the recurrent appearance of p & ∼Bp and p & ∼Kp. Denying that operators iterate easily won’t block the original Fitch argument that was given for knowledge. ‘‘Knows that’’ is certainly a factive operator so both: K (2) K (1) p 1 ⊃ K (1) p 1 and: K (2) ∼ K (1) p 1 ⊃ ∼K (1) p 1 surely hold. There is a different aspect of the original Fitch argument about knowledge and the two variants about belief, the Hintikka argument, and my ‘‘Fitch-B,’’ to which one might object after paying attention to types. This objection is motivated by thinking about what an agent might come to know or believe in ideal conditions. I repeat the original Fitch argument, only now with type indices: The Fitch Argument with Types 1 1) K (2) (p 1 & ∼K (1) p 1 ) Assumption (2) 1 (1) 1 (2) 1 (2) (1) 1 ∼ K p Axiom 2) K (p & ∼K p ) ⊃ K p & K 1,2, Prop Logic 1 3) K (2) p 1 & K (2) ∼ K (1) p 1 T Axiom 4) K (2) ∼ K (1) p 1 ⊃ ∼K (1) p 1 3,4, Prop Logic 1 5) K (2) p 1 & ∼K (1) p 1 1–5 reductio? 6) ∼K (2) (p 1 & ∼K (1) p 1 ) My suggestion is that we can also reject step (6) on the grounds that there is no literal contradiction at step (5) to produce a reductio argument. We need not in general accept the principle K (2) p 1 ⊃ K (1) p 1 . The antecedent should certainly be considered well formed, for we do want to express sentences like (1) and also use the distribution principle, but it is not clear that the antecedent always implies the consequent. Think of an idealized agent developing beliefs in order, in some sort of ideal epistemic conditions. There is no reason to believe that what is known at one level is also known at the next higher, or that what is known at a higher level must be known at lower levels. These inferences between knowledge claims fail as a matter of logic, and hence are open to objection on the grounds that they misrepresent the idealized notion of knowledge or belief being formalized. There is no incoherence in not knowing p at the lower level and knowing it at the next higher level. What is known is frequently a function of other beliefs and knowledge. If knowledge is distinguished by types, what is known at one type will be a function of other propositions of the same type which may be evidence or defeators for knowledge claims. Thus a blindspot at one level may be resolved when considered at a higher level, at which one

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knows p after all. Again we don’t even have to accept the validity of the Fitch argument. A similar problem arises with the Hintikka argument. The instance of the ‘‘Distribution Axiom’’ that is relevant will have type assignments as follows: B (2) (p 1 & ∼B (1) p 1 ) ⊃ B (2) p 1 & B (2) ∼ B (1)p

1

The problem is that the ﬁrst conjunct in the consequent is not a simple step of iteration away from producing a believed proposition that will contradict the second. Rather, one will need a version of the ‘‘KK’’ thesis for belief of the following sort: B (2) p 1 ⊃ B (2) B (1) p 1 While well formed, this is certainly even more contentious than its untyped version. The Fitch argument for belief, my ‘‘Fitch B,’’ faces exactly the same problem with types as the original Fitch argument. Thus, while the two arguments for belief require strong principles of iteration and ‘‘disbelief ’’ that may be seen as especially contentious when formulated with types, all three run into further problems involving types in the distribution step. The results of the foregoing discussion can illuminate the question of the logical status of veriﬁcationism, and in particular the relevance of Fitch style arguments to veriﬁcationism. In his paper ‘‘Truth as Sort of Epistemic’’ (Wright (2000)), Crispin Wright has proposed a new formulation of the veriﬁcationist view that to be a truth is to be knowable. After considering some counterfactual formulations of the view that what is true is what would be veriﬁed if tested under ideal epistemic conditions, one of which will be discussed below, Wright proposes that an internal realist adopt what he calls ‘‘Provisional Biconditionals.’’ These require a notion of ideal epistemic conditions Q, perhaps relativized to a proposition p under consideration, Qp , and a notion of what would be believed under those conditions Z (p). The biconditional proposed then is: (o∗ )Q → (p is true ↔ Z (p) ) This is modeled on Carnap’s notion of a ‘‘reduction sentence,’’ such as that which would analyze dispositional predicates such as ‘x is soluble’ with a conditional ‘If x is placed in water then x is soluble if and only if x dissolves.’¹⁴ The reduction sentence replaces the simpler conditional ‘If x is placed in water and dissolves then x is soluble’ for it makes all objects not in water be soluble.¹⁵ To leave the connection ¹⁴ Carnap (1936–7). Carnap sees reduction sentences as giving ‘‘partial deﬁnitions’’ for in cases where the antecedent test condition does not obtain, in this case x is not put in water, the sentences are silent about the term ‘‘soluble.’’ He envisages test conditions being built up to progressively approach a genuine deﬁnition. ¹⁵ The motivation for (o∗ ) in fact treats the ﬁrst conditional as counterfactual. In keeping with the spirit of the account being designed to handle the solubility of substances never put into water, we also want to know about the truth of propositions that are in fact never submitted to

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between truth and veriﬁcation open in so many cases is to retreat from the general veriﬁcationist position. Wright is forced into this by the failure of a number of stronger conditional connections, some of which will be considered below. Wright suggests that using his preferred (o∗ ) veriﬁcationism will have a response to the ﬁrst Fitch argument, as follows. An agent in an ideal epistemic situation faced with the apparent combination p & ∼Zp will have to either stick with the evidence for p and give up ∼Zp, or else stick with ∼Zp and thus drop p. An agent in an ideal situation would resolve the blindspot, or to put the point differently, for agents in ideal epistemic situations there are no blindspots. There may be some things that such an agent doesn’t know, but when the agent is put in an ideal epistemic situation with regard to that fact, the blindspot will be resolved. There is no truth that couldn’t be known by the agent in an ideal epistemic condition for that truth. The response to the Fitch argument, then, is to accept it as sound, but deny that it is an objection to veriﬁcationism. It may thus be true that I cannot know both that p and that I don’t know that p, but for the good reason that in an ideal epistemic situation I would abandon one or the other conjunct. Granted then, this is something that could be true but not known by me (at least), but that is so only because in ideal circumstances it would not be true. One use of the account of iterated belief that I have discussed is to make Wright’s proposal ﬁt easily with a logic of belief. We can treat the operator B in Fitch-B as expressing what would be believed by an agent in an ideal epistemic situation, and show then how the argument is blocked. Beliefs of the ﬁrst level, propositions of type p 1 that are believed, as expressed by propositions B (1) p 1 , will be beliefs about ordinary, non-epistemic, states of affairs. There will also be beliefs about such beliefs, expressed by B (2) (B (1) p 1 ). One interpretation of the operator B is to describe an idealized belief, what a believer after sorting out logical consequences and unhampered by limitations of memory and concentration might be said to at least tacitly believe. Might one not also build into this notion of an ideal believer some feature of being a believer in an ideal epistemic situation? Then we can easily represent the general idea of Wright’s response to Fitch arguments as holding that they are cases where we have more than the usual reason to reject both the ‘‘BB’’ principle and the Disbelief principle. On this account the second level belief operator will describe those beliefs that an ideal agent, or agent in an ideal epistemic situation, can have in the logical position of also holding certain ﬁrst level propositions and beliefs about those propositions. That may differ from what beliefs the agent might be able to have when considering simply the ﬁrst order propositions. There is no reason to hold, as a matter of logic, that what is believed as expressed by ‘B (1) ’ is therefore believed in the way expressed by ‘B (2) ’, or vice versa. veriﬁcation procedures, but which might have been. Using a counterfactual conditional does not make a difference to the sort of objections being considered here.

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A counterexample to Hintikka’s BB thesis might consist of the following situation; B (1) p 1 at the ﬁrst ‘‘stage’’, but then ∼B (2) B (1) p 1 at the second due to some different surrounding beliefs of the same type, indeed, at this stage p 1 is not believed, i.e., ∼B (2) p 1 . The second level beliefs will be either a reﬁnement and improvement or simply an extension of the ﬁrst level beliefs. Similarly for the ‘‘Disbelief ’’ principle. We may well have a situation of revised beliefs where B (2) ∼ B (1) p 1 at the second level, while B (1) p 1 yet, for some reason again, ∼B (2) p 1 . The ﬁrst belief is indeed false, but understandable, especially given the last. Thus not only can one argue that the conclusion of the Fitch-B argument or the Hintikka version is not a problem for veriﬁcationism, one might actually object that these arguments are invalid. On my account a belief might be missed or dropped when moving up a type, precisely because it is part of a blindspot. A belief might be added, as it is reconciled with a larger group of beliefs, including some beliefs about beliefs. At each type there may well be logical connections between beliefs, as expressed with the distribution and conjunction axioms, but all principles involving iteration of belief are up for grabs, and most likely false in certain instances. This notion of resolving blindspots in ideal conditions helps to answer a challenge from T. Williamson ((2000a): 281) presented in his account of the Fitch arguments. Williamson asks how it could be that K (2) p yet not K (1) p. He suggests that this could only be through a convoluted path. ‘‘Perhaps a claim could be known at level i + 1 but not at level i if the route to knowing it involved claims about knowledgei , even though the target claim did not, but it would be bizarre if such contrived cases were crucial to a defence of weak veriﬁcationism.’’ I think that Wright’s notion of resolving blindspots in ideal circumstances presents just such a justiﬁcation. Williamson points out that the ‘‘canonical’’ veriﬁcation of some p would result in ﬁrst level knowledge, K (1) p, while the veriﬁcation of a conjunction K (1) q & p would yield K (2) (K (1) q & p). Deriving that latter proposition by conjoining knowledge claims would require ﬁrst establishing K (2) p rather than K (1) p. It is true then, as he observes, that the canonical way of verifying a certain conjunction would not be by means of canonical veriﬁcations of the conjuncts. A sentence conjoined with a statement about knowledge or belief would have to be veriﬁed in a different way than if it were confronted in isolation. I leave it to veriﬁcationists to determine amongst themselves whether this is out of keeping with the spirit of their program. Williamson has presented another objection to this ‘‘type’’ response to the Fitch argument.¹⁶ The type of an operator like K , he points out, is determined by the content of the propositions to which it applies, i.e. their logical type. ¹⁶ In discussion at the conference on ‘‘The Limits of Warrant’’ held at the University of Waterloo in May of 2001 from which this paper derives.

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The justiﬁcation for the reply to the Fitch argument, however, seems to rely on claims about the procedure employed by an epistemic agent in resolving potential ‘‘blindspots.’’ The objection seems to be that one can’t make claims about the logical types of propositions which rely on a speculative account of how some agent might come to have certain attitudes towards those propositions. My response is that the use of epistemic logic requires a certain amount of idealization in its application. We must decide what idealized epistemic agents are to be represented with a given formalism. In the application I propose, we consider agents that observe type distinctions in their beliefs, i.e. epistemic states are determined in ‘‘stages,’’ whether temporal or not, in which beliefs about beliefs depend on beliefs of lower order. If one application of epistemic logic is to model certain sorts of epistemic agents, this seems to be a legitimate sort of such modeling. Propositions are indeed assigned to types by their contents, but attitudes towards them will depend on each other in ways that do reﬂect a procedure for determining epistemic states. Not every general argument of this sort against every form of veriﬁcationism makes use of iterated operators, however. There are other forms of veriﬁcationism that fall prey to other objections and which just look as though they involve self-reference or at least some sort of variation of type, but in fact do not. Consider a version of the notion that what is true is what would be conﬁrmed under ideal conditions Q, which Wright also discusses: (o) p is true ↔ (Q → Z (p) ) Some time ago Alvin Plantinga (1982) gave an argument against this proposal. Plantinga’s argument is that if one assumes (o) then it will be impossible for some p to obtain without Z (p), in other words, the notion of truth and actual veriﬁcation collapse, provided that one identiﬁes truth with veriﬁcation under ideal conditions. Here is yet another formulation of this argument: The Plantinga Argument 1 1) p is true ↔ (Q → Z (p) ) Assumed account of truth 1 2) Q is true ↔ (Q → Z (Q ) ) 1, Instantiation 1 3) (Q is true ↔ (Q → Z (Q ) )) 2, Necessitation 4 4) ♦(Q & ∼Z (Q ) ) Assumption 4 5) ♦(Q is true & Q & ∼Z (Q ) ) 4, (A is true ↔ A) 4 6) ♦(Q is true & ∼(Q→ Z (Q ) ) 5, counterfactual logic 1 7) (Q → Z (Q ) ) 4–6, reductio 1 8) (Q→ Z (Q ) ) 7, counterfactual logic 1 9) (Q is true) 8, 3, (S5) modal logic 1 10) Q 9, RE, (A is true ↔ A) 1 11) p is true ↔ Z (p) 10, 1, counterfactual logic The course of the argument is as follows: It is ﬁrst shown that it is necessary that if Q obtains that it is veriﬁed, Z(Q), for if at some world the ﬁrst were true

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but not the second, Q would be true and the counterfactual connecting its truth and its veriﬁability (step 2) would be false. But if Q necessitates its veriﬁcation the assumed account of truth yields the conclusion that Q is necessarily true. But then Q occurs vacuously in the ﬁrst conditional (1) and so the account collapses into a straightforward identiﬁcation of truth with veriﬁcation. The earlier tactic of reformulating operators as properties of propositions and thus expressing the counterfactual conditional as a two place relation between propositions does not reveal any of the type differences that occur in the Fitch argument. If one treats the Lewis or Stalnaker possible world semantics for counterfactuals as an analysis of a relation between propositions, and thinks of propositions as sets of worlds, that relation between the antecedent of a counterfactual and its consequent will be of a high order, involving considerable quantiﬁcation over worlds and classes of worlds, etc. As well, possibility may either be thought of in terms of quantiﬁcation over worlds and those as classes of propositions, or just directly as a property of propositions. Fixing on precise type indices for and → will not be easy. What is crucial for the analysis I have proposed for the Fitch argument is the issue of whether those predicates and relations are applied to propositions of different type levels in the course of the argument and whether, when seen that way, the inferences remain justiﬁed. There are such changes of level, as in the move from (7) to (8), where necessity is ﬁrst applied to a relatively low level material conditional and then to a much higher level counterfactual conditional. But these moves do not threaten the validity of the argument as one might argue the shift of levels of belief does affect the validity of the Fitch argument. There might seem to be a whiff of self-reference to the Plantinga argument (although neither Plantinga nor Wright suggest there is any real self-reference here). The issue is rather that if one thinks that truth coincides with what would be veriﬁed in ideal conditions, then turning that analysis on the ideal conditions themselves and wondering whether one can verify that they obtain (in ideal conditions) seems to require some reﬂexive veriﬁcation. But it doesn’t actually do so. All that is required is universal instantiation to apply the condition in (1) to those ideal conditions Q. (1) strictly speaking ought not to be called a deﬁnition. Although p appears on both sides of the biconditional, the notion of truth appears only on the left. However the argument does assume that the Tarski style equivalence (A is true ↔ A) is not only true, but necessarily true, so that ‘A’ and ‘A is true’ are readily interchangable, so (1) comes very close to being circular. Rather, (1) should be called simply an axiom or analytic truth. If it is in order logically then so is the instance that applies to Q. That is as close as this argument comes to involving self-reference. A version of the above argument will still work when operators are treated as higher order predicates and type restrictions are observed. This is not all that one might say about the Plantinga argument from the perspective of type theory, however. Part of the motivating conception of

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type theory is the notion that certain totalities are ‘‘illegitimate.’’¹⁷ The totality of propositions must be seen as only a ‘‘potential’’ inﬁnite, approached by a series of increasingly more inclusive classes of propositions, but never completed. It is not possible to have propositions about all types. But then the notion of a settled state of affairs in which all propositions can be evaluated, whether seen as being ideal conditions for verifying any statement, or as the ‘‘end of scientiﬁc enquiry,’’ does not ﬁt with this perspective. In speciﬁc, then, the proposition Q above will be limited to a speciﬁc type. What type will that be? While formally the argument allows it to be of a low type, it might seem that a proposition describing a situation in which any proposition can be veriﬁed would have to be of a higher type, and hence even impossible because it would have to have an arbitrarily high type. However this may be, the argument, at least for a speciﬁc order of Q, is clear of the sort of type violations considered earlier. As another potential formulation of a knowability thesis for the veriﬁcationist, Wright (2000) considers that a defender of (o) might say that the ideal conditions need not be the same for every proposition, especially when that proposition can be the description of those very ideal conditions. Rather the ideal conditions may vary according to the proposition under consideration, thus for p the ideal conditions will be Qp : (o) p is true ↔ (Qp → Z (p) ) This proposal runs afoul of one of the sort of ‘‘conditional fallacies’’ that Robert Shope (1978) describes. What if it is possible that the obtaining of Qp itself inﬂuences the obtaining of p counterfactually, say Qp →∼ p? Then we could have Qp , Z (p) and ∼ p true in the same possible world if Qp → Z (p) obtains as well (and p is true in the actual world). The ‘‘fallacy’’ is in asserting an ‘‘analysis’’ like (o) when a counterfactual like Qp →∼ p might be true. Again all of this does not require any use of iterated operators or predicates, and so is not subject to the same response as was the Fitch argument and its relatives. This survey of our family of related arguments suggests the following conclusions. The ‘‘Fitch style’’ arguments, both for knowledge and belief, which make use of principles about iterated operators such as BB and Disbelief, may be faulted on the truth of those principles, but not for some sort of illicit self reference in the very iteration. They are not arguments that ‘‘this very truth’’ cannot be known. The Fitch argument for knowledge, can, however, be faulted for a violation of the principles of logical types, though not an explicit self reference. Arguments against accounts that use the notion of ideal epistemic conditions which question how we might know whether such conditions obtain, such as Plantinga’s, also do not make illicit use of self reference or other violation of the theory of types. The same seems to hold about qualms about counterfactual analyses of truth ¹⁷ See Whitehead and Russell, Principia Mathematica, p. 37, and the discussion of the vicious circle principle.

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in terms of veriﬁcation where the truth of a counterfactual hypothesis interferes with veriﬁcation conditions. Church was right to notice that his ‘‘Fitch Argument’’ was ‘‘suggestive of the liar and other epistemological paradoxes.’’ An overview of various epistemological arguments does reveal similarities, but also differences. Some arguments against veriﬁcationism can be met by denying their premises, others rely on dubious principles of epistemic logic, while another seems to violate principles of type theory. Other arguments, however, cannot be so easily dismissed by the veriﬁcationist.

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Pa r t V C a r t e s i a n Re s t r i c t e d Tr u t h

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12 Tennant’s Troubles Timothy Williamson

First, some reminiscences. In the years 1973–80, when I was an undergraduate and then graduate student at Oxford, Michael Dummett’s formidable and creative philosophical presence made his arguments impossible to ignore. In consequence, one pole of discussion was always a form of anti-realism. It endorsed something like the replacement of truth-conditional semantics by veriﬁcation-conditional semantics and of classical logic by intuitionistic logic, and the principle that all truths are knowable. It did not endorse the principle that all truths are known. Nor did it mention the now celebrated argument, ﬁrst published by Frederic Fitch (1963), that if all truths are knowable then all truths are known. Even in 1970s Oxford, intuitionistic anti-realism was a strictly minority view, but many others regarded it as a live theoretical option in a way that now seems very distant. As the extreme veriﬁcationist commitments of the view have combined with accumulating decades of failure to reply convincingly to criticisms of the arguments in its favour or to carry out the programme of generalizing intuitionistic semantics for mathematics to empirical discourse, even in toy examples, the impression has been conﬁrmed of one more clever, implausible philosophical idea that did not work out, although here and there old believers still keep the ﬂame alight. A diffuse philosophical tendency cannot be refuted once and for all by a single rigorous argument. Nevertheless, such an argument can severely constrain the forms in which the tendency is expressed. The tendency labelled ‘anti-realism’ and Fitch’s argument together constitute a case in point. My ﬁrst publication (1982) was a response to the Fitch argument. I argued that it was intuitionistically invalid, and therefore did not show intuitionistic anti-realism to be committed to the absurd claim that all truths are known. Naturally, my aim was not to endorse intuitionistic anti-realism; I found it as deeply implausible then as I do now. But that does not distinguish it from other forms of anti-realism, and such Thanks to Peter Milne and Peter Pagin for comments on a draft of Williamson (2000b), which have in turn beneﬁted the present paper.

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dispositions are not invariant across persons. My aim was rather to assess what forms of anti-realism must be argued against in some way beyond Fitch’s. The advantage in ﬁnal plausibility of those other forms of anti-realism over the brazen assertion that all truths are known is tenuous at best: but it is still worth getting clear about the logical situation. Some of my later work on the Fitch argument (1988b, 1992, 1994a) reﬁned the envisaged response to the Fitch argument and established its formal stability. In The Taming of the True (1997), Neil Tennant objects to the speciﬁc intutionistic anti-realist response to Fitch that I had envisaged, and proposes his own alternative responses, still of a broadly intuitionistic anti-realist kind. In response (2000b), I argued that both Tennant’s objections and his alternatives fail, and that the result illustrates a more general point: that moderate forms of anti-realism tend to be the least stable. Tennant replied at length (2001a). For some time I thought that the problems with his 2001 reply were sufﬁciently evident to make any further response from me unnecessary. Later experience has taught me otherwise. The purpose of this paper is to show that Tennant’s reply fails completely to meet the difﬁculties that I raised in 2000. Since his reply engages with many details of that paper (2000b), while missing the relevance of some of the most crucial ones, the most efﬁcient course is to rehearse the arguments of that paper, interspersing them with discussion of Tennant’s objections as they arise. Thus the present paper constitutes a self-standing critique of Tennant’s treatment of the Fitch paradox that properly includes its predecessor.¹

I The ﬁrst task is to expound the Fitch argument in a form suitable for the subsequent discussion. Anti-realists argue that truth is epistemically constrained. Their arguments are too complex, elusive and at least vaguely familiar to formulate here. We can gesture at them thus: a sentence s as uttered in some context expresses the content that P only if the link between s and the condition that P is made by the way speakers of the language use s; their use must be sensitive to whether the condition that P obtains; that requires of them the capacity in principle to recognize that it obtains, when it does so; thus P only if speakers of the language can in principle recognize that P. In brief: all truths are knowable. We can formalize the anti-realist conclusion in a schema: (1) ϕ → ♦Kϕ ¹ Where appropriate, sentences or paragraphs from Williamson (2000b) have been absorbed into the present text.

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Here ♦ and K abbreviate ‘it is possible that’ and ‘someone sometime knows that’ respectively; ϕ is to be replaced by declarative sentences.² Presumably, the relevant sense of ‘possible’ is not merely epistemic, because the anti-realist takes sensitivity to whether a condition obtains to require a genuine recognitional capacity (a metaphysical possibility of knowing), not a mere incapacity to recognize one’s ignorance (an epistemic possibility of knowing). According to (1), the truth really could have been known. Much in the anti-realist arguments deserves to be questioned. Fitch (1963) introduced a direct objection to their conclusion with an apparent reductio ad absurdum of (1). It requires two highly plausible principles about knowledge: only truths are known, and known conjunctions have known conjuncts. More formally: (2) Kϕ → ϕ (3) K(ϕ ∧ ψ) → (Kϕ ∧ Kψ)³ Principles (2) and (3) jointly entail that nothing is ever known to be an always unknown truth. For (2) yields K(ϕ ∧ ¬Kϕ ) → (ϕ ∧ ¬Kϕ ) and therefore K(ϕ ∧ ¬Kϕ ) → ¬Kϕ, while (3) yields K(ϕ ∧ ¬Kϕ ) → (Kϕ ∧ K¬Kϕ ) and therefore K(ϕ ∧ ¬Kϕ ) → Kϕ, so together they give: (4) ¬K(ϕ ∧ ¬Kϕ ) Principles (2) and (3) are intended as necessary constraints on knowledge, and the propositional logic used to derive (4) from (2) and (3) is necessarily truthpreserving, so by a variant of the rule of necessitation in modal logic we can conclude that what (4) says is not could not have been: (5) ¬♦K(ϕ ∧ ¬Kϕ ) Now consider the special case of (1) with ϕ ∧ ¬Kϕ in place of ϕ: (6) (ϕ ∧ ¬Kϕ ) → ♦K(ϕ ∧ ¬Kϕ ) By (5) and (6): (7) ¬(ϕ ∧ ¬Kϕ ) In classical logic, (7) is equivalent to: (8) ϕ → Kϕ ² Tennant brieﬂy considers other possible readings of K. He complains (1997: 270) that ‘Williamson [ . . . ] appears not to have anticipated the possibility’ of interpreting K in Fitch’s arguments as ‘it is known at t that’ for a particular time t. He has overlooked the discussions of such readings of the argument at Williamson 1982: 204, 1988b: 425–8 and 1994a: 141, 144. It is unnecessary to add to them here. ³ For difﬁculties facing the attempt to evade Fitch’s argument by rejecting (3) see Williamson 1993 and 2000a: 275–85.

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Of course, (8) is deeply implausible. It says in effect that any truth is known. As one instance, it says that there is a fragment of Roman pottery at a certain spot only if someone sometime knows that there is a fragment of Roman pottery there. The corresponding instance of (1) for the same value of ϕ is quite plausible: there is a fragment of Roman pottery there only if it could have been known by someone sometime that there was a fragment of Roman pottery there. But, according to (8), there is a fragment of Roman pottery there only if the possibility of knowing is actualized; that claim is quite unwarranted. Although, by (4), the attempt knowingly to identify a particular example of an unknown truth would be self-defeating, we surely have ample evidence of a less direct sort that not every truth is ever known. Although some believe that an omniscient god makes (8) true, that issue is not very pertinent here. For if we restrict the substitutions for ϕ to sentences of our language, the anti-realist motivation for (1) allows us to read ‘someone’ in the deﬁnition of K as ‘some member of our speech community’; the links between sentences of English and their contents are made by human speakers of English without divine intervention. ‘There is a fragment of Roman pottery at that spot’ is a sentence of our language; surely not every truth expressible in our language will ever be known by some member of our speech community. If you think it matters, give K that restricted reading. We should thus reject schema (8). On the assumption that (8) was derived from (1) using uncontentious principles, we should therefore reject schema (1) too. The seminal presentation of the case for (1) is Michael Dummett’s (1959b, 1975 and elsewhere). Notoriously, he integrates it with a case for a comprehensive anti-realist reconception of meaning in terms of veriﬁcation-conditions rather than truth-conditions, which, he argues, will invalidate classical logic, in particular the law of excluded middle, and justify its replacement by something like intuitionistic logic. The latter was originally proposed as the logic of intuitionistic mathematics, and its intended semantics reﬂect that role, being formulated in terms of the notion of proof. Since the mathematical notion of proof is inappropriate to empirical statements, Dummett envisages a generalized intuitionistic semantics in which a broader notion of veriﬁcation plays the key role.⁴ Even granted that his arguments are not compelling, it is pertinent to ⁴ It is sometimes claimed that one can meet Dummett’s demand simply by treating the notion of truth in a truth-conditional compositional semantics for empirical discourse as veriﬁability. That is a mistake. The key notion in the intuitionistic compositional semantics for mathematical language is ‘ is a proof of ϕ’, not ‘ϕ is provable’ (‘Something is a proof of ϕ’); for example, the semantic clause for → concerns the transformability of proofs of the antecedent into proofs of the consequent, which makes no sense in terms of an undifferentiated notion of provability. Thus the key notion in an analogous veriﬁcation-conditional compositional semantics for empirical discourse is ‘ is a veriﬁcation of ϕ’, not ‘ϕ is veriﬁable’. The truth-conditional clause for negation, ‘¬ϕ is true if and only if ϕ is not true’ (or something similar), cannot be interpreted as a constraint on veriﬁcations, for if something is not a veriﬁcation of ϕ, it does not follow that it is a veriﬁcation of ¬ϕ. The idea of Dummett’s original argument is that the key notion in a compositional semantics should be one

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ask how the attempted reductio ad absurdum of (1) fares under intuitionistic logic, which we may provisionally suppose the anti-realist defender of (1) to have adopted.⁵ The argument from (1) to (7) is intuitionistically acceptable, but the step from (7) to (8) is not. Intuitionistically, (7) is equivalent to: (9) ¬Kϕ → ¬ϕ Intuitionistically, we cannot reach (8) from (9), deleting the negations.⁶ Intuitionists can consistently accept (9) while denying that all truths are known. Since (7) generalizes, they must deny that there is an unknown truth: on their constructivist understanding of the existential quantiﬁer, one could in any case verify that one could never verify that existential claim, because (by (4)) one could never verify a particular instance of it. Intuitionistically, to verify that one could never verify ψ is to verify ¬ψ. Denying that there is an unknown truth does not commit one intuitionistically to asserting that all truths are known. Given (9), the intuitionist can consistently deny the universal generalization of (8) but cannot consistently deny any particular instance of (8), for (9) is intuitionistically equivalent to the double negation of (8). In this respect, the intuitionistic status of ϕ → Kϕ given (9) is exactly like that of the law of excluded middle. For the intuitionist can consistently deny the universal generalization of ϕ ∨ ¬ϕ but cannot consistently deny any particular instance of it, because ¬¬(ϕ ∨ ¬ϕ ) is to which speakers’ use is sensitive, and that their use is sensitive to a given condition in a given context only if they can decide in that context whether it obtains. On such a view, they can decide in a given context whether they have a proof or veriﬁcation of ϕ in that context (for an argument against this decidability claim, see Williamson 2000a: 110–13), but no notion of provability or veriﬁability that would constitute a not wildly subjectivist notion of truth is decidable within the limitations of every given speech context. Thus no notion of provability or veriﬁability that would constitute a not wildly subjectivist notion of truth meets Dummett’s constraints on the key notion in a compositional semantics, even if it is the existential generalization of a notion of proof or veriﬁcation that does meet Dummett’s constraints. ⁵ Dummett’s own response to Fitch (2001) does not appeal to intuitionistic logic; rather, it restricts the knowability principle (1) to atomic sentences. This restriction is hard to reconcile with Dummett’s original motivation for the knowability principle, a motivation that applies to complex sentences just as much as to atomic ones. It will not do to say that the use of complex sentences is indirectly epistemically grounded because their atomic constituents are. For connectives such as conjunction and negation are used as constituents of complex sentences, not by themselves. Thus any epistemic grounding of the use of connectives must derive from an epistemic grounding of complex sentences in which they occur, not vice versa: yet Dummett’s strategy against Fitch is just to avoid any such direct epistemic grounding of the use of complex sentences. Thus his anti-realism unravels. Note also that his original (1959b) examples of sentences that the realist contentiously treated as veriﬁcation-transcendent involved complex constructions such as universal quantiﬁcation and the counterfactual conditional: ‘A city will never be built on this spot’ and ‘If Jones had encountered danger, he would have acted bravely’ are not atomic sentences. See Brogaard and Salerno (2002) (some points of which are anticipated at Williamson 1990: 300) and Tennant (2002) for criticism of Dummett’s response to Fitch and Rosenkranz (2004) for more discussion. ⁶ Williamson (1992) proves model-theoretically that schema (8) is not derivable from schemas (1)–(7) and (9) even in a very strong system of intuitionistic modal epistemic logic.

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intuitionistically valid.⁷ Call an anti-realist position on which (7) is valid and (8) invalid moderately hard anti-realism. It is opposed to very hard anti-realism, on which both (7) and (8) are valid, and soft anti-realism, on which both (7) and (8) are invalid.⁸ A consistent moderately hard anti-realist view can be worked out in some detail within the framework of intuitionistic logic (Williamson 1982, 1988b, 1992). Moderately hard anti-realists may regard the Fitch argument as a further reason for anti-realists in general to adopt intuitionistic rather than classical logic, since their response to the challenge depends on a distinction available within intuitionistic but not classical logic—although of course this was not Dummett’s reason for proposing intuitionistic logic as an appropriate logic for anti-realism. Obviously, (9) itself is a deeply problematic consequence of (1). According to (9), what is never known is not true: thus if no one ever knows that there is a fragment of Roman pottery at a certain spot, there is no fragment of Roman pottery there. That sounds as bad as (8). But there are differences. Schema (8) eliminates the logical distinction between ϕ and Kϕ, for (8) and its uncontentious converse (2) jointly yield ϕ ↔ Kϕ, and therefore (ϕ ) ↔ (Kϕ ) for any sentential context ( ) deﬁned with the standard intuitionistic connectives (in particular, of course, ¬ϕ ↔ ¬Kϕ ). Although (9) eliminates the logical distinction between ¬ϕ and ¬Kϕ, since (9) and (2) jointly yield ¬ϕ ↔ ¬Kϕ, and therefore (¬ϕ ) ↔ (¬Kϕ ), it does not eliminate the logical distinction between ϕ and Kϕ, as the underivability of (8) from (9) shows. The moderately hard anti-realist loses fewer distinctions than does the very hard anti-realist. Indeed, the former can consistently deny the universal generalization of (8), while the latter must assert it. Thus the moderately hard anti-realist, unlike the very hard anti-realist, can assert a gap between what is true and what is ever known. For reasons not peculiar to anti-realism, the acknowledgement of the gap is essentially general; it cannot be made at the level of an individual sentence, because one cannot knowingly present a speciﬁc instance of a never known truth. But how can the moderately hard anti-realist mitigate the implausibility of particular instances of (9)? If one knew that human life was about to be eliminated by a huge meteorite, might one not be entitled to assert that no one will ever know that there is a fragment of Roman pottery at that spot without being entitled to ⁷ See Williamson (1982: 206). Tennant (1997: 267–8) objected that ¬(ϕ → Kϕ) is not intuitionistically inconsistent given (1) (and the other principles used to derive (9)) unless it intuitionistically implies both ϕ and ¬Kϕ, and that, intuitionistically, although it implies ¬Kϕ and ¬¬ϕ it does not in general imply ϕ (it does in the special case when ϕ is decidable, but then ϕ ∨ ¬ϕ holds and the analogy is not useful). The objection rests on an error. Intuitionistically, ¬Kϕ and ¬¬ϕ imply ¬(¬Kϕ → ¬ϕ), the negation of (9); thus ¬(ϕ → Kϕ) is intuitionistically inconsistent given (1) (and the other principles used to derive (9)), although (without those principles) it does not intuitionistically imply ϕ. Tennant (2001a: 277–9) concedes and ampliﬁes this criticism of his objection. ⁸ This adapts Tennant’s terminology (1997: 261).

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assert that there is no fragment of Roman pottery there?⁹ Such examples suggest that the intuitionistic operator ¬ does not correctly formalize the negation that we apply to empirical sentences, such as those of the form Kϕ. Dummett himself distinguishes the negation of intuitionistic mathematics from empirical negation (1977: 337). That is not to say that intuitionistic negation cannot meaningfully be applied to empirical sentences; rather, another negation operator may be needed as well to interpret ‘not’ in empirical discourse. Empirical negation must itself behave non-classically if moderately hard anti-realism is not to collapse into very hard anti-realism, for otherwise empirical negation could be substituted for ¬ throughout the derivation of (8), but it must not behave non-classically in exactly the same ways as ¬, otherwise it would offer no advantage. Major difﬁculties face the attempt to add such an operator (Williamson 1994a). In what follows, I continue to assume that the anti-realist employs intuitionistic negation; I do not seek to minimize the associated problems. Moderately hard anti-realism remains a deeply problematic position. Nevertheless, it avoids the most drastic consequences of very hard anti-realism, and a full critique of it will not simply cite Fitch. Of those engaged in the reﬁnement of Dummett’s programme and the attempted generalization of intuitionistic semantics to empirical discourse, one of the most active has been Neil Tennant (1987, 1997). In his 1997 analysis of the Fitch problem, he argues that the envisaged moderately hard anti-realist line is unstable in the crucial test cases for the Fitch argument, and suggests an alternative soft anti-realist strategy based on restricting the knowability principle (1) (1997: 245–79).¹⁰ Section II below shows that his objection to the envisaged moderately hard anti-realist line is fallacious, and that his later attempt to defend his objection merely changes the subject. Section III shows that under rather general conditions his restricted version of (1) still implies (7) and (9), so that his would-be soft anti-realism collapses into the moderately or very hard view; his later attempt to defend his restricted version of (1) results in its trivialization.

II Tennant objects to the moderately hard anti-realist treatment of the Fitch argument that I described that it is unstable, because in what I presented as ⁹ For related points see Percival (1990). Other relevant discussions of Fitch’s argument in the context of intuitionistic logic include Wright (1993: 427–30), Cozzo (1994), Pagin (1994), Usberti (1995: 65–6, 121–8) and DeVidi and Solomon (2001). ¹⁰ Tennant (1997: 276–8) also rejects the attempt in Edgington (1985) to reconstrue the knowability principle by means of something like an ‘actually’ operator, for reasons given in Williamson (1987, 2000a: 290–301) and Wright (1993: 426–32). See also Percival (1991). Edgington’s idea is developed rigorously by Rabinowicz and Segerberg (1994), Lindström (1997) and Rückert (2004), but none of these papers fully answers the philosophical objections.

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crucial test cases the distinction between (7) and (8) collapses (1997: 268). He correctly points out that in the examples I use to illustrate the argument, I instantiate ‘ϕ’ in (7) and (8) with a decidable sentence: we have a decision procedure whose application would result in a veriﬁcation or falsiﬁcation of ϕ (‘There is a fragment of Roman pottery at that spot’). This is the simplest and most vivid form of the problem that Fitch raises, and a crucial test for any adequate treatment. Tennant also correctly points out that (7) is intuitionistically equivalent to (8) when Kϕ is decidable.¹¹ He then attempts to argue that Kϕ is decidable if and only if ϕ is decidable. If he is right, the envisaged moderately hard anti-realist line fails the crucial test. Before proceeding, I note an obvious condition of adequacy on Tennant’s critique. On pain of irrelevance, it must not depend on a reading of the operator ‘K’ other than that intended by the object of his criticism. It is to be read as ‘someone sometime knows that’, where ‘knows’ itself is understood in its predominant ordinary sense. In this sense, I may fail to know whether a given very large natural number (as presented by a corresponding numeral) is prime, even though I have a decision procedure for ﬁnding out. Of course, I know many things that I am not currently thinking about; my knowledge is stored. But I do not know something merely in virtue of its being routine for me to ﬁnd out, if I do not in fact ﬁnd out, or merely in virtue of its being a logical consequence of other things that I know. This is the understanding of ‘know’ that is evidently in play throughout my presentations of the moderately hard anti-realist line (1982, 1988b, 2000) and in most other discussion of Fitch’s argument. It is a very natural understanding in the context of that argument, which concerns the relation between potential and actual knowledge, a contrast obscured if ‘K’ is itself understood as meaning something potential. In order to preserve the relevance of Tennant’s critique, I will therefore take it for the time being in terms of the usual reading of ‘K’. Another reading will be considered later. Tennant’s objection is sustained if he can demonstrate that Kϕ is decidable whenever ϕ is; the converse does not concern us here. I will argue that his supposed demonstration that the decidability of ϕ implies that of Kϕ is fallacious. Here it is: Suppose that ϕ is decidable. Then here is a decision method for Kϕ: apply the given decision method for ϕ. If you thereby determine that ϕ is true, then you know that ϕ. So you have determined that Kϕ is true. If, on the other hand, you determine that ϕ is false, then you have determined that Kϕ is false, because no one could ever know a falsehood. So if ϕ is decidable, then so is Kϕ. (1997: 262) ¹¹ Tennant claims that ‘the validity of ¬(ϕ ∧ ¬Kϕ) guarantees the validity of ϕ → Kϕ if, but only if, Kϕ is decidable’ (2001a: 265). The ‘only if’ direction, which is inessential to his argument, is an error. The validity of ¬¬Kϕ → Kϕ is easily seen to be sufﬁcient for the validity of ¬(ϕ ∧ ¬Kϕ) to guarantee the validity of ϕ → Kϕ but is intuitionistically a weaker condition than the decidability of Kϕ.

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Our possession of a decision procedure for Kϕ entitles us to assert:¹² (10) Kϕ ∨ ¬Kϕ Intuitionistically, (7) and (10) jointly entail (8). If Tennant is right, the difference between moderately and very hard anti-realism disappears in paradigms of the cases where it was supposed to help. Why does Tennant suppose that our possession of a decision procedure for Kϕ entitles us to assert (10)? He is relying on a principle about the assertibility of disjunctions: (DIS) Our possession of a method whose application will either verify ϕ or verify ψ entitles us to assert ϕ ∨ ψ. (DIS) allows us to assert the disjunction in advance of actually applying the method.¹³ Since the application of Tennant’s decision procedure will supposedly either verify Kϕ or verify ¬Kϕ, by (DIS) our mere possession of the decision procedure, in advance of actually applying it, entitles us to assert Kϕ ∨ ¬Kϕ. At ﬁrst sight, (DIS) looks very plausible. It is surely correct when ϕ and ψ are mathematical. Nevertheless, we can easily see that Tennant’s argument must be unsound on the present reading of ‘K’. For if it were sound, there would also be a sound argument for a much stronger conclusion, namely, that our possession of a decision procedure for ϕ entitles us to assert this: (11) Kϕ ∨ K¬ϕ We can argue for this conclusion in Tennant’s style: Suppose that ϕ is decidable. Then apply the given decision method for ϕ. If you thereby determine that ϕ is true, then you know that ϕ. So you have determined that Kϕ is true. If, on the other hand, you determine that ϕ is false, then you know that ¬ϕ. So you have determined that K¬ϕ is true.

The reasoning for the case where ϕ is true is in Tennant’s own words; the reasoning for the case where ϕ is false in effect merely substitutes ¬ϕ for ϕ throughout that reasoning. By (DIS), we can conclude that our mere possession of the decision procedure for ϕ entitles us to assert (11). But it is utterly implausible to claim that whenever ϕ is decidable, someone sometime will know whether ϕ holds, in the usual sense of ‘know’. Mere possession of the decision procedure does not entitle us to assert that anyone will ever have that knowledge. For in advance of applying the procedure, we may have no reason to think that it will ever be applied; indeed, we may have reason to think that it will never be applied. ¹² Tennant (1997: 268) is explicit that if ϕ is decidable then ϕ ∨ ¬ϕ is intuitionistically acceptable. ¹³ The phrase ‘possession of a method whose application will . . . ’ is to be read as implying ‘recognition that application of the method will . . . ’.

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Perhaps its application is costly in time and other scarce cognitive resources and ϕ is a proposition whose truth-value is unlikely ever to be of interest to anyone. Moreover, it may be unlikely that anyone will ever come to know whether ϕ holds without applying such a decision procedure. Alternatively, we may know that the meteorite is about to strike. Let us provisionally accept the Tennant-style argument for the proposition that if ϕ is decidable then we have a method whose application will either verify Kϕ or verify K¬ϕ. Nevertheless, our mere possession of that method does not entitle us to assert Kϕ ∨ K¬ϕ. Thus the problem lies with (DIS). What has gone wrong is that the application of the decision procedure for ϕ brings about the state of affairs expressed by (11). That is why our mere possession of the method is not enough. To assert (11), we need some reason to think that someone sometime will apply the method. Exactly the same problem affects Tennant’s assumption that we are entitled to assert (10) whenever ϕ is decidable. For, if ϕ is true, the application of the decision procedure for ϕ brings about the state of affairs expressed by Kϕ. Our mere possession of the method, in advance of actually applying it, does not entitle us to assert (10). If, as Tennant assumes, our possession of a decision method for ψ always entitles us to assert ψ ∨ ¬ψ, then it has not been shown that our possession of a decision method for ϕ puts us in possession of a decision method for Kϕ. Alternatively, if we did count the Tennant-style procedure as a decision method for Kϕ, then it has not been shown that mere possession of a decision method in that weak sense for Kϕ would entitle us to assert Kϕ ∨ ¬Kϕ, and Tennant’s argument would still fail because we could not bridge the gap from (7) to (8).¹⁴ For a more dramatic example of the fallacy, consider a paradigm of a potentially undecidable sentence, ‘A city was, is or will be built on this spot’ (compare Dummett 1959b). Assume that no city has ever been built on the spot, and there is no present plan to build one, but equally no special reason why one should never be built there. For the Dummettian anti-realist, we are not entitled to assert ‘Either a city was, is or will be built on this spot or no city was, is or will be built on this spot’, for we have no procedure for determining which disjunct holds. Now imagine someone claiming: We do have a decision method for the sentence. For you can in principle build a city on this spot. Having done so, you will have determined that a city was, is or will be built on this spot.

Although we have the capacity in principle to build a city on the spot, it does not put us in possession of a decision procedure in the relevant sense, for by intuitionistic standards it does not entitle us to assert ‘Either a city was, is or will ¹⁴ As observed in n. 11, we do not need (10) to get from (7) to (8); the weaker ¬¬Kϕ → Kϕ would sufﬁce. But the same considerations apply; the purported decision procedure would entitle us to assert ¬¬Kϕ → Kϕ only by entitling us to assert (10).

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be built on this spot or no city was, is or will be built on this spot’ in advance of exercising the capacity. Indeed, the problem for (DIS) is even worse, for we can take both ϕ and ψ in it to be ‘A city was, is or will be built on this spot’. We possess a method (building a city) whose application will verify ‘A city was, is or will be built on this spot’. Therefore, by (DIS) we are now entitled to ‘Either a city was, is or will be built on this spot or a city was, is or will be built on this spot’. Since even in intuitionistic logic ϕ ∨ ϕ is trivially equivalent to ϕ, we are now entitled to assert ‘A city was, is or will be built on this spot’, in advance of building one or even planning to do so. That is absurd. More generally, such an argument would conclude that whenever one has the power in principle to make ϕ true, one is entitled to assert ϕ in advance of exercising that power or even planning to exercise it. That conclusion involves one in contradictions, since one often has both the power to make ϕ true and the power to make ¬ϕ true, for example when ϕ is ‘I shall count to a thousand by midnight’.¹⁵ By (DIS), one is now both entitled to assert ϕ and entitled to assert ¬ϕ, irrespective of one’s intentions. Evidently, (DIS) can fail for sentences whose truth-values depend on our will; more speciﬁcally, it fails when the the truth-value of ϕ or of ψ depends on whether the method is actually applied. Of course, this problem does not arise when ϕ and ψ are mathematical. If (DIS) does not govern the assertibility of disjunctions, what does? Intuitionistic semantics relies on some notion of canonical veriﬁcation (Dummett 1977: 389–403). One is entitled to make an assertion for which one lacks a canonical veriﬁcation, if one knows that such a veriﬁcation exists. The existence of the veriﬁcation does not consist in its being possessed by anyone; nevertheless, since the intuitionist conceives it as essentially capable of being possessed by someone, it does not import any sort of platonism inconsistent with the intuitionistic view.¹⁶ The natural suggestion is then that a canonical veriﬁcation of a disjunction consists of a canonical veriﬁcation of a disjunct. We may be entitled to assert a disjunction without being entitled to assert any disjunct, because we know that a canonical veriﬁcation exists for some disjunct without knowing which. We are in that position with respect to ϕ ∨ ¬ϕ when we have a genuine decision procedure for ϕ but have not yet applied it. But if we possess the purported decision procedure for Kϕ without having applied it, we do not thereby know that a canonical veriﬁcation for Kϕ or a canonical veriﬁcation for ¬Kϕ exists, even though we know how to bring such a canonical veriﬁcation ¹⁵ Any incompatibility between free will and determinism is irrelevant here. That I am causally determined not to apply a method is compatible with my possession of it in the relevant sense. ¹⁶ For more detail see Williamson (1982: 206–7, 1988: 429–32), where the idea is used to show the invalidity of an argument for ϕ → Kϕ from the intuitionistic semantics of → and the premise that Kϕ is veriﬁable whenever ϕ is veriﬁable (see Hart 1979: 165; Wright 1993: 430); the existence of a canonical veriﬁcation of ϕ does not imply the existence of a canonical veriﬁcation of Kϕ. Tennant describes this objection as ‘compelling’ (1997: 264).

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into existence. Consequently, we do not know that a canonical veriﬁcation for Kϕ ∨ ¬Kϕ exists; so by intuitionistic standards we are not entitled to assert the disjunction. Tennant’s response to this critique is to insist on a different reading of ‘K’ (2001a: 273–7). In his terminology, he understands ‘know’ to mean virtually or implicitly know rather than occurrently know, with a corresponding difference in the interpretation of ‘K’ (2001b: 109). In this sense, in merely possessing a decision method for primeness we already know whether any given natural number is prime. Unfortunately, he never squarely faces the obvious problem that this makes his discussion prima facie quite irrelevant to the point at issue, namely, the stability of the moderately hard anti-realist response to Fitch as I proposed it, which involves a logical distinction between (7) and (8) for decidable ϕ on the usual reading of ‘know’, not Tennant’s. We have just seen that his arguments do not work on the usual reading. To make his arguments relevant, Tennant would have to show that the usual reading of ‘know’ was somehow unavailable to the moderately hard anti-realist. The boldest, least plausible strategy would be to argue that such a reading makes no sense. But it is hopeless for the anti-realist to pretend that actually applying a decision procedure makes no cognitive difference at all. Indeed, Tennant describes in his own terms what cognitive difference it makes: The whole point of having a decision procedure is to discover the canonical form of expression of a proposition that, at the outset, can be identiﬁed only by description: as the result of applying the decision procedure. (2001a: 276)

We are supposed to discover the canonical form by applying the decision procedure. We thereby discover the canonical form only if we did not already know what it was. But in Tennant’s special sense we did already know what it was, since we had a procedure for ﬁnding out. Thus the point of applying the procedure is to gain knowledge in the usual sense of the canonical form. Consequently, Tennant himself relies on the coherence of something like the usual reading.¹⁷ Indeed, he speaks of it as of a genuine sense of ‘know’ (2001a: 276, 2001b: 109). Although he writes favourably of a conceptual reform that would impose his special reading on ‘know’, not all moderately hard antirealists need feel bound by such a reform; moreover, eliminating all unreformed epistemological terminology would leave us unable to articulate the point of applying a decision procedure. ¹⁷ Tennant says of a passage in which I emphasize the cognitive difference that applying a decision procedure makes that it ‘displays a vestige of realist thinking’ (2001a: 274); if so, Tennant has not explained how he himself can do without that vestige of realist thinking. Tennant’s further complaint concerning the conditional ¬Kϕ → ¬ϕ and ‘a curious asymmetry (between truth and falsity)’ (2001a: 275) raises an issue that is discussed in a more acute version in Williamson (1994), to which the reader is referred. That issue is separate from Tennant’s main argument.

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Tennant points out that his special reading of ‘know’ is better suited to epistemic logics that assume logical omniscience. He claims that ‘if this idealizing assumption were disallowed, the original Fitch argument would not go through’ (2001a: 274). But that is to ignore crucial differences. The only aspect of logical omniscience used in the argument is principle (3), that knowledge of a conjunction implies knowledge of its conjuncts. But one can accept that very weak closure principle even for knowledge in the usual sense without accepting logical omniscience in general. The modest idea that in knowing a conjunction one knows its conjuncts does not commit one to the extravagant idea that in knowing anything one knows anything that it entails. Moreover, there are revisions of the Fitch argument that do not even rely on (3).¹⁸ Nothing that Tennant says compels the moderately hard anti-realist to formulate their knowability principle (1) in terms of a reading of ‘know’ that satisﬁes logical omniscience. Thus the effect of Tennant’s insistence on his special reading of ‘know’ is that his arguments completely fail to engage with the version of moderately hard anti-realism that he was supposed to be attacking. Indeed, they fail to engage with just about any form of anti-realism that endorses the knowability principle (1) on the usual reading of ‘know’, since the most pertinent version of the Fitch argument will then involve that reading throughout. To refrain from endorsing the principle that every truth is capable of being known in the usual sense is to take a signiﬁcant step back from full-blooded anti-realism. Do Tennant’s arguments establish something of interest on his preferred reading of ‘know’, even though they do not establish what they were supposed to? On such a reading, (11) is no longer obviously absurd when ϕ is decidable but not occurrently decided. His purported decision procedure for Kϕ is at much less risk of violating the constraint, which he accepts, that ‘[p]roper decision procedures do not interfere with the states of affairs’ that they are supposed to determine (2001a: 276–7). Nevertheless, crucial unclarities remain. First, Tennant’s argument that the decidability of ϕ implies the decidability of Kϕ appears to commit him to a version of the highly controversial principle that when one knows, one knows that one knows. For the key passage is this: If you [ . . . ] determine that ϕ is true, then you know that ϕ. So you have determined that Kϕ is true. (1997: 262)

The principles invoked in the two sentences can be formalized by schemas (D1) and (D2) respectively, with Dy for determination by you as true and Ky for your knowledge: (D1) Dyϕ → Kyϕ (D2) Kyϕ → DyKϕ ¹⁸ See n. 3.

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By substitution of ‘Kϕ’ for ‘ϕ’ in (D1): (D3) DyKϕ → KyKϕ By transitivity from (D2) and (D3): (D4) Kyϕ → KyKϕ In other words, if you know something, you know that someone sometime knows it. But that principle is extremely doubtful, even on a special reading of ‘know’ that satisﬁes logical omniscience (Williamson 2000a: 114–34). In terms of epistemic logic: a knowledge operator can satisfy deductive closure without satisfying the S4 principle. An argument that relies on assumptions that are jointly as strong as (D4) is some way from establishing its conclusion. Second, it is not altogether clear what Tennant means by ‘decidable’. He is attracted by a version of moderately hard anti-realism different from the one I envisaged. On his version, one accepts schema (7) and rejects schema (8), but accepts this restricted version of (8) (2001a: 266–7): (8a) [ϕ∧ (ϕ is decidable)] → Kϕ If one accepts the Tennant-style argument from the decidability of ϕ to (11) on his special reading, (8a) is a simple consequence given principle (2) (the factivity of knowledge), since K¬ϕ implies ¬ϕ. Clearly, if (8a) is not to collapse into (8), one must reject: (8b) ϕ → (ϕ is decidable) For of course (8a) and (8b) jointly entail (8). Thus ‘ϕ is decidable’ cannot be regarded as a notational variant of ϕ ∨ ¬ϕ, as it often is in intuitionistic writings, for that would immediately validate (8b). Moreover, that reading disqualiﬁes undecidability as a genuine option, as Tennant takes it to be (2001a: 266), since ¬(ϕ ∨ ¬ϕ ) is intuitionistically inconsistent. At ﬁrst sight, it is unclear how an intuitionist can reject (8b). For suppose that we have a proof of ϕ. Then that proof decides ϕ, and thereby constitutes a proof of its decidability. Thus (8b) seems to be intuitionistically provable, because we can transform any proof of its antecedent into a proof of its consequent. Presumably, the way to block (8b) is to insist that its consequent concerns our actual possession of a decision method, not the mere possibility in principle of possessing one. Attempts to validate (8b) by means of the intuitionistic semantics for the conditional can then be blocked like similar attempts to validate (8).¹⁹ But then a more than virtual notion of possessing a decision procedure is doing the crucial work. To clarify his envisaged version of moderately hard anti-realism, Tennant needs to explain how it is supposed to reconcile virtual and non-virtual aspects of cognition. ¹⁹ See n. 16.

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Tennant’s alternative version of moderately hard anti-realism remains undeveloped. On an understanding of the language relevant to the version that I mooted, the decidability of ϕ does not imply the decidability of Kϕ in any sense conducive to (10). Tennant fails in his attempt to collapse (7) into (8) for decidable ϕ. He has located no instability in the envisaged version of moderately hard anti-realism.

III Tennant also proposes another response to Fitch’s argument, this time a soft anti-realist one, by constructing and defending a modiﬁed knowability principle. Having deﬁned a sentence ϕ to be Cartesian if and only if the contradiction ⊥ does not follow from Kϕ, he endorses this restricted variant of (1), formulated as a rule of inference:²⁰ (♦KC)

ϕ; ergo ♦Kϕ, where ϕ is Cartesian

Informally: (♦KC) says that truth entails knowability except when Fitch’s problem occurs. Tennant admits that it may be undecidable whether a given step is an instance of a rule like (♦KC), because it is undecidable whether ⊥ follows from Kϕ. He claims that this does not matter, on the grounds that we will apply it only when we do know that the condition is met. Since (♦KC) is intended for use only in a few philosophical arguments, not for systematic application in mathematics or science, the undecidability is supposed not to defeat its purpose. Tennant’s rule looks desperately ad hoc. He replied in detail (2001b) to a similar charge from Michael Hand and Jonathan Kvanvig (1999). However, his reply depends on a misunderstanding of the nature of the charge. He writes as though what is wrong with ad hoc principles is that they are restricted to the point of total or partial triviality. For instance, he argues that the restriction in (♦KC) is analogous to restrictions in other principles that are nevertheless ‘substantive, informative and important’ (2001b: 110, 111, 113).²¹ He seems to assume that if a principle P∗ is more restricted than a principle P, then P is ad hoc only if P∗ is also ad hoc.²² He assumes that ‘[a]d hoc emendations to general ²⁰ For (♦KC) to be a well-deﬁned rule, the notion of consequence used to deﬁne ‘Cartesian’ should be given independently of (♦KC) and cannot be assumed to be closed under it (see Tennant 1997: 275). This does not affect the argument in the text. ²¹ The examples are unconvincing. They too look ad hoc. Moreover, his restricted thesis about truth treats ‘This sentence is false’ as a premise of the Liar paradox (2001b: 110), whereas the mere well-formedness of the sentence is what makes trouble. His restricted thesis about wondering (2001b: 113) results from a derivation that makes the radically idealizing assumption (SI) that ‘a rational thinker is one whose attitudes are self-intimating to the thinker himself’ (1997: 248) so that I rationally wonder whether something is the case only if I believe that I so wonder (ibid.: 255). ²² Some such assumption is required for the relevance of his claim that ‘The restricted thesis about truth [Tarski’s] to which almost every philosopher subscribes is in fact even more restricted

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laws in natural science’ detract from their applicability (2001b: 112). However, imagine a scientist who has always maintained that emeralds are always green, but at time t is confronted with a blue emerald, and adopts the revised theory that emeralds are always grue, where something is grue at a time if and only if either it is green and the time is before t or it is blue and the time is not before t. The gruesome, gerrymandered revised theory is clearly ad hoc, even though it is just as general, substantive, informative and important (if correct) as the old theory. The problem is not triviality but ill-motivated complexity. Although the new theory predicts the same data before t as the old theory, and improves on the old theory with respect to the datum at t, the previous evidential support for the old theory does not transfer to the new theory, even before counterexamples to the new theory emerge. Analogous problems face (♦KC). If Fitch’s argument forces anti-realists to restrict the original knowability principle (1), then something is wrong with their original meaning-theoretic arguments for (1). Until we have an adequate diagnosis of the fallacies in those arguments, we cannot assume that such considerations confer any support whatsoever on (♦KC) or any other attempted approximation to (1) that does not immediately succumb to the Fitch argument. A subtle fallacy in an argument can easily mean that it establishes nothing of interest whatsoever. (♦KC) is a gruesome principle. We need not dwell on the charge that (♦KC) is ad hoc, for that is not the worst of its problems. The point of restricting (♦KC) to Cartesian cases is to enable the soft anti-realist to avoid asserting (8), that something is true only if known, even for decidable sentences. Since Tennant holds that (7) collapses into (8) when ϕ is decidable, his soft anti-realist rejects (7) too. The restriction on (♦KC) blocks the original derivation of (7) from (1). But Tennant overlooked a more complex derivation of (7) from (♦KC) and some plausible assumptions. The argument will be conducted in Tennant’s preferred background logic, which is not the standard intuitionistic one but his weaker system IR of intuitionistic relevance logic (Tennant 1997: 343–4). The differences do not affect the arguments below. For deﬁniteness, let ϕ be the decidable sentence ‘There is a fragment of Roman pottery at that spot’ (we assume a suitable context). Introduce a proper name ‘n’ by the stipulation that it is to designate (rigidly) the number of books actually now on my table. Thus ‘n’ is not a numeral such as ‘9’ but rather a name whose reference is ﬁxed by an empirical description. Let ‘E’ be the predicate ‘is even’. We ﬁrst argue that the conjunction ϕ ∧ (Kϕ → En) is Cartesian. For suppose that K(ϕ ∧ (Kϕ → En) ) is inconsistent, in the sense that ⊥ follows from than’ a thesis about truth that Tennant wants to show not to be ad hoc, given that ‘Tarski can hardly be accused of making an ad hoc restriction to his disquotational Thesis about truth’ (2001b: 111).

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K(ϕ ∧ (Kϕ → En) ) according to the logic adverted to in the deﬁnition of ‘Cartesian’. Then this story contains an inconsistency: STORY: I ﬁnd a fragment of Roman pottery at this spot and identify it correctly; I thereby come to know that there is a fragment of Roman pottery there. I also count the books actually now on my table and discover that the number is even; I deduce that if someone sometime knows that there is a fragment of Roman pottery at that spot then n is even.²³ By putting the two pieces of knowledge together, I acquire the knowledge expressed by the conjunction ϕ ∧ (Kϕ → En). Thus K(ϕ ∧ (Kϕ → En) ) holds.

But STORY is obviously consistent; we cannot exclude its truth on purely logical grounds (just try!). Of course, n may in fact be odd, in which case, since that number could not have been even, STORY expresses an impossible state of affairs. Nevertheless, STORY itself is still consistent; we cannot discern by reason alone that the description which ﬁxes the reference of ‘n’ picks out an odd number. Someone who asserts En because he failed to see one of the books is not guilty of an inconsistency. Although we might produce an inconsistent story by substituting for ‘n’ throughout STORY a numeral with the same reference as ‘n’, it does not follows that STORY itself is inconsistent. More precisely, the result of substituting a coreferential numeral for ‘n’ in the sentence K(ϕ ∧ (Kϕ → En) ) is a different sentence; the inconsistency of the latter does not imply the inconsistency of the former. Thus ⊥ does not follow from K(ϕ ∧ (Kϕ → En) ), so ϕ ∧ (Kϕ → En) is Cartesian. For the rest of the argument, let be the consequence relation of a system of modal epistemic logic based on IR with the additional rule (♦KC) and the axiom schemas (2) and (3).²⁴ Since its condition is met in this case, (♦KC) gives: (12) ϕ ∧ (Kϕ → En) ♦K(ϕ ∧ (Kϕ → En) ) Moreover: (13) ϕ ∧ ¬Kϕ ϕ ∧ (Kϕ → En) For ¬α α → β holds even in IR (Tennant 1997: 344); from ¬Kϕ Kϕ → En we can derive (13) by the rules for ∧. Now (12) and (13) yield: (14) ϕ ∧ ¬Kϕ ♦K(ϕ ∧ (Kϕ → En) ) ²³ β α → β in IR (Tennant 1997: 342). ²⁴ may differ from the consequence relation used to deﬁne ‘Cartesian’. Tennant uses the weaker set of inference rules K(ϕ ∧ ψ) Kϕ and from , ϕ ⊥ to , Kϕ ⊥ in place of (3) and (2) respectively (1997: 259–60). Nevertheless, his discussion makes clear that on his view we can reasonably treat instances of (2) and (3) as theorems of epistemic logic.

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The next step is a Fitch-like argument for: (15) K(ϕ ∧ (Kϕ → En) ) En For (3) and ∧-elimination yield K(ϕ ∧ (Kϕ → En) ) Kϕ, while (2) and ∧elimination yield K(ϕ ∧ (Kϕ → En) ) Kϕ → En, and even in IR we can then move to (15).²⁵ Since the rules used to derive (15) are truth-preserving in all possible situations, not just the actual one, if the premise of (15) expresses a possibility, so does its conclusion (if α → β is a theorem of a normal modal logic, so is ♦α → ♦β): (16) ♦K(ϕ ∧ (Kϕ → En) ) ♦En Now (14) and (16) yield: (17) ϕ ∧ ¬Kϕ ♦En Uncontentiously, it is not contingent whether n is even. Since I can count the books on my table, it is decidable whether n is even; hence n is either odd or even. But if n is odd, it could not have been even, for the mathematical properties of numbers are not contingent. Thus n could have been even only if it is even. We can symbolize that as the argument En ∨¬En, ¬En → ¬♦En, ♦En En, since ‘n’ is a rigid designator.²⁶ Thus, treating the uncontentious auxiliary assumptions concerning En as part of the background logic, we can strengthen (17) to: (18) φ ∧ ¬Kφ En Since the two cases are symmetric, we can now repeat the argument for (18) with ‘odd’ in place of ‘even’ to derive: (19) φ ∧ ¬Kφ ¬En But (18) and (19) together yield: (20) ¬(φ ∧ ¬Kφ ) That is to make (7) a theorem. But the point of Tennant’s restricted knowability principle (♦KC) was precisely to enable the soft anti-realist not to assert (7) for decidable φ (as in the present case), since on Tennant’s view (7) collapses into ²⁵ The cut rule used to chain inferences together does not hold unrestrictedly in IR, but fails only in case of a redundant premise or conclusion, which (15) does not contain. ²⁶ For the reason sketched in section I, the argument assumes a nonepistemic reading of ♦. The mere epistemic possibility that n is even does not entail that n is even. (♦KC) is in any case quite unpromising on an epistemic reading of ♦. If ♦ is read as ‘for all we know’, the principle will be unacceptable to soft anti-realists of the sort for whom Tennant seems to intend (♦KC), since, according to them, we sometimes know empirically that a given decidable proposition will never in fact be decided. If ♦ is read as ‘for all we know a priori’, (♦KC) is more or less trivialized because ♦Kφ then says little more than that φ is Cartesian.

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(8), the principle deﬁnitive of hard anti-realism, for decidable φ. Thus Tennant’s restriction is futile. Evidently, the foregoing critique of Tennant’s principle (♦KC) depends on careful respect for the distinction between the logical notion of inconsistency and the metaphysically modal notion of impossibility. One or other of En and ¬En expresses a metaphysical impossibility, but each of them is logically consistent, since the reference of ‘n’ is ﬁxed empirically. Similarly, the sentence ‘George W. Bush = Tony Blair’ is logically consistent, even though it expresses an impossibility. Unfortunately, Tennant’s reply to the critique displays a startling insensitivity to the distinction. In expounding (♦KC), he writes: It should be clear to anyone with a sympathetic understanding of the spirit of the proposed restriction that for a proposition to be Cartesian one ought to be unable to derive absurdity from it modulo any necessarily true propositions. It is a logical convention of long standing that mention of theorems as premises can be suppressed. (2001b: 264)

This passage conﬂates necessary truth and theoremhood. Of course, given the cut rule, the use of theorems of a given logic as premises of derivations in that logic does not enable one to reach any conclusions that could not be reached without those premises. But it is certainly not ‘a logical convention of long standing’ that mention of necessary truths as premises can be suppressed. The undecidable G¨odel sentence for ﬁrst-order arithmetic is a necessary truth, but that does not mean that mention of it as a premise can be suppressed in the proof theory of ﬁrst-order arithmetic. Similarly, one cannot declare ‘George W. Bush = Tony Blair’ logically inconsistent just on the grounds that its negation expresses a necessary truth. Nevertheless, Tennant is quite explicit in his notion of a Cartesian proposition that: To say that absurdity is not derivable from Kφ is equivalent to saying that absurdity is not derivable from Kφ in conjunction with any set X of necessarily true propositions. (2001a: 269–70)

We had best understand Tennant as deﬁning ‘Cartesian’ in terms of a special consequence relation for which, by stipulation, all necessities are theorems. For the time being let us read his term ‘proposition’ as equivalent to ‘sentence’, since elsewhere he treats the constituents of arguments in the logically standard way as linguistic (1997: 313–15); in his reply (2001a) he does not object to my treatment of premises and conclusions as sentences. We will reconsider the talk of propositions later. According to Tennant, ‘We are dealing primarily with logico-mathematical possibility and necessity here’ (2001a: 269). On Tennant’s proposal, if n is odd then ¬En is a logico-mathematical necessity because ‘n’ is a rigid designator; since En follows from K(φ ∧ (Kφ → En) ) by (15), absurdity is derivable from K(φ ∧ (Kφ → En) ) in conjunction with the necessity ¬En; thus φ ∧ (Kφ → En) is not Cartesian after all. By parallel reasoning, if n is even

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then φ ∧ (Kφ → ¬En) is not Cartesian. Either way, one half of my argument is supposed to break down. Tennant seems to assume that logico-mathematically necessary truths are knowable a priori, for he glosses the account quoted above of what it is for absurdity not to be derivable from Kφ thus: Whether this deﬁnition calls for the consideration only of sets X all of whose members are knowable a priori, or calls for the consideration also of sets X some of whose members might be knowable only a posteriori, is an issue of principle on which we are not at present forced to take a stand. (2001a: 270)

Tennant is not forced to take a stand on the issue of principle only if he is entitled to assume that if n is odd then ¬En is knowable a priori and if n is even then En is knowable a priori. But recall that the reference of ‘n’ was ﬁxed by the description ‘the number of books actually now on my table’. Thus we cannot know a priori whether n, so presented, is even! That was the crux of my argument. En and ¬En are not sentences of the language of mathematics, because ‘n’ is not a term of that language: although it refers to a number, it does so in a non-mathematical way. Tennant describes ‘n is even’ as ‘a mathematical proposition’ on the grounds that ‘n’ is a rigid designator (2001a: 270), but it is unclear what he means by ‘mathematical proposition’. At any rate, his discussion ignores the signiﬁcance of the empirical way in which the reference of ‘n’ was ﬁxed.²⁷ In order to make Tennant’s discussion relevant to the argument that I presented, we should understand him as deﬁning ‘Cartesian’ in terms of a special consequence relation for which, by stipulation, all necessary truths may occur as premises in the derivation of absurdity, whether or not they are knowable a priori. The result does not constitute a formal system, but never mind. Consider the following variant of (♦KC): (♦KC∗ )

ϕ ; ergo ♦Kϕ , where ¬¬Kϕ holds

We can derive (♦KC) from (♦KC∗ ) by showing that if ϕ is Cartesian, ¬¬Kϕ holds. Suppose that ¬Kϕ holds. Then ¬Kϕ is permitted to occur as a premise in the derivation of absurdity, in the sense used in the deﬁnition of ‘Cartesian’. So absurdity is derivable from Kϕ in that sense. Therefore ϕ is not Cartesian. Thus if ¬Kϕ holds, ϕ is not Cartesian. By an intuitionistically valid form of contraposition, if ϕ is Cartesian, ¬¬Kϕ holds. Thus (♦KC) is a simple consequence of (♦KC∗ ); one can also argue for the converse, although that is not our present concern. But (♦KC∗ ) is not a distinctively anti-realist principle. For a realist who accepts classical modal logic, (♦KC∗ ) is trivially truthpreserving, since ♦ is equivalent to ¬¬ simply by the duality of the two modal operators. Of course, it is odd to present ♦Kϕ as derived from the ostensible premise ϕ rather than from the condition for the applicability of (♦KC∗ ), that ²⁷ In his examples, Tennant replaces ‘n’ by a numeral (2001a: 264, 271), thereby obscuring the vital point.

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¬¬Kϕ holds, which is what really guarantees the truth of where ♦Kϕ. But that oddity is harmless once one appreciates that the connection signalled by ‘ergo’ outside the scope of the ‘where’ clause need not be knowable a priori. For exactly the same reason, Tennant’s principle (♦KC) as now interpreted, a corollary of (♦KC∗ ), is also not distinctively anti-realist; for a realist who accepts classical modal logic, it is trivially truth-preserving. The two principles may not be quite so innocent from an intuitionistic perspective, since the inference ¬¬α to ♦α is structurally analogous to the intuitionistically invalid inference from ¬∀x¬α to ∃xα. However, that extra piece of logical content from an intuitionistic perspective is, if anything, a slight concession to classicism, not the articulation of a distinctively anti-realist claim about the possibility of knowledge. Thus Tennant’s stipulations about the notion of derivability in the deﬁnition of ‘Cartesian’ are relevant to my original objection to (♦KC) only by voiding (♦KC) of all interest as a formulation of an anti-realist principle of knowability. I had in mind considerations of the kind above when I wrote in my original critique ‘A more liberal interpretation of inconsistency might trivialize ♦KC; it is not what Tennant intends’ (2000b: 110). Perhaps it was what Tennant intended, and he has indeed fallen into the trap that I warned against. Does it make a difference if we suppose that by ‘proposition’ Tennant means something signiﬁcantly more coarse-grained than a sentence, even though he does not mention the distinction between sentences and propositions in this connection? Let ‘q’ be a numeral with the same reference as ‘n’. If proper names are directly referential, then the two sentences Eq and En express the same proposition, even though the sentences do not have the same cognitive signiﬁcance for us. If knowledge is a relation to propositions, it follows that knowing Eq a priori is knowing En a priori, although one can be in a position to express one’s knowledge by one sentence without being in a position to express it by the other. Such a view might ﬁt what Tennant says in his discussion of decidability about different forms of expression of a given proposition (2001a: 276–7). Would this approach enable Tennant to restrict the use of necessarily true propositions in the derivation of absurdity to those knowable a priori under some mode of presentation or other (such as Eq)? The appeal to a directly referential semantics for proper names deals with only one class of examples. An analogue of my argument can be developed for any decidable sentence ϕ whatsoever, by substituting for En the sentence Aϕ, where A is the rigidifying ‘actually’ operator, so that Aϕ is a priori equivalent to ϕ, but ♦Aϕ entails Aϕ and ♦¬Aϕ entails ¬Aϕ. To extend the approach just sketched to such examples, Tennant would need to argue that Aϕ expresses the same proposition as some sentence that one can use to express a priori knowledge. It is hard to see what principled justiﬁcation there could be for such a claim, short of the identiﬁcation of all necessarily equivalent propositions. But on that approach there is just one necessary truth, which is known a priori under the

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mode of presentation ‘0 = 0’. We can then reproduce the derivation of (♦KC) from (♦KC∗ ), and regain just the trivialization of Tennant’s principle that the appeal to coarse-grained propositions was supposed to avoid. Thus Tennant’s response to the critique of (♦KC) serves only to emphasize his difﬁculties; his attempt to locate a fallacy in it merely trivializes his own view. Should Tennant’s soft anti-realist seek, instead of (♦KC), a knowability principle with a stronger restriction to avoid (7)? The natural suspicion is that such a restriction would have to be very draconian indeed, and thereby risk trivialization again. But the search is in any case ill-motivated, for reasons already indicated in the discussion of Tennant’s unsuccessful response to the charge that (♦KC) is ad hoc. The original knowability principle (1) was the outcome of an anti-realist argument (albeit a very dubious argument). If (1) has false consequences, then something must be wrong with the argument. If the argument is irreparably fallacious, one has lost one’s reason for postulating any knowability principle at all, restricted or unrestricted. If the argument can be repaired, the nature of the repairs should dictate the nature of the restrictions on the resultant knowability principle.²⁸ Tennant does not indicate any form of argument for anti-realism that would motivate a principle restricted in the manner of (♦KC) on a non-trivializing interpretation. To say that ϕ is non-Cartesian is not to explain on anti-realist terms how ϕ could be unknowably true, how speakers’ use of ϕ could be sensitive to a condition they could not in principle recognize to obtain or how ϕ could express its content without such sensitivity; it is merely to say that broadly logical considerations do (not not) rule out knowledge of its truth. Without such an explanation, from the perspective of principled anti-realism it is quite premature to endorse anything like (♦KC). If Fitch’s argument does not by itself refute all forms of anti-realism, it certainly shows how much would have to be done before there was a working anti-realist semantics for empirical language, even in the toy examples that we have been considering. The attempts on behalf of anti-realism to deal with the Fitch problem give every sign of a degenerating research programme. ²⁸ A failure to ﬁt the philosophical arguments for anti-realism may also affect the revisions of knowability principle proposed in Edgington (1985), Melia (1991), Rabinowicz and Segerberg (1994), Kvanvig (1995), Lindström (1997) and Rückert (2004), although the point cannot be argued here; see also the comments on Dummett (2001) in n. 5. The upshot of the treatment of Fitch’s argument in Usberti 1995 is a much more drastic restriction of ϕ in (1) to mathematical sentences, which excludes those containing K. The proposal is grounded in Usberti’s anti-Dummettian analysis of the arguments for an intuitionistic approach (see also Williamson 1998). For a discussion of Fitch’s argument in the context of classical logic see Williamson (2000a: 270–301, 318–19).

13 Restriction Strategies for Knowability: Some Lessons in False Hope Jonathan L. Kvanvig

The knowability paradox derives from a proof by Frederic Fitch in 1963. The proof purportedly shows that if all truths are knowable, it follows that all truths are known. Antirealists, wed as they are to the idea that truth is epistemic, feel threatened by the proof. For what better way to express the epistemic character of truth than to insist that all truths are knowable? Yet, if that insistence logically compels similar assent to some omniscience-like claim, antirealism is in jeopardy. Response to the paradox has drifted toward a common theme, a theme I will argue is a non-starter in resolving the paradox. Seeing this point will also make clear the philosophical inadequacy of simply viewing the paradox as a refutation of a wide range of antirealisms. Re s p o n s e s t o t h e Pa r a d o x One way to respond to this problem for antirealism is to question the proof itself, and there have been a number of questions raised about the proof. Such questioning seems to lead nowhere, however. The simplest form of the proof goes as follows. Where we understand the operator K as ‘it is known by someone at some time that’, we begin by assuming (1) K(p & ∼Kp). If we distribute the K operator across the conjunction, we get (2) Kp & K∼Kp. In thinking about the issues discussed here, I have been helped immensely in a number of conversations and blog interchanges with the following, whom I would like to thank: Mike Beaty, Bryan Frances, Michael Hand, Stephen Hetherington, Carrie Jenkins, Robert Johnson, Matt McGrath, Julien Murzi, Joe Salerno, Fritz Warﬁeld, Jonathan Weinberg, and Tim Williamson.

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Since knowledge implies truth, K∼Kp implies ∼Kp; hence (2) implies (3) Kp & ∼Kp, allowing us to prove by reductio (4) ∼ K(p & ∼ Kp). Since (4) is a theorem, we can derive by the Rule of Necessitation (5) ∼ K(p & ∼ Kp), which is equivalent to (6) ∼ ♦K(p & ∼ Kp). This claim, however, is pretty obviously inconsistent with the claim that all truths are knowable. All that is needed is for the value for p in (6) to be a truth that nobody knows, in which case p & ∼Kp is a truth. By the knowability principle, it must be knowable, i.e., (7) ♦K(p & ∼ Kp). Since (7) contradicts (6), we learn that not all truths can be known (or that all truths are known, for those who enjoy the bizarre, half-baked, inane, and philosophically barmy¹). The options for ﬁnding a logical ﬂaw in this proof are quite limited. The only rules of inference it employs beyond those of propositional logic are these: (K-Dist) K(p & q) Kp & Kq (KIT) Kp p and the metalinguistic rule (RN) (p) ⇒ ( p). Of these rules, the ﬁrst is the most likely candidate to be challenged, but Timothy Williamson (1993) has shown how to generate paradox without relying on (K-Dist) at all. Given this result, the only hope for avoiding the paradox is to deny that knowledge implies truth or to deny the rule of necessitation.² In light of the well-entrenched character of these rules, it is not surprising to ﬁnd the literature on the paradox turning in a different direction in attempting to save antirealism from the dark force of the knowability paradox. The dominant ¹ Do not say here: if you understand what we mean by the claim that all truths are known, it is not a bizarre or barmy claim. It’s a sentence of English; we all speak the language; you don’t get to reinterpret into your favored alternative idiolect or dialect. ² This is not to say that other strategies have not been tried. One common attempt is to reinterpret the disappointing results in intuitionistic language, i.e., to hold that, in that language, it is not so bad a thing to have to deny that there are unknown truths. I shall not comment here on this strategy beyond pointing out how philosophically strained it is.

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strategy has become to deny that the antirealist commitment to the epistemic character of truth involves any commitment at all to the claim that all truths are knowable. Instead, the heart of antirealism is to be found in some weaker claim. Dorothy Edgington (1985) proposes that antirealists need only hold that all actual truths are knowable. Michael Dummett (2001) insists that knowability is required only for basic statements, and Michael Hand (2003) provides a sophisticated defense of a similar point of view. Cesare Cozzo (1994) develops an alternative to the knowability claim in terms of idealized arguments, and Neil Tennant (1997: ch. 8) argues that the only truths that must be knowable are those for which the assumption that they are known is logically consistent, leading to a cottage industry regarding whether his approach is a complete non-starter.³ The body of literature pursuing such restriction strategies—strategies for coping with the paradox that deny that the idea that truth is epistemic commits the antirealist to the claim that all truths are knowable—comprises an enormous percentage of the writing on this subject, and has become the favored approach among antirealists for disarming the paradox. When one takes into account the additional literature aimed at undermining such strategies, the dominant issue displayed by recent literature on the knowability paradox is whether any such restriction strategy can successfully disarm the paradox. Nowhere in this body of literature is the strategy itself questioned. Instead, the questions are two: is the strategy faithful to, and theoretically sustained by, the antirealist commitment to the epistemic character of truth, and does the restriction yield a claim that avoids Fitch’s result? We do not need answers to these questions, however, if such approaches to the paradox are red herrings, and that is what I claim here. To defend this point, I will ﬁrst explain a different approach to the paradox that is clearly a red herring. In the process, we will learn more about the heart of the knowability paradox, enough to see clearly that restriction strategies are simply lessons in false hope.

T h e i s m a n d K n ow a b i l i t y The knowability paradox is typically thought of as deriving from two assumptions. The ﬁrst is that all truths are knowable and the second is that some truths are unknown. Upon generating a contradiction from these two assumptions, we are required to discharge, leaving us to conclude that the knowability claim is false (and antirealism thereby threatened). In the presentation given above, however, a different characterization of the paradox might be given. The presentation above could be characterized as a proof ³ See, e.g., Hand and Kvanvig (1999); Williamson (2000a); DeVidi and Kenyon (2003). Replies and further discussion can be found in Tennant (2001a), (2001b), and (2002); and Brogaard and Salerno (2002).

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that there have to be unknown truths, for it begins by assuming that a particular truth is known (the truth that p is an unknown truth), and derives from that claim the impossibility of knowing this particular truth. For philosophers of a theistic bent, this characterization of the paradox may disturb, since it threatens the idea that there is an omniscient being. Given such a disturbance, theistic philosophers may see themselves as having a strong reason to ﬁnd some ﬂaw in the proof, hoping thereby to prevent the knowability paradox from refuting their theistic perspective. Such a response, however, is confused. After more sober reﬂection, the theistic philosopher may see the ﬂaw in this reaction to the paradox. The theistic philosopher may come to see that the above proof is no threat to theism unless it is true that there is some unknown truth, for, if there is no unknown truth, then omniscience does not require that it be known that there is some claim that is both true and unknown. Yet, if there is an omniscient being, then there aren’t any unknown truths! Hence, if one thinks there is an omniscient being, the above proof can be dismissed as a challenge to that viewpoint. It is no more interesting a challenge to theism than any argument that presumes an omniscient being must know what is false. So the theistic philosopher can move on to other interesting areas of philosophy, knowing that the knowability paradox is of no concern. So characterized, the supposed reaction by the theistic philosopher is both right and wrong. It is right in that the proof above does not threaten the claim of omniscience, but it is wrong in supposing that nothing paradoxical remains about which the theistic philosopher need be concerned. One way to see this point is to notice that the paradox does not depend simply on whether one accepts the two assumptions in question, the knowability assumption and the non-omniscience assumption. The central perplexity involved in the paradox does not depend on the idiosyncrasies of one’s favored philosophy, but rather on a perplexing lost logical distinction between what is actually the case and what might be case. It is obvious that knowledge implies the possibility of such, but what is not obvious is what the Fitch proof attempts to demonstrate: that, to put it carelessly, possible knowledge implies actual knowledge. Should that distinction disappear, it would be ﬁtting to ﬁnd ourselves in a state of perplexing philosophical stupor. How could it be that there is no logical distinction between actuality and possibility in this way? We might try for equilibrium by reminding ourselves that there are philosophical domains in which the distinction between actuality and possibility is lost. For example, modal logicians have long been comfortable with the idea that what is actually necessary is not logically distinct from what is possibly necessary. The comfort experienced by this thought will not last long, however. We are comfortable with the lost distinction in this domain because we have a semantical theory to which to appeal to explain why there is no logical distinction here, and we became comfortable with denying the distinction here only after the development of the semantical theory that makes intelligible the loss of such a

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distinction.⁴ Merely formulating S5 systems with the distinctive axiom central to the proof that eliminates the logical distinction between actual and possible necessity is not enough. As the history of the philosophy of modal logic shows, it took a semantical explanation to motivate the contemporary orthodoxy in favor of S5. Proof rules for the modal operators do not, by themselves, yield the degree of understanding necessary to rid us of the philosophical puzzle; only something more, such as is provided by Kripke semantics for quantiﬁed modal logic will help.⁵ Nothing similar can be said when we return to the context of the knowability paradox, however: we have no semantical basis whatsoever for being sanguine about a lost distinction between actual and possible knowledge. So, even if our theistic philosopher should dissent from the assumption that there are unknown truths, said philosopher has as much reason as anyone to view Fitch’s proof as establishing a very troubling conclusion. There is nothing about theism that yields an explanation as to why actual and possible knowledge are not logically distinct. It is for this reason that the theistic response to the paradox is a red herring, even though such a philosopher can take refuge in holding that theism itself is not at stake. The imagined theistic philosopher denies that there are unknown truths, and thereby achieves serenity in the face of the paradox. Such serenity is warranted, however, only if the theistic perspective does more. It will need to explain why, to speak again in the loose and popular vernacular, there is no logical distinction between actual and possible knowledge.

A n t i re a l i s m a n d K n ow a b i l i t y Antirealists, I maintain, do something similar to what the imagined theist has done. The imagined theist denies the assumption of non-omniscience, thereby claiming to avoid any perturbation from the paradox. The now-dominant antirealist strategy is to deny the knowability claim, substituting for it some careful emendation with weaker implications, also thereby claiming to avoid the reach of the paradox. Yet, if the theistic response to the paradox is a red herring, one should wonder why the antirealist restriction strategy isn’t as well. What reason can an antirealist give on behalf of a restriction strategy that will render respectable such a response to the paradox in contrast to the theistic response? Here is what antirealists will need to say in response to the claim above that the heart of the paradox concerns a lost logical distinction between actuality ⁴ Note here how rare it is to ﬁnd a defender of S5 prior to a development of the semantics in question by Kripke, and the corresponding paucity of deniers of S5 after this development. ⁵ There is a way that the remarks in the text are a bit misleading, for it is not the mere fact of having a formal semantics that does the trick here. What is important is that the pure semantics connect up with ordinary meaning, which then gives us the explanation we seek. For more on the distinction between these two kinds of semantics, see Alvin Plantinga’s (1979) distinction between pure and depraved semantics.

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and possibility. They can insist on more precision; they can say that we need to speak with the sophisticates rather than the vulgar. The lost distinction is not one between actual and possible knowledge, for even false (contingent) claims are objects of possible knowledge (in worlds where they are true). Fair enough; so let’s try to be more careful. The antirealist is likely to put the careful point this way: the lost distinction is a lost distinction between actual known truths and possible known truths. That is, a careful presentation of the lost distinction is: (LD) ∀p( (p & ♦Kp) ⇔ (p & Kp) ). The proof from p & Kp to p & ♦Kp is trivial, depending only on the modal principle that what is actual is possible. So, the antirealist can claim, the heart of the paradox is found in demonstrating that p & Kp follows from p & ♦Kp. That proof, however, requires assuming that all truths are knowable.⁶ So (LD), the careful expression of the heart of the knowability paradox in terms of a lost logical distinction between actuality and possibility, is derivable only on the assumption that all truths are knowable. Hence, a perfectly respectable strategy in responding to the paradox is to weaken the knowability assumption in such a way that (LD) can no longer be derived. So long as the resulting restriction still expresses the idea that truth is epistemic, the antirealist has bested the theist above, for not only can the antirealist claim that their position is not undermined by the paradox but also that the paradox has been disarmed since the lost distinction at the heart of the paradox follows only by assuming a claim that is false and necessarily so. In case we needed reminding, we might be reminded as well that strange consequences often follow from necessarily false assumptions. It is time for a philosophical lament here, however. This response to the theist analogy is valuable because of its demand for precision regarding the lost logical distinction at the heart of the paradox. It is mistaken, however, in claiming that (LD) is the proper formulation of that distinction. The paradox is generated from two assumptions, the assumption that all truths are knowable and the assumption that some truths are not known. The proof from the latter assumption to the former is trivial; the proof from the former to the latter is just Fitch’s proof. Given these two proofs, the obvious formalization of the lost distinction is not (LD) but (LD∗ ) ∀p(p → ♦Kp) ⇔ ∀p(p → Kp). Whereas (LD) is not a theorem, but instead can be proven only by assuming that all truth are knowable, (LD∗ ) is a theorem so long as Fitch’s proof is valid. I will express (LD∗ ) in ordinary English by saying that there is no logical distinction between universally knowable truth and universally known truth. This more ⁶ Once we get to ∼ ♦K(p & ∼ Kp), as above, we can only get a contradiction by noting that p & ∼Kp is true and hence knowable by the knowability principle.

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careful articulation still codiﬁes a lost logical distinction between actuality and possibility with respect to what is known. With this more careful formulation, the analogy with the theist above is restored. The theist is both right and wrong in being undisturbed by the paradox, and the antirealist is in the same boat. If the idea that truth is epistemic doesn’t require that all truths are knowable, then the paradox does not threaten to undermine antirealism any more than it threatens to undermine theism. Neither theism nor antirealism of this restricted variety has anything to say that is relevant to the paradoxicality engendered by Fitch’s proof. Each view denies a different assumption in Fitch’s proof, but the paradoxicality involved in (LD∗ ) depends in no way whatsoever on the truth of the assumptions used to generate the paradox. Antirealists may still ﬁnd comfort in undermining (LD) by pursuing a restriction strategy, but they should not pretend that undermining (LD) solves the paradox. Im p l i c a t i o n s Prior to encountering the literature on the topic, the taking of (LD) as the proper careful articulation of the threat of the paradox should strike one as surprising. The obvious careful articulation is (LD∗ )—after all, Fitch’s proof is a derivation of the left side of (LD∗ ) from its right side. There is a larger point to note as well. Critics of antirealism, such as Williamson, view the paradox as a refutation of (most versions of ) antirealism, with Fitch’s proof simply a display of a surprising logical result to this effect.⁷ Such approaches to the paradox, however, leave the paradoxicality in question unresolved. What is paradoxical here is not that Fitch has discovered a proof that threatens antirealism, but rather that Fitch has discovered a proof that threatens a logical distinction between actuality and possibility. One way to put this point is to notice that the omniscience-like claim, though not likely to be thought true (especially when we envision the quantiﬁers restricted to ﬁnite minds), is not obviously impossible. Contrast this point with the fact that the knowability claim, if true, is supposed to be necessarily true: it is a purported implication of a proper understanding of the nature of truth. Yet (LD∗ ) claims that the two are logically equivalent, which they cannot be without having the same modal status. A satisfactory response to the paradox cannot simply swallow this result without explanation. We have already seen one example of a satisfying response to a similar situation, where we have a semantic explanation of the lost logical distinction between actual and possible necessity. If we could have the same here, we’d have a solution to the paradox. It may be there is some alternative ⁷ A point he made most recently to me at the Modalism and Mentalism conference in Copenhagen at the end of January 2004, and most clearly made in print in Williamson (1987) and (1993).

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explanation as well that is not semantic in character, but it is hard to see at this point what such a weaker explanation might look like. I want to consider the issue of whether there might be a non-semantical explanation of the lost distinction in a moment, but I ﬁrst want to emphasize the paradoxicality of the lost distinction by contrasting it with results that are merely surprising but not paradoxical to prevent some deﬂationary approach to the paradox that claims that the result is merely surprising. Consider, for example, Gödel’s incompleteness results. These results are surprising, and threaten important philosophical perspectives, such as Hilbert’s formalism. These results themselves are not paradoxical, however. They present no challenge to anything like the edicts of common sense or the viewpoint of received opinion. That makes these results quite different from a lost logical distinction between actuality and possibility. One might disagree here with my characterization of the Gödel results, arguing that they involve real paradox, but that point can be granted without implying that there is nothing paradoxical resulting from Fitch’s proof. If I’m wrong that the Gödel results do more that threaten important philosophical perspectives, then they may be paradoxical; but if so, they join the class of things already including the results of Fitch’s proof rather than showing that Fitch’s proof is merely surprising. Consider for another example Vann McGee’s (1985) apparent counterexample to modus ponens. The result of McGee’s arguments is not merely surprising, but paradoxical. Modus ponens is so well-entrenched a part of our ordinary view of things that our reaction to his arguments is that they must contain a mistake. Suppose, however, that we are wrong. If we are wrong, and McGee is right, some explanation is in order. We need to know how it could be that our ordinary view of things could be so mistaken. It is worth noting that McGee attempts just such an explanation: logical rules should be thought of as more akin to generalizations and lawlike statements in science which can be useful and instructive even if not always completely accurate. My intention in citing McGee’s explanation is not to endorse it, nor to endorse his arguments that modus ponens is not an exceptionless logical rule.⁸ The point is only that when a proof conﬂicts with ordinary understandings, a further explanatory burden must be shouldered. So it is not enough simply to accept the surprising character of Fitch’s result. One must also shoulder the ⁸ My own view of the matter is that it is preferable to abandon importation/exportation in response to his arguments. If his example is put in counterfactual form, this response becomes obvious: to say that if Reagan were to lose, then if Anderson were to lose, Carter would win, is to say something false; whereas to say that if both Reagan and Anderson were to lose, Carter would win, is to say something true. Only the former, however, is of any use to McGee’s argument. McGee’s argument, of course, involves indicative conditionals rather than counterfactual ones. As a result, more argument is needed to get around his claims. The extra arguments needed, I believe, involve refusing to adopt the assertibility condition semantics he employs, but since that is beyond the scope of the present essay, I will leave that topic for another time and place.

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philosophical burden of explaining how the proof could be correct since it implies a lost distinction between actuality and possibility. Even worse would be to respond as follows. ‘‘We’ve carefully considered the rules involved in the paradox, the rules of (K-Dist) and (KIT), plus the metalinguistic rule of (RN) are so inherently plausible that the conﬂict they create with the intuitive logical distinction between possibility and actuality is not paradoxical at all. The results are surprising and unanticipated, but not paradoxical.’’⁹ Someone is living in logical denial. No argument can conclusively show that this approach is mistaken, since the difference between what is paradoxical and what is merely surprising is, perhaps, only a difference in degree and not in kind. Even so, there is a distinction to be drawn here between the unanticipated and the seemingly contradictory, and Fitch’s proof engenders the latter experience and not simply the former. It is not merely surprising when we are told that what looks like a modal truth is logically equivalent to what looks like a non-modal truth (especially when, given ordinary assumptions, the ﬁrst would apparently be necessarily true if true at all and the second would apparently not be). We can’t simply afﬁrm the rules, and say, ‘‘I guess we were wrong; non-omniscience really is impossible.’’ That’s simply not an adequate explanation of what’s gone wrong; more accurately, it is not an explanation at all. As we have seen, the paradigm example of a satisfying explanation in this regard is a semantical one. Moreover, as already noted as well, a purely syntactic explanation in terms of axioms of a system and proof rules for it, is particularly unsatisfying. Suppose, for example, that in the modal domain, we showed multiple contexts in which the introduction of a possibility operator on a formula yielded nothing logically distinct from the original formula. Such is the case, if S5 is to be believed, for logical necessities. It is also true of other modal systems, however. For example, obligation statements and possible obligation statements cannot be distinguished logically in certain deontic systems. We learn to accept such results, if we do, by being told a semantic story. In the deontic case, if we interpret obligation statements in terms of ideal worlds, worlds where everything is done properly, then we can see actual and possible obligations collapse. An actual obligation statement takes us to what is true in an ideal world, and a possible obligation statement takes us to another world where the obligation statement is interpreted with reference to ideal worlds. On the assumption that all worlds are accessible from every world, the class of ideal worlds will be the same, whether accessed from our world or some other possible world, generating a logical equivalence between possible obligation and actual obligation. If we accept this semantical story, we understand the lost distinction here. The important point to note is that understanding is not achieved merely by multiplying contexts ⁹ This response is in the spirit of Williamson’s approach to the paradox both in print and in conversation, as well as that of Carrie Jenkins’s contribution to this volume.

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in which there is a similar lost distinction. All such multiplication would do is to replace a rather speciﬁc paradoxicality with a more general one. What is needed is some understanding of why an apparent logical distinction is lost, and, without such understanding, we cannot say ‘‘paradox lost.’’ This approach to the paradox in terms of trying to accustom us to the loss by generalizing on the syntactic features involved in the paradox has a history going back to J. L. Mackie’s (1980) early paper on the paradox. Seeing the failure of the strategy in other contexts should make us suspicious here, and it is worth taking a look at the details to see why this suspicion is correct. Here’s an attempt along these lines.¹⁰ Consider the operator ‘‘it is written on my blackboard that’’ and the operator ‘‘it is true that,’’ and the idea that anything true might be written correctly on my blackboard. If we call these operators W and T respectively, we won’t be able to get an analogue of the knowability paradox out of them, since only the latter is factive. To get a paradox, we’d have to generate a contradiction from this assumption: WT(p & ∼WTp). From this formula, we can get WTp & WT ∼ WTp, since, we assume for now, both operators distribute over conjunction. We also assume that the operators can be split by the following rule: WTp Wp & Tp. Using this rule, we can get T ∼ WTp from the second conjunct (and &-Elim), and then get ∼WTp from this formula given the factive character of truth: Tp p. We may also wish to put the two operators together into a single operator WT . Intuitively, this operator is supposed to mean something like ‘‘written truthfully on my blackboard,’’ but, formally speaking, the crucial idea is that WT is both distributive and factive—that is, it borrows distributivity from the W operator and factivity from the T operator. Because it is both factive and distributive, we can generate the analogue of the contradiction crucial to the knowability paradox a bit more quickly: WT (p & ∼ WT p) WT p & WT ∼ WT p (by distribution) WT p & ∼ WT p (by factivity) ¹⁰ I owe a great deal to Michael Hand regarding this approach. In fact, I think it fair to say that I simply would not have seen the possibility or signiﬁcance of proof-theoretic insight without the long discussions we have had together.

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So it appears that not everything true can be correctly written on my blackboard, and thus that WT is an analogue of the K operator. Both are distributive and factive, and hence the knowability paradox is but a special case of a more general phenomenon: ﬁnd any distributive and factive operator, and the crucial contradiction in the knowability paradox will follow. There are some niggling problems with this attempt at generalizing so as to provide a syntactic explanation of the lost logical distinction that constitutes the heart of the knowability paradox. These problems are not my fundamental reasons for rejecting this strategy, but they place important limitations on any attempt to pursue this generalization strategy. The ﬁrst problem to note is that neither of the ways above of demonstrating a contradiction is quite adequate. To see the problem, let us generalize here beyond W and T and the combined operator WT and think in terms of p and p, for any operators and . The idea above is to allow the combined operator to inherit the preferred formal features of the individual operators themselves, but there is no guarantee that this result can be achieved. Suppose we construct a new operator that is a combination of knowledge (‘‘it is known by someone at some time that’’) and necessity (‘‘it is necessary that’’). Call this the ‘‘known-to-be-necessary’’ operator. Assume as well that the combined operator inherits formal features from the individual operators of which it is a combination. Since &-I works within the context of necessity and since knowledge implies belief, we can infer from the governing of p and the governing of q by this new operator that the conjunction of the two is believed by someone at some time. This inference is faulty, however, since the ﬁrst could be known by someone and the second known by someone, but the conjunction believed by no one. For another example, combine obligation-for-everyone and knowledge-bysomeone, and you get knowingly obligatory (known-by-someone-to-be-obligatory-for-everyone). Being obligatory preserves &-I, so if both p and q are knowingly obligatory, then someone believes both p and q (because knowledge implies belief ). This inference is obviously absurd, however. The lesson here is that one can’t combine operators and expect to be able to apply the usual rules for either of the individual operators that were put together to form the combined operator. Notice further in the example about writing truly that if we keep the operators separate, we can’t prove the contradiction. Consider how to try. We begin by representing the claim that a speciﬁc truth is not truthfully written on my blackboard as: Tp & ∼(Wp & Tp). Then the reductio assumption will have to be: W(Tp & ∼(Wp & Tp)) & T(Tp & ∼(Wp & Tp)).

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If we then distribute the W and T operators, respectively, we get: WTp & W ∼ (Wp & Tp) & TTp & T ∼ (Wp & Tp). Since truth is factive, we get from the latter two conjuncts: Tp & ∼(Wp & Tp), from which we can get ∼Wp. This point is somehow supposed to contradict the ﬁrst conjunct WTp, but since we can’t apply the factivity claim about truth inside the W operator, we can’t demonstrate the contradiction. We can avoid this problem by eliminating the truth operator in our representation: p & ∼(Wp & p). The assumption for reductio can then be W(p & ∼(Wp & p)) & (p & ∼(Wp & p)). From this claim we derive by the distributivity of W: Wp & W ∼ (Wp & p) & p & ∼(Wp & p). The latter two conjuncts give us ∼Wp, which contradicts the ﬁrst conjunct of this formula. There are two points to note about this derivation. First, there is no factive operator in this proof. Even so, it is reminiscent of Fitch’s proof since we represent the claim that p is an unwritten truth as the claim that ∼Wp & p, just as we represented the idea that p is an unknown truth in Fitch’s proof as the claim that ∼Kp & p. To claim, however, that we have an analogue of Fitch’s proof that mirrors its dependence on an operator that is factive and distributive is mistaken. The only operator in this representation is the W operator, and it is not factive. This point is not important when we are looking for non-syntactic generalizations of the knowability paradox, for the above proof is reminiscent enough of Fitch’s proof that any explanation of the lost distinction engendered by Fitch’s proof will shed light on this proof as well. The syntactic strategy, however, hopes that mere duplication of syntactic form will relieve our perplexity, and there is no such duplication here. The second point to note will take a bit longer to develop, but it too casts doubt on the idea that we have a syntactic analogue of knowability here. Any syntactic explanatory value for the above derivation depends on the interpretation

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assumed of the operator in question, since no one should have an issue with the idea that there are distributive operators ψ for which the formula (p & ∼p) implies a contradiction. Neither should anyone think that the existence of such operators appropriately addresses the knowability paradox. Instead, if any help is found in syntactic mimicry here, it comes from the assumed interpretation of the W operator in terms of what is written. Once we begin thinking carefully about the concept of writing, however, problems appear. To see them, let’s think ﬁrst about asserting. To assert a claim is not just to utter a sequence of phonemes that conventionally expresses the proposition in question. Instead, to assert is to express that very proposition. When speaking only of the sequence of phonemes and the associated noises, I will term the act in question an act of uttering; when the proposition itself is expressed by such uttering, I will term such an act of asserting. What I mean by the term ‘proposition’ here is simply a bearer of truth-value. Thus, I leave open whether propositions are sentences or abstract objects of some sort, and claim that there is a speech act involving the uttering of phonemes by which bearers of truth-value are expressed—namely, the speech act of asserting. We should note that uttering has formal properties that asserting does not. For one thing, I can’t avoid uttering ‘‘it is raining’’ by uttering ‘‘I believe it is raining,’’ but I do not (always) assert it is raining by asserting I believe it is raining. Furthermore, a string of phonemes is not itself a bearer of truth-value, since propositions have that property exclusively. As a result, the string of phonemes that, when uttered, express a bearer of truth-value is not itself a bearer of truthvalue, but only a vehicle by which a bearer of truth-value is expressed. The lesson is that if we wish to consider operators that are both factive and distributive, we will not be able to appeal to the utterance operator, but only to the assertion operator, since only the latter governs items capable of being true or false. Suppose then that we focus on the assertion operator. Once we notice this difference between uttering and asserting, the claim that asserting has the distributive property is in doubt. When I utter ‘‘p and q’’ I clearly utter ‘‘p’’ and I clearly utter ‘‘q’’. But when I assert p and q, do I assert p? Well, when I assert I believe p, I don’t assert p, so if you think that assertion distributes across conjunction, what’s the difference? Once the question is formulated, the answer is obvious: the difference is that I’ve logically committed myself to p by asserting p and q. So, in order to preserve the distributive character of assertion, we have to take the concept of asserting to include not simply what propositions one expresses by uttering a string of phonemes that make up a simple declarative sentence. We’ll also have to count you as asserting at least some claims to which you are logically committed in virtue of the assertions you make by saying declarative sentences. Which ones? Trying to sort among logical consequences leads to enough of a mess that the road most easily traveled takes us to the point of including all of them.

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That’s a mistake. I can assert that Fermat’s last theorem is unprovable even though everything I assert logically commits me to the truth of that theorem (since it has been proven). Antirealists are wont to appeal to idealizations, so it wouldn’t be surprising to ﬁnd some appealing here to the concept of what is assertible by an ideal rational agent, substituting for the concept of assertion the concept of what a logically omniscient being is committed to in virtue of what s/he asserts. Too much idealizing, methinks. The being would have to be quite unlike us, capable of knowing an uncountably inﬁnite number of things and propositions with uncountably inﬁnite components. If we want to speak of God here, theists like myself will have no problem with the discourse, but to think of such a being in terms of some ﬁnite extension of our own abilities and capacities is intolerable. These same points hold for the concept of what is written. Everything written is inscribed, but sometimes only a string of morphemes is inscribed and sometimes the writing expresses a proposition as well. In my terminology to inscribe a sentence is the scribal form of uttering a string of phonemes, and writing relates to a proposition in scribal form in the way asserting relates in a vocal form to a proposition. As before, we’ll have the same reasons to focus on the concept of writing rather than inscribing, since what is inscribed is not itself a bearer of truth-value; but when we consider the writing operator W, we ﬁnd that it is distributive only if the operator includes a reference to the logical consequences of what is written, and then the operator is not that of writing. For it is one thing to write down a claim, and it is another thing for what one has written to commit one logically to some further claim. This problem about the W operator is not likely to detain the proof-theoretician for long. For one thing, there is no reason we can’t interpret the W operator as ‘‘logically implied by what is written.’’ Such an operator would purportedly show the falsity of the intuitive idea that anything true can be logically implied by something written truthfully on my blackboard. Even so, there are costs to the syntactic generalization strategy. The W operator, on this understanding, is now logically complex, requiring reference to some correct logic for its interpretation. By contrast, in the knowability paradox, the rules of inference are intrinsic to the formal shorthand for the ordinary concept of knowledge. The more complex the operator, the more tempting it is to attribute the perplexing result of the proof to the complexity of the operator and the difﬁculty in processing this complexity. That is, the temptation is to treat it like we do barber sentences (‘‘there is a barber who shaves all and only them who do not shave themselves’’): once we see the implications, we relieve our perplexity by simply reminding ourselves of the logical complexity of the sentence, so that the appearance of possibility is misleading. Interpreting the W operator in this complex way suggests the same kind of response, but it is a response that is not appropriate for the knowability paradox. Given these disparate reactions to two proofs that can be formally represented in the same way, it is not clear that

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sameness of formal representation has any power here to relieve our perplexity at the particular lost logical distinction resulting from Fitch’s proof. This last claim raises a more general point about such a syntactic generalization strategy of ﬁnding analogues of knowability by searching for operators that mimic the distributivity and factivity features upon which Fitch’s proof relies. To see the issue, note that we can generalize the form of Fitch’s proof with a number of operators, such as the ‘‘true belief’’ operator, the ‘‘truly wished for’’ operator, the ‘‘truly imagined’’ operator, the ‘‘truly desired’’ operator, etc. In each case, the existence of a thing not truly X-ed will purportedly be incompatible with the idea that any truth can be truly X-ed. Note here that we don’t need to idealize to some logically omniscient X-er nor do we need to talk of the logical implications of what is X-ed in order to ﬁnd an analogue of the knowability results, so no distinction between the arguments is naturally brought to mind by the degree of complexity of one operator over another. Does this variety of operators somehow explain the paradoxicality of a lost logical distinction between possible and actual universally known truth? I can’t see why. A more plausible response to such generalizing is simply to characterize the more general paradoxicality in question. Instead of saying that there’s a knowability paradox, we’d say instead that there is a paradox about any mental state operator that is factive and distributive. Syntax generalized is just paradox generalized, not paradox lost. To generate paradox lost, my preferred explanation would be semantic. Since I’ve given no argument that no other explanation is possible, it is worth considering what other non-semantic approaches might look like besides the syntactic generalization strategy just rejected. A pragmatic explanation might appeal to the notion of structural interference, claiming that the process of X-ing when applied to a conjunction can cause problems since in X-ing the ﬁrst conjunct, one may affect the truth-value of the second.¹¹ Well, not quite, since the claim in question is an eternal truth if a truth at all (it quantiﬁes over all individuals who X and all times), so a more careful claim would be that the X-ing of the ﬁrst conjunct entails the falsity of the second. These claims may be correct, but I have some questions about it. In order to carry the explanation through, one will have to distinguish, in the case of the knowability paradox, between the executability of the basic steps of a procedure for generating knowledge (say, for knowing a conjunction) and the executability of the entire procedure itself, holding that only the basic steps are required to be executable, leaving unanswered why the basic steps are assumed to be executable.¹² Surely more is required here than simply assuming that these steps are executable. ¹¹ For the latter perspective on the paradox, Michael Hand (2003). ¹² See Hand (2003). Hand uses the analogy of recursion theory to show how a formal system can be developed by informal reference to procedures that might be executed, when the formal construction itself should not be thought to require such. Carrying the analogy through would lead to the conclusion that not even the basic steps of a veriﬁcation procedure need to be thought of as executable.

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This point doesn’t by itself cast doubt on this pragmatic approach to the paradox, however, so let’s assume that this problem can be overcome. Deeper problems can be seen, though, if we simply attend to what the account is good at explaining and what it is not. Put pithily, it is good at explaining why antirealists shouldn’t say that all truths are knowable, and it is not good at explaining the lost logical distinction expressed by (LD∗ ). It is clearly designed for the former purpose. Given this approach, one should deny that all truths are knowable by noting that those with antirealist sympathies should only go so far as to say that all truths for which structural interference is not an issue are knowable. To go further is to risk refutation by Fitch’s proof. The notion of structural interference is not designed to explain the lost logical distinction at the heart of the knowability paradox, however, and it is not very successful when turned to that purpose. To see the problem, I want to compare the omniscience-like conclusion of Fitch’s proof with a more famous claim of the same sort, the claim that God exists. One’s intuitive, pre-philosophical attitude toward this claim should be that it is a contingent matter whether there is a God (just as it should be contingent whether all truths are known). There is a plausible path of reasoning to the denial of the contingency claim, however. It begins by claiming that God is the most perfect being, that He exempliﬁes maximal greatness. We thus identify the claim that God exists with the claim that maximal greatness is exempliﬁed. The ﬁnal steps toward a denial of the contingency assumption about God’s existence is to clarify what maximal greatness involves (it is to display the maximal amount of any great-making property that has an intrinsic maximum) and argue that modal stability is itself a great-making property whose intrinsic maximum is existence in all possible worlds. There are two quite natural responses to this threat to the contingency of the theistic claim. The ﬁrst is to question the proof itself, to question the implications of the concept of maximal greatness, especially to doubt whether maximal modal stability is itself a great-making property. In doing so, one may look for analogues of the property, or one may simply construct formal notions that are claimed to have the modal stability property of being necessarily instantiated if instantiated. This strategy has a long history of threatening the ontological argument, from Gaunilo’s perfect island to Arnauld’s existent lion. This approach is like the syntactic generalization strategy rejected earlier. It is, however, a more promising approach here, since the examples used do not simply mimic the problematic proof, but constitute reductios of it. What they show is that the proof contains a mistake, even if we cannot identify exactly where the mistake occurs, thereby reafﬁrming our intuitive sense of the contingency of the theistic hypothesis. This ﬁrst response to the ontological argument thus could be used only to try to resurrect the syntactic generalization strategy, not to make sense of the pragmatic explanation relying on the notion of structural interference. Since we are now only considering the latter issue, we should move past this ﬁrst response

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to look at the second kind of response. This response questions the account of the theistic claim itself. Why should we think that the claim that God exists is logically equivalent to the claim that maximal greatness is exempliﬁed? After all, it’s not as if the meaning (sense) of the term ‘God’ is the same as the meaning (sense) of the term ‘maximally great being.’ Seeing what defenders of the argument do at this point shows why the pragmatic approach to the paradox is unsatisfying. Defenders of the ontological argument sometimes simply stipulate an understanding of ‘God’ in terms of maximal greatness. Such an approach leaves untouched the intuitive sense of the contingency of the theistic hypothesis, and thus provides no useful model for the pragmatic approach to the knowability paradox to emulate. What is needed instead is some way of explaining away some apparent contingency. An alternative to this stipulation approach to the issue tries to argue against the contingency claim. The argument takes the form of a reductio of the denial of contingency, beginning with the supposition that there is a God and also that there is a distinct being greater than God. The argument then proceeds by asking what understanding of God one might have that would call for allegiance, or worship, or religious commitment to the lesser being. This line of argument could be resisted by insisting that a proper conception of God has no religious signiﬁcance whatsoever,¹³ but that escape route will strike most as fairly extreme. My point here, however, is not to defend this approach, but to show it can be used to explain away the apparent contingency of the theistic hypothesis. The apparent contingency is resisted by pointing out that we think in these terms because there is nothing about the meaning or sense of the claim that God exists that yields a denial of contingency, and yet there is an argument for this conclusion. Put in the language of the analytic/synthetic distinction, the claim is not necessary because analytic, but it is necessary nonetheless. The important point to note is that the argument for this claim is not simply the original argument for the necessity of a maximally great being. Applying this point to the context of the knowability claim, we can see that the pragmatic approach to Fitch’s proof in terms of structural interference is not plausibly taken at all as providing an analogue of this way of defending the necessity of the claim that God exists. To function in the same way, the pragmatic approach would need to provide a basis to argue, independently of Fitch’s proof, that the omniscience-like conclusion of that proof is necessarily false. Even the most superﬁcial understanding of this approach shows that it would be complete pretense to assert that it can provide such an argument. Once we appreciate the design plan of the structural interference approach, we can see why that approach seems so irrelevant to the lost logical distinction in question. The reason is that it wasn’t designed to answer the question of how ¹³ See, e.g., Richard Taylor (1982). Taylor endorses arguments for the existence of God, but takes this result to have no religious signiﬁcance at all.

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there could be such a loss. It was designed to answer the question of whether an epistemic conception of truth requires afﬁrming that all truths are knowable. Looked at from this perspective, it is a powerfully promising idea. It holds out the promise of being able to explain why ‘‘I don’t exist’’ can’t be veriﬁed even though it might be true, and how ‘‘no thinkers exist’’ can’t be conﬁrmed even though possible. Such conclusions are essential to an adequate defense of semantic antirealism, and it is a mark in favor of the philosophical fecundity of the idea of structural interference that it blocks these problems for antirealism while at the same time showing why Fitch’s proof fails to refute the view. The lesson to learn here is that it is sometimes best to let tools be used for what they were intended, rather than to try to force them to accomplish a job for which they were not intended. So independently of any use to which antirealists might put the concept of structural interference, we ﬁrst need a solution to the paradox itself.

Conclusion In short, the paradox should disturb us all, antirealists and realists alike. It is true that the difference between a paradox and a merely surprising logical result is often not a difference in kind but only a difference in degree. Even so, there are distinctive marks of each that we look for when assessing what kind of a result we have achieved. If, for example, the result is merely one that we had no reason to think was true, we should classify such a result as a surprise. Or, again, if the result is merely one that threatens a particular philosophical perspective, such as antirealism, we should still classify the result as merely surprising. But when the result threatens some aspect of received opinion, especially received opinion on logical matters themselves, we should not classify the result as merely surprising. In the present case, the lost logical distinction is part of a ﬁrmly entrenched understanding of the nature of the modalities of necessity, possibility, and actuality. It is not a partisan distinction that only certain philosophical perspectives could endorse, and in this way, it is paradoxical to face a derivation that undermines the distinction, in the same way it is paradoxical to be told that two grains of sand constitute a heap or that motion is impossible. The perplexity engendered by Fitch’s proof is paradoxical, and the paradox cannot be addressed either by embracing Fitch’s proof as a refutation of antirealism or by ﬁnding a version of antirealism that involves no commitment to the knowability claim itself. What we need is either an explanation of the failure of Fitch’s proof or an explanation of the lost logical distinction between actuality and possibility that it implies—nothing short of that constitutes a proper philosophical response to the paradox.

14 Revamping the Restriction Strategy Neil Tennant

Ab s t r a c t This study continues the anti-realist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the anti-realist’s knowability principle ‘ϕ , therefore ♦K ϕ ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of ϕ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before applying the proposed restriction. A similar restriction is proposed for the converse inference, the so-called Rule of Factiveness ‘♦K ϕ therefore ϕ ’. The proposed restriction appears to block another Fitch-style derivation that uses the KK -thesis in order to get around the Cartesian restriction on applications of the knowability principle. Korean saying: Joong-i je meo-ri mot kkak-neun-da. Translation: A (buddhist) monk cannot shave his own hair.¹

1 . In t ro d u c t i o n In The Taming of The True a restriction was proposed on the anti-realist’s Knowability Principle, which can be expressed as a rule of inference in natural deduction as follows: ϕ ♦K ϕ This paper would not have been written without the stimulation, encouragement and criticism that I have enjoyed from Joseph Salerno, Salvatore Florio, Christina Moisa, Nicholaos Jones, and Patrick Reeder. ¹ Thanks to Sukjae Lee for the motto.

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It was proposed that this principle be limited, in its applications, to Cartesian propositions ϕ. A proposition ϕ is Cartesian just in case K ϕ ⊥. So the restricted Knowability Principle would be ϕ where K ϕ ⊥ ♦K ϕ This way of restricting the Knowability Principle may well be suspected of being overly ‘local.’ It might be advisable to have a more ‘global’ restriction. In general, a step of inference from ϕ to ♦K ϕ (with or without an extra condition on ϕ) takes place within a proof which will have some set undischarged assumptions: ϕ ♦K ϕ If we limit ourselves to the ‘local’ restriction, we ignore the contribution of the set of assumptions, focusing instead on the fact that, via , we have just reached the conclusion ϕ, whatever our starting point might have been: ϕ ♦K ϕ

where K ϕ ⊥

But if our grounds for ϕ are indeed , then the inferred possibility of knowing that ϕ surely presupposes the possibility of knowing that . Indeed, if it were impossible to know the joint truth of the assumptions in , how could one be conﬁdent in inferring from the intermediate conclusion ϕ to the knowability claim ♦K ϕ? These considerations lead to the thought that the restriction strategy, instead of looking down at ϕ within should rather look up at . The proposal, then, is that the restricted Knowability Principle should take the form of the following rule of inference, with a rather more exigent pre-condition for its applicability: Globally Restricted Knowability Principle where K ⊥ ϕ ♦K ϕ Here K is deﬁned in the usual Frobenian way as {K δ|δ ∈ }. When K ⊥, we shall say that is Cartesian. In logics whose relation of deducibility is not effectively decidable, the correctness of applications of the globally restricted rule is accordingly not effectively decidable. This is a modest price to pay, however, when one is concerned to avoid Fitch’s paradox.

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The main purpose of this study is to explain, investigate and defend this new proposed global restriction on applications of the Knowability Principle. A similar proposal will be made concerning its converse, the Rule of Factiveness for the compound operator ♦K : ♦K ϕ ϕ As will become clear from details that will emerge below, we need to restrict Factiveness too, in order to avert a different proof of Fitch’s paradox, which exploits the KK -thesis but does not fall foul of the global restriction on its application of the Knowability Principle.

1.1. A retraction, for the record The present author’s claim, made in Tennant (2000), at p. 829 and repeated in Tennant (2002), at p. 140, to the effect that ♦K ϕ is factive, was incautious. While it is valid so long as the sentence ϕ in question concerns only non-epistemic facts, the inference is not guaranteed always to preserve truth if we allow ϕ to contain occurrences of K (and adopt the KK -thesis). A related claim, however, still stands: to the extent that ♦K is factive, ♦ is not to be analyzed as the familiar alethic modal operator. Its contribution to truth- or assertability-conditions of sentences in which it is preﬁxed to K will have to be elucidated in terms of possibilities of investigative outcomes, at future times, within the actual world. Those possibilities will be strongly constrained by relevant contingencies in the actual world. It is this feature of the possibilities adverted to within ♦K that make the use of an ordinary alethic ♦ inappropriate.

1.2. Global restrictions on rules of inference A ‘global’ restriction on a rule of natural deduction is one that imposes some pre-condition for applicability by adverting to syntactic features of the proof other than the forms of the sentences standing as the immediate premises, or as the conclusion, of applications of the rule. As soon as any ‘global’ restriction is proposed on a rule of natural deduction, the possibility arises that its strictures can be rendered toothless by applying other rules of inference in a roundabout fashion that creates an artiﬁcial deductive context that meets the pre-condition in the letter, but not in the spirit, of the proposed restriction. The most obvious way to do this is by constructing proofs that are not in normal form.² Thus the most obvious prophylactic ² Such proofs involve inferring a sentence as the conclusion of an application of an introduction rule, and then treating that sentence as the major premise for an application of the corresponding

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against such deductive chicanery is to insist that, when determining whether the pre-condition is met, the proof that is to result from the contemplated application of the restricted rule should be in normal form. With this said by way of foreshadowing, we shall defer detailed illustrative examples to their most natural points of entry below. 2 . T h e Ne w Re s t r i c t i o n o n K n ow a b i l i t y

2.1. How the new restriction works on Fitch’s original proof We recall the proof of the Fitch paradox as given in Tennant (1997), at pp. 260–1.³ (1) K (j ∧ ¬Kj) ¬K j (∧I ) j ( K ) j ∧ ¬K j ⊥ (1) ( ⊥ ) K (j∧ ¬Kj) ⊥ where the embedded proof is

(I ) K (j ∧ ¬Kj)

j ∧¬Kj ¬Kj ⊥

(1)

(K ∧) K (j∧ ¬Kj) Kj ⊥ (1)

The reader will easily verify that the new, global restriction blocks this proof of Fitch’s paradox at its application of the rule (♦K ). For the premise-set for that step is {ϕ , ¬K ϕ}. Hence K = K {ϕ , ¬K ϕ} = {K ϕ , K ¬K ϕ}. And the latter set is not Cartesian: K ¬K ϕ ¬K ϕ Kϕ ⊥ We see, then, that the global restriction can do the old work required of the local restriction. elimination rule. Such sentence-occurrences within a proof are called maximal, and their presence is what prevents the proof from being in normal form. By contrast, a proof in normal form is one that contains no maximal sentence occurrences. ³ Natural deductions will be set out in tree form below. The reader unfamiliar with this format for proofs is advised that with applications of so-called ‘discharge rules’ the parenthetically enclosed numeral ‘(i)’ has an occurrence labeling the step at which the indicated assumption-occurrences higher up at ‘leaf nodes’ of the sub-proof(s) are discharged by applying the rule in question. A discharged assumption no longer counts among the assumptions on which the conclusion of the newly created proof depends.

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2.2. How the new restriction works on a proof of Salerno The strengthened restriction also does important new work that the local restriction cannot do. The following is a short proof of a result brought to my attention by Joe Salerno (albeit using a different proof).⁴ Consider the following proof of ψ from the set of assumptions {p, ¬Kp}. Note that the application of (♦K ) uses the old, ‘local’, restriction. (2) Kp

¬Kp ⊥ (1) K (1) (2) K ( p ∧(Kp →K )) p K p →K p ∧ (K p →K ) K ( p ∧ (K p →K )) Kp →K Kp K K( p ∧( K p →K )) ⊥ p ∧(Kp →K ) K K( p∧ (K p →K )) (1) K :

Note that the step labeled (1) is an application of the following rule of inference in modal logic: (i)

( )

where

is the sole assumption of the subordinate proof

(i)

Note also that the ﬁnal step of the proof : ♦K ψ ψ is an application of the aforementioned Rule of Factiveness of ♦K . It is worth stressing that its application here would be correct even if it were subject to the restriction that ψ should not contain any occurrence of K . (We shall return to this rule later.) Given the pattern of occurrences of ψ within , we could evidently take any contingent atomic proposition q (not involving K ) and complete a new proof of Fitch’s paradox as follows: {p, ¬Kp} {p, ¬Kp}

[ψ/q] [ψ/¬q] q ¬q ⊥ ⁴ Personal communication. See also Brogaard and Salerno (2006).

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Remember that these two substitution instances of involve the old, ‘local,’ restriction on (♦K ). But those steps of (♦K ) will not go through when we impose the new, ‘global,’ restriction. For at the point where (♦K ) is applied, the subordinate proof of p ∧ (Kp → Kq) (resp., p ∧ (Kp → K ¬q)) has as its set of assumptions the non-Cartesian set {p, ¬Kp}.

2.3. A possible objection to the new restriction A possible objection to the new global restriction is that it is all too easy to comply with. The thought might be that one could still construct a Fitchian reductio by sneaking around the restriction to Cartesian in the Globally Restricted Knowability Principle. The trick would be to successively discharge all the members of a non-empty, non-Cartesian by means of terminal applications of →-introduction within the subordinate proof in the proof-schema below. One would thereby reduce to the empty set the set of assumptions of the resulting subordinate proof for globally restricted (♦K ); in which case the restriction in question—that the set of assumptions be Cartesian—would be trivially met: (1) (2) 1, 2,

(n−1)(n ) , n−1, n

j (1) →j 1 (2) 2 → ( 1 →j)) (n−1) ( 2 →( 1 →j)) ) (n ) )) n → ( n−1 →( ( 2 →( 1→j)) K ( n → ( n−1 → ( ( 2 →( 1→j)) ))) n−1 → (

The objector who takes this line this far will not, however, be able to press it much further. For now we see that any use of ♦K (δn → (δn−1 → (. . . (δ 2 → (δ 1 → ϕ ) ) . . .) ) ) as a premise for a rule that involves stripping away the preﬁx ♦K (or, ﬁrst stripping away ♦, and thereafter stripping away K ) will presumably result in the multiply nested conditional eventually being exposed. Therewith arises a need to assume δ 1 , . . . , δn in order to winkle out ϕ for whatever Fitchian mischief is up the objector’s sleeve—mischief which would have been blocked by the new global restriction before the currently contemplated stratagem of multiple →-introductions. Yet there is no guarantee that these forced extra assumptions δ 1 , . . . , δn will be of the required form (enjoying a dominant occurrence of

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or of K , say) that might be called for in the application of whatever rule(s) might have been used to strip away both ♦ and K .

2.4. The importance of normal forms for the new restriction As foreshadowed earlier, in order for the new global restriction to be effective, we must insist that it be applied only within the context of a proof in normal form. The reason for this is that by resorting to proofs that are not in normal form, one can ‘hide from view’ the full set that provides the genuine grounds for the possible knowledge-claim ϕ. Only in the context of a proof in normal form will all those grounds be displayed as undischarged assumptions on which the premise ϕ (for an application of the Knowability Principle) depends. An example illustrating this point would be the following, which I owe to Salvatore Florio (Florio, unpublished). It exploits the rule called (λ) in Tennant (2000), at p. 837:

K, Kj ( )

Kj

⊥

⊥

(i)

(i)

The proof using (λ) is as follows. (2) p∧¬Kp Kp ¬Kp (3) K (p∧(Kp→¬Kq)) ⊥ (3) (2) ¬Kq p∧¬Kp p∧(Kp →¬Kq) K (p∧(Kp→¬Kq)) (1) p Kp→¬Kq (4) Kp→¬Kq Kp Kq ¬Kq p∧(Kp →¬Kq) p∧(Kp→¬Kq) ∃j(j∧¬Kj) ∃j (j∧(Kj →¬Kq)) K(p∧(Kp →¬Kq)) ⊥ (2) (3) ( ) ∃j(j∧(Kj →¬Kq)) ⊥ (4) ⊥ (1)

This proof, absurdly, reduces an arbitrary proposition of the form Kq to absurdity, on the basis of there being an unknown truth. The proof, however, is not in normal form, for the major premise of the ﬁnal step of ∃-elimination had been derived two steps earlier as the conclusion of a step of ∃-introduction. By means of the resulting abnormality, this proof is able to harbor an application of the Knowability Principle without, apparently, violating the condition that the ultimate premises for that application should be Cartesian. This can be seen by normalizing the foregoing proof. We do so by applying the appropriate reduction procedure in order to get rid of the maximal occurrence of the sentence ∃ϕ (ϕ ∧ (K ϕ → ¬Kq) ):

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(3) p ∧ ¬Kp Kp ¬Kp (2) ⊥ K (p ∧(Kp → ¬Kq)) (2) (3) ¬Kp K (p ∧(Kp→¬Kq)) p∧(Kp →¬Kq) p ∧¬Kp (1) Kp→¬Kq Kp p Kp→¬Kq Kq ¬Kq (†) p∧(Kp →¬Kq) K(p∧(Kp →¬Kq)) ⊥ (2) ( ) ∃j(j ∧ ¬Kj) ⊥ (3) ⊥ (1)

We ﬁnd that in the normalized proof the Knowability Principle is, after all, being incorrectly applied at the step marked ( † ); for its ultimate premise (at this application) is p ∧ ¬Kp. And this, as we already know, is not Cartesian. It took conversion into normal form to detect this violation of the global restriction on the Knowability Principle. 3 . Re s t r i c t i n g t h e Ru l e o f Fa c t i ve n e s s It was pointed out earlier that the Rule of Factiveness ♦K ψ ψ could be restricted in its applications to sentences ψ that do not contain any occurrences of K . Call the Rule of Factiveness of ♦K , restricted in this way, F . The restriction in question is very well motivated for the anti-realist who is also an internalist about knowledge. For if, as some internalists do, one subscribes to the so-called KK -thesis: Kψ KK ψ an unrestricted Rule of Factiveness U F will instate the unwanted Fitch inference, even for Cartesian propositions ψ. For consider the following proof that ψ implies K ψ.⁵ ( K) :

K KK K KK K

(1) (KK ) (1) ( )

⁵ Compare Brogaard and Salerno’s proof of the KK-knowability paradox in Brogaard and Salerno (2002).

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Given this proof, one will be able, disastrously, to inﬂate possibility to actuality (for Cartesian propositions ψ),⁶ by means of the following proof . (2)

(2) Π:

K K

(2) ( )

( K) , i.e.

(1) K (K K ) K K K (1) ( ) KK K (2) ( ) K

The way to avoid this madness, without losing a proper grip on knowability, is to refuse to grant the unrestricted factiveness of ♦K . It should be possible to hold the KK -thesis without making every truth known (via ), and without inﬂating the possible truth of a Cartesian proposition to its actual truth (via ). Suppose one holds that a knower’s knowing is always reﬂectively accessible to the knower, so that if x knows that ϕ, then x knows that x knows that ϕ. It follows that if it is known that ϕ, then it is known that it is known that ϕ. That is, the KK -thesis holds. Suppose now that ψ is true. The anti-realist wants to say that it is possible, in principle, for someone to know that ψ. That envisaged in-principle possibility will, if and when actualized (say at time t), bring with it the knowledge (at t) that ψ is known (at t). So, it is possible also that it be known that ψ is known (i.e., ♦KK ψ). But does this intuitively imply that ψ is actually known, or will ever actually be known (i.e., K ψ)? Of course it does not!—for the envisaged possibility might never be actualized. There is a degree of serendipity in empirical (and even mathematical) inquiry, which even the anti-realist must recognize. So we cannot treat ♦K as reliably factive when applied to propositions, such as K ψ, that have K dominant. Our formal rules of inference must answer to our pre-theoretic intuitions. We cannot issue the carte blanche of the unrestricted rule F .

3.1. Two proposals for restricting Factiveness How, though, might F be restricted? Clearly, it is the rot within the proof that has to be stopped. The pre-theoretic intuitions mulled through above tell us that this much of is in order:

:

( K)

K K KK KK

(1) (KK ) (1) ( )

⁶ This was observed by Brogaard and Salerno in Brogaard and Salerno (2006).

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It is only the ﬁnal step of that should give us pause: ♦KK ψ ( UF ) Kψ We have envisaged circumstances in which one would be warranted in asserting the premise ♦KK ψ, without being warranted in asserting the conclusion K ψ. So: what is wrong with this application of the rule U F ? What might be the general defect exhibited? Could we restrict applications of the rule so as to avoid just those would-be applications that are defective in this way?

3.1.1. The ﬁrst proposal for restricting Factiveness A ﬁrst stab at the problem might be to insist that applications of the rule ♦K ϕ ϕ may be made only when the sentence ϕ is ‘about basic, non-epistemic facts.’ One could give here a myriad examples from the language of physics, mathematics, chemistry, biology etc., of sentences that are about basic, non-epistemic facts. The proposal would be that sentences like these could be substituends for ϕ in applications of the rule of factiveness. For such sentences, surely, the only reason why it might be possible to know that they are true is that they are indeed true. This is not the case, however, with ‘epistemic’ sentences θ. Here, there can be reason to hold it possible to know that the truth of θ is known, without this being a reason for holding that the truth of θ is indeed (or will ever be) known. Our inquiries might, by the heat-death of the universe, never have taken the turns required, even though, had we conducted our investigations otherwise, we could have come to know the truth of θ. Restriction to K -free ϕ would certainly block the ﬁnal step of , since it depends on taking K ψ as a substituend for ϕ in the statement of the rule. But this seems rather drastic as a proposed logical inoculation against the possibility of incurring -type rot.

3.1.2. A second proposal for restricting Factiveness Another way to formulate a restriction that would render the proof ill-formed (at its ﬁnal step) would be to observe that the premise for that would-be application of the Rule of Factiveness stands as the conclusion of the application, labeled (1), of the rule ♦ , in whose subordinate proof there had been an application of the rule KK : (1) K ( K) (KK ) K K (1) ( ) K KK K

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We can perhaps make more vivid the reason why one might wish to formulate the restriction this way. Let us identify the different occurrences of K by means of italics and boldface. The proof would then look like this: (1) K (K K ) ( K) K K K (1) ( ) KK K The idea would be that ♦K is not factive, whereas ♦K is. ♦K is not factive because K was introduced by an application of the rule (KK ).

3.1.3. The importance of normal form, again As with the global restriction on the Knowability Principle, this restriction will work only when we have ensured that the proof is in normal form. In order to illustrate this, let us remind ourselves how proofs not in normal form characteristically arise. They come from joining together two proofs, the conclusion of one of them being an undischarged assumption of the other. The occurrences of the sentence serving as such a ‘point of accumulation’ can thereby be maximal: a point of locally increased, and—given the overall context—unnecessary logical complexity. Such local complexity can be eliminated by applying a suitable reduction procedure. The result of applying the appropriate reduction procedure might well be a new proof in which other sentences now have maximal occurrences. But these will be of lower complexity than the original one. By repeatedly applying the appropriate reduction procedures, the proof will eventually be transformed into one in normal form. In the context of epistemic logic, here is a simple proof—call it —that appears to be entirely in order. It does not even sin against the newly proposed restriction on the Rule of Factiveness of ♦K (the one that is framed in terms of earlier applications of the rule ♦ ). (1) K (j ∧ ) Kj Φ: ( ) K (j ∧ ) (1) Kj j Below is another proof—call it . The reader ought to look ahead at in order to follow the explanatory comments about to be entered. In , the sentence ψ is indicated as having been proved outright, from the empty set of assumptions. Since we are idealizing the logical abilities of our knowers, we may accordingly infer K ψ—for a logical saint knows every logical theorem. In we also employ the rule Kθ Kχ K (θ ∧ χ)

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This rule can be justiﬁed, in its application within , by appeal to the more obvious rule aK θ aK χ aK (θ ∧ χ) which employs a logical form making explicit provision for the knower a. (Claims of the form K ϕ are really short for ∃x(xK ϕ ).) The latter rule, applied to the materials involved in , would mediate the inference aKK ϕ aK ψ aK (K ϕ ∧ ψ) —for, since any logical saint knows the truth of the theorem ψ, she will know also the conjunction θ ∧ ψ, for any truth θ that she knows. (Here, the truth θ takes the form K ϕ.) Here at long last is the proof . ∅ Ω:

(2) Kj j KKj K Kj K (Kj ∧ ) (2) ( ) K (Kj∧ )

Let us substitute K ϕ for ϕ within the proof , so as to obtain the following proof . (1) K(Kj ∧ ) KKj (1) Φ: ( ) K (Kj ∧ ) KKj Kj The time has come now to join together the proofs and . Note that the conclusion of is the undischarged assumption of . When we make that sentence ♦K (K ϕ ∧ ψ) the point of accumulation, we obtain the following proof, which is not in normal form, and which purports to establish the Fitch result ϕ K ϕ: ∅ (2) Kj K j K Kj K (K j ∧ Kj ) ( ) (2) ( ) K (Kj ∧ ) KKj Kj

K (Kj ∧ )

(1)

KKj (1)

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The advocate of restricting the Rule of Factiveness so that it yields only nonepistemic conclusions will already have objected that the substitution-instance of sins against this restriction. Moreover, he might (mistakenly, as it happens) complain that the other bruited restriction, banning applications of the rule KK within subordinate proofs for applications of the rule ♦ , does not work: witness the proof just formed, in which the latter restriction does not appear to be violated. Appearances, however, can be deceptive. Note that the proof in question is not in normal form. It turns out that, if we normalize it, it is transformed into a normal-form proof in which that second proposed restriction is violated. To see this, proceed as follows. First, apply the obvious reduction procedure to the proof just constructed, so as to get rid of the maximal occurrence of ♦K (K ϕ ∧ ψ). The result is ∅ (1)

Kj KKj K j K (K j ∧ ) KKj (1) Kj KKj Kj

in which there is a new maximal sentence occurrence, namely the occurrence of K (K ϕ ∧ ψ). Upon applying the appropriate reduction procedure in order to get rid of this occurrence, we obtain the following proof in normal form: (1) Kj ( K) j (KK ) Kj KKj j: (1) ( ) KKj Kj —which of course we had already faulted earlier, on the basis of our second proposed restriction on applications of the rule of factiveness.

3.1.4. Blocking an S4-like route to Fitch Consider the following purported proof , involving a Cartesian proposition ϕ:⁷ (1) j j Kj (1) -E Kj S4 Kj j ⁷ This proof is due to an anonymous referee.

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The topmost step, an application of the Knowability Principle, complies with the global restriction. The purported result is unacceptable: that every Cartesian proposition, if possible, is true. Indeed, so is the inference from ♦ϕ to the claim ♦K ϕ, the penultimate conclusion of the proof . This ‘proof’ highlights a point made in §1.1. There are two possibility operators at work, and they need to be distinguished. The one that is introduced by applications of the Knowability Principle adverts not to metaphysical possibilities (which may be contrary to actual fact), but rather to the possibility of an agent coming to know, in accordance with the contingent facts governing his own world, that a certain proposition is true. This epistemic possibility operator should accordingly be distinguished from the metaphysical possibility operator. We shall use and ♦ respectively. The ‘proof’ accordingly becomes (1) j j Kj (1) -E Ξ Kj S4 (??) Kj j in which the allegedly S4-like step is now clearly not valid. While the step ♦♦θ ♦θ is formally correct and valid for the metaphysical possibility operator ♦ of S4, and hence also its substitution instance ♦♦K ϕ ♦K ϕ is formally correct and valid, matters are very different with the step ♦

Kϕ Kϕ

By way of counterexample, consider the Cartesian proposition ‘Grass is purple’ as an instance of ϕ. It is metaphysically possible that grass be purple; but it happens, in our world, not to be. In any other possible world in which grass were purple, however, it would be possible (in that world) to know that it was purple. Hence such possible knowledge would also be a metaphysical possibility. That is, the premise ♦ K ϕ of the last displayed rule of inference, for this choice of ϕ, is true. Its conclusion K ϕ, however, is false—for grass’s being purple is not anything we could come to know in this world. Since the rule of inference in question is invalid, we can prohibit any application of it in proofs. The purported ‘proof’ is not a proof.

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3.1.5. Summary of discussion of how to restrict Factiveness We see, then, that there are still two ways of restricting the Rule of Factiveness, so as to avert the Fitch paradox in the presence of the KK -rule. The ﬁrst restriction is easy to apply: simply ban applications of the Rule of Factiveness that involve conclusions containing K . The second restriction, however, seems at this stage to be just as effective. But we need to bear in mind that it stops the rot only when we have converted the proof to be appraised into normal form. In this regard, the second restriction is like the global restriction proposed earlier for applications of the Knowability Principle. These applications, too, can be assessed for correctness only when the proof in question has been transformed into normal form. This is not the ﬁrst philosophical problem for which the technique of converting proofs into normal form has afforded useful insights. Dag Prawitz used normalization techniques to frame a fertile conception of intuitionistic consequence. (See Prawitz 1974 and Prawitz 1977.) Reduction procedures also form the centerpiece of Michael Dummett’s inferential theory of meaning, and his arguments in favor of intuitionistic logic as the right logic. (See Dummett 1977, the two famous essays ‘The Philosophical Basis of Intuitionistic Logic’ and ‘The Justiﬁcation of Deduction’ in Dummett 1978, and Dummett 1991b.) Normalization lies at the heart also of the present author’s characterization of relevance in deduction.⁸

4. Conclusion We have undertaken here only the most preliminary explorations of prooftheoretic measures designed to stave off certain threats of paradox in an anti-realist epistemic logic. We are still a long way, of course, from having a fully adequate proof-theory governing the interaction among ♦, and K (let alone a formal semantics, with respect to which one might be able to establish the soundness and completeness of whatever proof system is devised). The aim here has been to clear the way for an eventual proof (if such can be found) of a metatheorem to the effect that a suitable system of proof (in epistemic modal logic) embodying the globally restricted Knowability Principle is Fitch-free: that is, it affords no proof of ϕ from K ϕ. Establishing such a result, however, is beyond the scope of the present paper. This discussion has suggested a proof-theoretic path for the anti-realist to follow, without being Fitched. The way forward is to formulate the Cartesian restriction on the Knowability Principle by reference to the ultimate grounds that one could know for the truth of a proposition. As our discussion revealed, the ⁸ See Tennant, ‘‘Logic, Mathematics, and the Natural Sciences’’, in Dale Jacquette (ed.), Philosophy of Logic (Amsterdam: North-Holland, 2006), 1149–66.

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technique of normalizing proofs is crucial for detecting incorrect applications of the Knowability Principle. And the use of a natural deduction format affords us structural insights into proofs, and the resources by means of which one can frame some of the delicate but philosophically motivated restrictions that might be called for on applications of certain rules of inference. With an eye to such resources, other logical or epistemic principles, besides the Knowability Principle, can be adopted in more liberal or more exigent formulations, by way of cautious development of an anti-realist, epistemic logic. The guiding requirement will be that every truth be knowable, without implying that it need ever be known.

Pa r t V I Mod a l a n d Ma t h e m a t i c a l Fi c t i o n s

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15 On Keeping Blue Swans and Unknowable Facts at Bay: A Case Study on Fitch’s Paradox Berit Brogaard

In t ro d u c t i o n What has come to be called the knowability paradox was ﬁrst published by Frederic Fitch as Theorem 5 (1963: 139). It is equivalent to the claim that if every truth is knowable, then every truth is known: (T5) ϕ → ♦Kϕ ϕ → Kϕ where ♦ is possibility, and ‘‘Kϕ’’ is to be read as ‘‘ϕ is known by someone at some time’’. Let us call the premise the knowability principle and the conclusion near-omniscience.¹ Here is a way of formulating Fitch’s proof of (T5). Suppose the knowability principle is true. Then the following instance of it is true: (p & ∼Kp) → ♦K(p & ∼Kp). But the consequent is false, it is not possible to know p & ∼Kp. That is because the supposition that it is known is provably inconsistent.² The inconsistency requires us to deny the possibility of the supposition, yielding ∼♦K(p & ∼Kp). This, together with the above instance of the knowability principle, entails ∼(p & ∼Kp), which is (classically) equivalent to p → Kp. Since p occurs in none of our undischarged assumptions, we may generalize to get near-omniscience, ϕ → Kϕ. QED. (T5) is today considered by many to be a paradox for a number of related reasons, among others, that it threatens to show that the very thesis that is thought to discriminate a mature semantic anti-realism from a naïve idealism entails that I am indebted to Joe Salerno for invaluable discussion, and to Joe, John Divers, Michael Hand, David Jehle, Julien Murzi and two anonymous referees for Oxford University Press for helpful comments. ¹ I used to call the conclusion omniscience. But, of course, ϕ → Kϕ does not entail omniscience, i.e., that there is someone who knows all truths (∃∀), but only the weaker claim that all truths are known by someone (∀∃). Thanks to Michael Hand for suggesting that I rename it. ² If p & ∼Kp is known, then it is true, giving p & ∼Kp, and so ∼ Kp. Also, if p & ∼Kp is known, then the left conjunct is known, giving Kp.

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very idealism. A number of strategies have been developed to avert the paradox, and several of them have provoked signiﬁcant and interesting debate.³ What has rarely, if ever, been noted, however, is that Fitch-like paradoxes threaten to undermine not only semantic anti-realism, but also potentially a number of other anti-realisms with superﬁcial resemblance to semantic anti-realism. One form of anti-realism that is troubled by a Fitch-like paradox is what has come to be called strong modal ﬁctionalism.⁴ Strong modal ﬁctionalists hold that possible world talk, like literary ﬁction, is literally false. Nonetheless, they think the ﬁction provides the resources for an analysis of modal claims. In this note I develop a Fitch-like paradox for strong modal ﬁctionalism. I argue that the most promising strategy to avoid paradox is to reject the claim that modal claims are to be analyzed in terms of the contents of the ﬁction of possible worlds. It is hoped that by looking at the parallel case of modal ﬁctionalism light can be shed on the threat posed by Fitch’s paradox to semantic anti-realism. Mo d a l Fi c t i o n a l i s m Since Gideon Rosen’s centerpiece of 1990, modal ﬁctionalism has been taken seriously by many as a way to employ the resources of possible-world semantics without any of the usual ontological commitments. Modal ﬁctionalism holds that possible world talk is to be treated on a par with talk of ﬁctional objects, such as Sherlock Holmes. Like talk of ﬁctional objects, possible world talk is literally false (or untrue).⁵ It is literally false that there is a brilliant detective at 221b Baker Street. Likewise, modal ﬁctionalists say, it is literally false that there are merely possible worlds, and merely possible objects. Thus, while there might have been blue swans, there is no possible world where there are blue swans. What distinguishes ﬁctionalists from eliminativists is that ﬁctionalists hold that modal claims can be explicated in terms of possible worlds, as long as quantiﬁcation over possible worlds occurs within the scope of an implicit story preﬁx (e.g., ‘‘according to the possible worlds ﬁction’’). Quantiﬁcation within the scope of a story preﬁx is, familiarly, not existentially committing. The content of the possible worlds ﬁction is standardly taken to be David Lewis’s theory of possible worlds (Lewis 1968), including an encyclopedia, that is, a list of all literally true non-modal propositions. Following Rosen, p is a non-modal proposition just in case ‘‘it contains no modal vocabulary and entails neither the existence nor the non-existence of things outside our universe’’ (Rosen 1990: 335). ³ For an overview of the literature, see Brogaard and Salerno (2004). ⁴ This position originates in Rosen (1990). Rosen also considers an alternative position which he calls ‘‘timid modal realism.’’ As we will see, this position is salvageable from paradox (or at least, salvageable from this paradox). For a rich overview of the literature, see Nolan (2002). ⁵ More carefully: possible world talk that incurs a commitment to merely possible worlds or merely possible individuals is literally false (or untrue).

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Where p is a sentence of quantiﬁed modal logic, p∗ is a translation of p into the language of possible worlds, and W is a story preﬁx which reads: according to the Lewis story, Rosen’s formulation of modal ﬁctionalism may be given as follows: (Fic) p ↔ Wp∗ (Fic) says that a modal claim will be true iff its translation into the language of possible worlds is true in the Lewis story. So, for example, ‘‘there might have been blue swans’’ is true iff, according to the Lewis story, there is a possible world where there are blue swans. Likewise, ‘‘there are no blue swans’’ is true iff according to the Lewis story, in the actual world, there are no blue swans. A problem for modal ﬁctionalism was noticed independently by Rosen (1993) and Stuart Brock (1993).⁶ In a nutshell, it is that since in the Lewis story it is true at each world that there exists a plurality of worlds, we can derive, by (Fic), that necessarily there is a plurality of worlds. Since necessity entails truth, there in fact exists a plurality of worlds, which is not something the ﬁctionalist should tolerate. However, the objection has been shown to be unproblematic if careful attention is paid to the translation scheme offered by Lewis in 1968 for translating sentences in the language of quantiﬁed modal logic into sentences in the language of counterpart theory.⁷ In the 1968 translation scheme, the sentences of quantiﬁed modal logic translate into sentences that quantify over worlds and their parts. Thus, where ‘‘U’’ means world, and I is the existing-wholly-in (or parthood) relation, a modal sentence of the form: ♦∃xFx translates as: ∃x∃y(Uy & Ixy & Fx) This says that there is a world of which something that is F is part. Assuming the letter of the 1968 translation scheme, ‘‘necessarily, there is a plurality of worlds’’ translates as: ∀x(Ux → ∃y∃z(Iyx & Izx & Uy & Uz & y = z)) This says that all worlds have at least two worlds as parts. But this is false, according to the Lewis story, since no world has any other world as a part. Hence, the objection fails. However, too careful attention to the 1968 scheme leads to Hale’s dilemma (see Hale 1995).⁸ The ﬁctionalist cannot say that modal realism is possible. For ⁶ For a much more substantial treatment of the history of the debate and alternative solutions to problems, see Nolan (2002). ⁷ The 1968 translation scheme is the one found in Lewis (1968) [1983]. For discussion, see also Nolan (2002). ⁸ I am simplifying the ﬁrst horn of the dilemma.

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assuming the letter of the 1968 translation scheme ‘‘it is possible that there is a plurality of worlds’’ translates as: ∃x(Ux &∃y∃z(Iyx & Izx & Uy & Uz & y = z)) This says that some world has at least two worlds as parts. But this is false, according to the Lewis story. Nor can he say that modal realism is impossible. For, Hale argues, the ﬁctionalist must provide some analysis of the story preﬁx ‘‘according to the Lewis story.’’ Most plausibly this explication will involve a non-material conditional of the form: if modal realism is true, then p. But if it is necessarily false that there is a plurality of worlds, then this conditional will be trivial; ‘‘according to the Lewis story, p’’ will be true for any p. Hence, the ﬁctionalist will be committed to the truth of any modal claim.⁹ The same problem arises if the ﬁctionalist holds that the story preﬁx is primitive. The ﬁctionalist ought to accept some closure principle of the form, Wp, p ⇒ q Wq. But where p is impossible (e.g. ‘‘there is a plurality of worlds’’), p entails any claim. So, Wq obtains for any q. If q is ‘‘there is no world where there are blue swans,’’ we can derive (by Fic), the far-fetched claim ‘‘ ∼♦(there are blue swans)’’. John Divers has offered the following resolution of Hale’s dilemma (Divers 1992). When the realist assents to ‘‘there is a plurality of worlds’’ his intention is that the ‘‘conventional world-restriction of quantiﬁcation should not apply’’ (Divers 1999a: 323; see also Divers 1999b). But, Divers argues, if restricted to alethic modality, the T axiom, p ♦p, ‘‘approaches the status of analyticity’(1999b: 218). Hence, the realist must assent to ‘‘it is possible that there is plurality of worlds.’’ In such cases, Divers argues, the operand modality must be read as redundant. That is, where p is unrestricted, the realist must assent to ♦p p.¹⁰ Where the realist translates the unrestricted possibility claim ♦p as p, the ﬁctionalist will thus do well to translate it as Wp. That is, the ﬁctionalist can assent to the following instance of (Fic): If p is read as an unrestricted possibility claim, then (♦p ↔ Wp) Since the ﬁctionalist is thus able to assent to the contingent falsehood of modal realism, Hale’s dilemma is blocked. ⁹ The suggestion can be found in Rosen (1995). Rosen also suggests that the ﬁctionalist could block Hale’s dilemma by, for example, rejecting ‘‘classical’’ semantics for non-material conditionals, or allowing truth-values to the statement involving the ‘‘world’’ predicate only when the predicate occurs within the scope of the preﬁx. Hale (1995) thinks these suggestions are desperate. Disallowing truth-values to the statements involving ‘‘free’’ occurrences of the ‘‘world’’ predicate would leave claims like ‘‘there is exactly one world, namely the actual’’ without a truth-value. Rejecting the claim that counter-possible conditionals are all trivially true is plausible for conditionals with metaphysically impossible but logically possible antecedents. But Rosen’s suggestion is the more radical one that even counter-possible conditionals with logically impossible antecedents may fail to be trivially true. For further problems with these strategies, see Divers and Hagen (2006). ¹⁰ p p trivially follows. S5 and ♦p p yields the trivial system (Tr), in which no signiﬁcant modal distinctions can be drawn.

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There is, however, a different path to disaster even assuming Divers’s counterpart-theoretic translation principles for possibility claims.

A Fi t c h - l i k e Pa r a d o x A collapse ensues owing to a Fitch-like proof of the following theorem (where p is an unrestricted locution):¹¹ (T2) p → Wp∗ (Wp → p) Like the logic of Fitch’s proof, the logic of the Fitch-like proof of T2 is modest: minimal, normal modal logic, and three intuitive story-preﬁx principles: (A) (B) (C) (D)

W(p & q) Wp & Wq WWp Wp Wp ∼W∼p ∼♦p ∼p

(A) is a distributivity principle that says that if in the Lewis story, p and q, then in the Lewis story, p, and in the Lewis story, q.¹² (B) says that if the Lewis story states that according to the Lewis story p, then the Lewis story states that p. (C) states: if in the Lewis story, p, then it is not the case that in the Lewis story, not-p. (D) is the inference of a necessary falsehood from an impossibility. If these resources are not already believable, we can say this. (A) is entailed by the uncontroversial assumption that conjunction in the Lewis story is classical. Denying (B) would yield the implausible result that the Lewis story may deny p yet assert about itself that it holds that p (certainly, Lewis would have disapproved of such a theory).¹³ (C) follows from the uncontroversial assumption that modal realism is classically consistent. (D) follows from the duality of the modal operators. The proof employs a theorem derived from these resources, Theorem T1—viz., for any p, it is not the case that according to the Lewis story, both not-p, and according to the Lewis story, p. (T1) ∼W(∼ p & Wp) ¹¹ The star is not required on the right-hand side of (T2). Starred statements are translations of the statements of quantiﬁed modal logic into statements of counterpart theory. But, given Divers’s translation scheme, unrestricted locutions of the form ‘‘♦p’’ translate as ‘‘p.’’ So, the star drops off. Thanks to an anonymous referee here. ¹² Of course, to avoid obvious counterexamples, we will ultimately need an account of tense operators occurring within the scope of a story preﬁx. Thanks to David Jehle here. ¹³ Notice, further, that (B) is validated on Divers’s (1999a) explication of the story preﬁx: ((Wp) ↔ (the Lewis story → p)), where the box is primitive. For if (the Lewis story → (the Lewis story → p)), then (the Lewis story → p). It is also validated on the following subjunctive explication of the story preﬁx: Wp ↔ if the Lewis story were true, p would be true. For if (if the Lewis story were true, then if the Lewis story were true, then p would be true), then (if the Lewis story were true, then p would be true).

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The proof of (T1) runs as follows. Suppose for reductio that W(∼p & Wp). Then, by (A), W∼p & WWp. The right conjunct, by (B), entails Wp. This, by (C), entails ∼W∼p. So, we derive: W∼p & ∼W∼p. Contradiction. Rejecting our assumption, by reductio, gives us ∼W(∼p & Wp). QED. Where p is an unrestricted locution (e.g. ‘‘there is a plurality of worlds’’), a Fitch-like proof of (T2) may be developed as follows:¹⁴ (1) (2) (3) (4) (5) (6)

∀p(p → Wp∗ ) ♦(∼p & Wp) → W(∼p & Wp) ∼W(∼p & Wp) ∼♦(∼p & Wp) ∼(∼p & Wp) (Wp → p)

from (Fic), left to right from 1, Divers instance of T1 from 2, 3 from 4, D from 5

We suppose at (1) that, for any unrestricted modal or non-modal locution p, if p is true, then according to the Lewis story, the translation of p into the language of counterpart theory obtains. (2) substitutes the unrestricted possibility claim, ♦(∼p & Wp), for p in (1). Following Divers’s realist translation schema for unrestricted claims, ♦p p, the modal realist translates ♦(∼p & Wp) as ∼p & Wp, and the ﬁctionalist translates it as W(∼p & Wp). Line (3) is an instance of the theorem, (T1). Line (4) follows trivially from lines (2) and (3). By (D), we derive line (5) from line (4). In classical logic line (6) is equivalent to line (5). Therefore, if ﬁctionalism is true, then every unrestricted claim that is true according to the Lewis story is true: T2 p → Wp∗ (Wp → p).¹⁵ According to the Lewis story, there is a plurality of worlds. By T2 and (Fic), we can derive that there is a plurality of worlds. This, of course, should not be acceptable to the ﬁctionalist.

K e e p i n g Re a l i s m a t Ba y How will the ﬁctionalist respond? Well, there is, of course, always the option of denying our initial logical resources. I will not rule out that this can be done in a principled manner. Assuming, however, that no such route is available to the ﬁctionalist, how can he avoid paradox? In this section I will look at four of the main strategies that have been employed in order to avoid Fitch’s original ¹⁴ In line 2 the star drops off as the result of applying Divers’s translation scheme for unrestricted locutions to an unrestricted locution. Thanks to an anonymous referee here. ¹⁵ Divers and Hagen (2006) think Divers’s solution to the problem of unrestricted claims is undermined because the T-theorem (i.e., truth entails possibility) holds for unrestricted claims. But this assumption is controversial. Lewis explicitly denies it (see 1968[1983]: 39–40). My argument does not rest on this assumption. Of course, it may be thought that if we deny the T-theorem for unrestricted claims, then the problem of possible unrestricted claims does not arise. But this is not so. For the ﬁctionalist wants to say that unrestricted claims are false in spite of being possible. So, for them, the T-theorem does not play a role in arguing for the possibility of an unrestricted claim.

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result, and try to determine whether the ﬁctionalist can avail himself of similar strategies.

The Intuitionistic Strategy Since Fitch’s proof is classically, but not intuitionistically, valid, the paradox can be avoided by rejecting classical logic.¹⁶ Like Fitch’s proof, the proof of T2 is classically, but not intuitionistically, valid. Without classical logic we cannot derive line (6) from line (5). An intuitionist, however, is committed to Wp →∼ ∼p. This entails: ∼p → ∼Wp. But the latter is evidently absurd from a ﬁctionalist stance. It reads: if it is not the case that p, then it is not the case that according to the Lewis story, p. So, if it fails to be true that there is a plurality of worlds, as the ﬁctionalist claims, then it is not the case that according to the Lewis story, there is a plurality of worlds.¹⁷

The Modal Fallacies Strategy Another important strategy that has provoked signiﬁcant and interesting debate is that offered by Jon Kvanvig (1995, and forthcoming). Kvanvig argues that Fitch’s result is invalid, owing to a fallacious substitution into a modal context. The problem, Kvanvig says, is that Kp is implicitly quantiﬁed. Explicitly it reads ∃x∃t(Kxpt), which says that there is someone x and a time t such that x knows at t that p. But, on Kvanvig’s neo-Russellian account of quantiﬁed expressions, quantiﬁed expressions cannot, in general, be legitimately substituted into modal contexts, hence, the failure of the substitution of the Fitch conjunction, p & ∼Kp, into the knowability principle. Kvanvig’s solution has been criticized on various fronts (see, e.g., Williamson 2000b: Chapter 12; Brogaard and Salerno 2007; Jenkins, Ch. 18 of this volume). However, even if it succeeds, the ﬁctionalist cannot avail himself of Kvanvig’s strategy. For the whole purpose of strong modal ﬁctionalism is to be able to analyze the sentences of quantiﬁed modal logic which, familiarly, allows for substitution into modal contexts. A closely related but equally unsuccessful strategy is this. For the Fitch-like proof to go through it is crucial that ♦(∼p & Wp) is an unrestricted possibility claim. So, might not the ﬁctionalist simply deny that ♦(∼p & Wp) can be read as an unrestricted possibility claim? Unfortunately, this is not an option. For the ﬁctionalist is prepared to say that it is possible that both ∼(there is a plurality of worlds), and according to the Lewis story, there is a plurality of worlds. That is, he is prepared to say: ♦(∼p & Wp). However, on a restricted reading, ¹⁶ For discussion of the intuitionistic strategy, see e.g. Williamson (1982); Wright (1993a [1987]). ¹⁷ Parallel reasons have been given for rejecting the intuitionist solution to Fitch’s paradox. See e.g. Percival (1990).

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♦(∼p & Wp) cashes out to the obviously false claim: ‘‘For some world w, w does not have a plurality of worlds as part, but according to the Lewis story, it does.’’ Thus, the substitutional fallacy move is unsuccessful.

The Rigidiﬁer Strategy A third strategy that has provoked signiﬁcant debate is that of Dorothy Edgington (1985). Edgington’s strategy is to bypass the knowability principle altogether. Instead, she requires of knowability the less general thesis: (AKP) Ap → ♦KAp where ‘‘Ap’’ is to be read ‘‘it is actually the case that p.’’ Since KA(p & ∼Kp) is not provably inconsistent, this strategy avoids paradox. The ﬁctionalist might, similarly, propose to replace ﬁctionalism with the following weaker thesis: (FicA) Ap ↔ WAp∗ There is, however, little reason to think the ﬁctionalist would want to do that. For if he accepts p Ap, and WAp Wp, which we can reasonably expect, then (FicA) entails p → Wp∗ . This is all we need to get the Fitch-like proof going.

The Restriction Strategy A more recent and widely discussed strategy to block Fitch’s original paradox is to restrict the universal quantiﬁer in ‘‘all truths are knowable.’’ Neil Tennant, for instance, favors what he calls the ‘‘Cartesian’’ restriction (Tennant 1997: 274).¹⁸ A proposition ‘‘p’’ is Cartesian just in case ‘‘Kp’’ is not provably inconsistent. Tennant’s Cartesian knowability principle may be stated thus: all Cartesian truths are knowable. (CKP) p → ♦Kp, where p is Cartesian. It should be apparent that the Cartesian restriction blocks Fitch’s paradox, since Fitch’s result requires the substitution ‘‘p & ∼Kp’’ for ‘‘p’’ in ‘‘p → ♦Kp’’. ‘‘p & ∼Kp’’ is not Cartesian, as K(p & ∼Kp) is logically impossible. The ﬁctionalist’s best option may be to follow Tennant’s lead and reformulate the ﬁctionalist principle as follows: (Fic∗ ) p ↔ Wp∗ , where Wp* is not provably inconsistent Provided that Wp∗ is provably inconsistent only when p is an unrestricted claim, we can offer the following more effective formulation of ﬁctionalism. (Fic∗∗ ) p ↔ Wp∗ , where the quantiﬁers in p are restricted to worlds. ¹⁸ A related proposal can be found in Dummett (2001).

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We, furthermore, propose that the ﬁctionalist treat the unrestricted modal operators as primitive S5 operators.¹⁹ On a non-redundancy interpretation of the unrestricted modal operators, the possibility of modal realism does not entail its truth. The ﬁctionalist can thus coherently deny ‘‘there is a plurality of worlds,’’ but assent to its possibility. But there is a potential danger in relying upon restriction to avoid paradox. One major obstacle to Tennant’s restriction strategy, for example, is that there are other Fitch-like paradoxes that are not averted by the restriction.²⁰ Tennant has subsequently promised to develop his restriction strategy to protect against these further paradoxes. Whether the ﬁctionalist who restricts is susceptible to similar criticism remains to be seen. If he is, there is then the option of following Tennant’s lead and developing more suitable restrictions. But there is a further worry about restriction strategies. Against Tennant’s strategy it has been argued that the restriction on knowable truth is unprincipled—that no reason has been given, other than the threat of paradox, to restrict the knowability principle (see for instance, Hand and Kvanvig 1999; DeVidi and Kenyon 2003; Hand 2003). A related charge against Tennant’s restriction strategy is that we must admit that however plausible the knowability principle is for a restricted class of sentences, it is to be rejected as a general principle. This is a stern confession on the part of the semantic anti-realist, who claims to have on offer an epistemic theory of truth. A related charge can be issued against the proposed restriction of ﬁctionalism, i.e. (Fic∗∗ ). Any reinterpretation of modal discourse must be inferentially adequate. The reinterpretation of modal discourse, for example, must inherit the inferential advantages of using discourse about possible worlds (Rosen 1990: 330; and Divers 2004: 665). However, the proposed reinterpretation of modal discourse does not inherit the inferential advantages of possible world semantics. For the modal realist can account for the validity of our standard modal inferences. For example, where p is world-restricted, the modal realist can account for the validity of p ♦p by translating p into an idiom of counterpart theory, inferring the ﬁrst-order consequence that there is a world in which p, and then translating this consequence back into an ordinary modal idiom. Since the unrestricted modal operators are redundant, validity is even easier to account for when p is unrestricted. By contrast, the ﬁctionalist can account only for the validity of p ♦p, where p is world-restricted. Hence, ﬁctionalism does ¹⁹ An unrestricted modal operators is a modal operator that embeds an unrestricted statement, for instance, the possibility operator occurring in ‘‘it is possible that there is a plurality of worlds.’’ The proposed strategy is entirely motivated. First, if Hale is right, then the ﬁctionalist cannot offer an adequate and comprehensive analysis of all possibility claims in non-modal terms. The ﬁctionalist will need to admit primitive modal operators in order to account for the meaning of the story preﬁx. Second, the modal realist, too, must acknowledge two kinds of modal operators. The modal realist analyzes the restricted modal claims of quantiﬁed modal logic in terms of possible worlds, but must, if Divers is right, treat unrestricted modal operators as redundant. ²⁰ See e.g. Williamson (2000b); Brogaard and Salerno (2002, 2006, 2007).

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not inherit the inferential advantages of using discourse about possible worlds without the ontological costs. We have been assuming strong modal ﬁctionalism. Strong modal ﬁctionalism holds that the ﬁction of possible worlds provides the resources for an analysis of modal claims, and so that modal claims depend on the content of the ﬁction of possible worlds. Some of the problems with ﬁctionalism can be avoided if, as Rosen suggests, we take ﬁctionalism to provide ‘‘not a theory of possibility, but merely a theory linking the modal facts with facts about the story PW’’ (1990: 354). The result is timid modal ﬁctionalism.²¹ Just like strong modal ﬁctionalism, timid modal ﬁctionalism licenses only a subset of transitions from modal idioms to idioms of counterpart theory. However, this may be unproblematic insofar as the timid ﬁctionalist does not regard (Fic∗∗ ) as purporting to shed light on the nature of modal truth, but merely takes it to link the sentences of quantiﬁed modal logic with the sentences of counterpart theory.

Ge n e r a l L e s s o n s Of the four solutions to Fitch’s result considered, only restriction strategies can be extrapolated to block the Fitch-like result developed above. This shows that restriction strategies are potentially more effective tools for avoiding Fitch-like paradoxes than are the other strategies we have considered. Unless it can be argued that different anti-realisms call for fundamentally different resolutions of Fitch-like paradoxes,²² the aptitude of restriction strategies to block Fitch-like paradoxes should count in their favor. The downside is that the restriction strategist cannot take the principles he is restricting as providing the resources for an analysis of truth. The result of restriction is thus timid anti-realism. Is timid semantic anti-realism an acceptable position? Well, one of the motivations for semantic anti-realism is Dummett’s manifestation argument (Dummett 1976; 1977: Chapter 4; 1978). The argument is often taken to be that, for reasons having to do with the manifestability of meaning, truth is to be understood epistemically, in terms of what our epistemic capacities allow us to verify in principle. But if it can be shown that manifestability ²¹ For a defense of timid modal ﬁctionalism, see Brogaard (2006). ²² Fitchy paradoxes may be symptomatic of a common malady in (a range of ) anti-realist positions. It would be interesting to ﬁnd out what the malady is. Is there a deep metaphysical point to be made about why these different positions give rise to this distinctive kind of paradox? On the face of things, it is not easy to see what that might be. Anti-realism and modal ﬁctionalism both consist in the non-acceptance of the objectivity of a certain subject matter (e.g. modality), but apart from this superﬁcial similarity, they are obviously very different. In a recent paper (Brogaard and Salerno 2005), it was argued that the malady in question is a kind of conditional fallacy. Perhaps, then, it is not that modal ﬁctionalism is similar in some deeper respect to semantic anti-realism, but rather that strong anti-realisms run into some kind of conditional fallacy. Thanks to John Divers here.

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considerations entail no such result, then it is open to argue for a timid semantic anti-realism. A timid semantic anti-realist could appeal to veriﬁability merely in order to counter the realist claim that even truths that can be consistently known may be beyond our epistemic reach.²³ In conclusion: the present case study indicates that Fitch-like paradoxes present a major obstacle, not only to semantic anti-realism, but also potentially to a number of other anti-realisms. The Fitch-like paradoxes give anti-realists reason to restrict. Restriction leads to timid anti-realism. Fitch’s proof can thus be construed as an argument for timid (as opposed to strong) anti-realism. ²³ This, in fact, may be all that Tennant is committed to. See Tennant (2001b).

16 Fitch’s Paradox and the Philosophy of Mathematics Ot´avio Bueno

In t ro d u c t i o n It is intuitively plausible to suppose that the sheer fact that something true can be known should not be sufﬁcient for it to be actually known. According to Fitch’s paradox, however, under very reasonable assumptions, we basically obtain this result (see Fitch 1963, and Brogaard and Salerno 2004). More precisely, we obtain a reductio of the claim that two apparently acceptable principles are consistent.¹ The ﬁrst principle is the knowability principle, according to which all truths can be known by somebody at some time: (KP) ∀p (p → ♦Kp), where K is the epistemic operator ‘‘it’s known by somebody at some time that,’’ and ♦ is the modal operator ‘‘it’s possible that.’’ The second principle states that we are non-omniscient; that is, there is a truth that nobody ever knows: (NonO) ∃p (p ∧ ¬Kp). It’s easy to see that (KP) and (NonO) entail ♦K(p ∧ ¬Kp).² However, we have independent reasons to believe that ¬♦ K(p ∧ ¬Kp).³ Thus, if we keep the claim My thanks go to Jody Azzouni, Joe Salerno, Amie Thomasson, and Ed Zalta for extremely helpful discussions. Thanks are also due to Joe Salerno and two anonymous referees for very perceptive comments on earlier versions of this work. Their comments led to substantial improvements. ¹ I follow here Berit Brogaard and Joe Salerno’s elegant presentation of the paradox (see Brogaard and Salerno 2004). ² Consider the following instances of, respectively, (NonO) and (KP): (1) (p ∧ ¬Kp) (2) (p ∧ ¬Kp) → ♦K(p ∧ ¬Kp) where (2) is obtained by substituting (p ∧ ¬Kp) for variable p in (KP). ♦K(p ∧ ¬Kp) then follows immediately. ³ In fact, the argument to this effect is based on four reasonably straightforward inferences:

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that all truths are knowable (KP), we have to deny that we are non-omniscient (NonO); that is, we have to assert that all truths are actually known: (O) ∀p (p → Kp). In other words, it looks as though, given the assumptions of the argument above, we are licensed to move from the claim that something true is knowable to the conclusion that it’s actually known.⁴ But, prima facie, this seems paradoxical. In what follows, I’ll take the paradox to be an inference that ultimately licenses us to conclude, given a true sentence p and the possibility of knowing p, that p is actually known; that is: (FP) p → (♦Kp → Kp). In this paper, instead of examining the paradox in a generic context, I’ll consider the impact it has on particular epistemological views about mathematics. I will assume therefore, for the sake of argument, that the reasoning leading to Fitch’s paradox is valid (which is indeed the case given the logical assumptions made). Having this focus provides a speciﬁc context to assess the nature and limitation of the paradox, while also showing the paradox’s signiﬁcance for current debates in the philosophy of mathematics. More speciﬁcally, I’ll examine two versions of Platonism (standard and full-blooded Platonism) and two versions of nominalism (mathematical ﬁctionalism and agnostic ﬁctionalism). And I’ll argue that, given the speciﬁc assumptions about mathematical knowledge that these views make, (FP) brings trouble for some, but not for all, of them. In particular, full-blooded Platonism and—to a certain extent—mathematical ﬁctionalism are in trouble with the paradox, but traditional Platonism and agnostic ﬁctionalism don’t seem to be. I conclude with a dilemma that this situation poses for Platonism, and the prospects it offers for nominalism. (a) (b) (c) (d)

If K(p ∧ q), then Kp ∧ Kq. (A conjunction is known only if the conjuncts are known.) If Kp, then p. (A statement is known only if it’s true.) If p is a theorem, then p. (All theorems are necessarily true.) If ¬p, then ¬♦p. (If it’s necessary that ¬p, then p it’s impossible that p.)

Assuming these inferences, here’s the argument for ¬♦K(p ∧ ¬Kp): (1) (2) (3) (4) (5) (6)

K(p ∧ ¬Kp) Kp ∧ K¬Kp Kp ∧ ¬Kp ¬K(p ∧ ¬Kp) ¬K(p ∧ ¬Kp) ¬♦K(p ∧ ¬Kp)

Assumption (for reductio) from (1), by (a) from (2), by (b) from (1)–(3), by reductio, discharging assumption (1) from (4), by (c) from (5), by (d).

⁴ The reason for this conclusion is that, as we saw, Fitch’s paradox establishes that the conjunction of (KP) and (NonO) leads to a contradiction. This, in turn, establishes the conditional: ∀p(p → ♦Kp) → ∀p(p → Kp), where the antecedent is (KP) and the consequent is the negation of (NonO), that is, (O). Now, if we assume (KP) and p (that is, if we assume that ‘p’ is true), we obtain the conditional: p → (♦Kp → Kp).

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Fi t c h’s Pa r a d o x a n d Fu l l - b l o o d e d P l a t o n i s m According to full-blooded Platonism (FBP), all mathematical objects that logically possibly exist actually do exist (Balaguer 1998: 53).⁵ The notion of possibility is here taken in its broadest sense, namely, as logical possibility (Balaguer 1998: 5–6 and 69–75). In other words, the picture of the mathematical reality that emerges is one of plenitude: all logically possible mathematical objects exist. From non-Cantorian sets to non-separable Hilbert spaces, from unusual metric spaces to yet unknown solutions to weird differential equations, everyone is welcome to this truly over-populous mathematical paradise. But why should anyone be happy with this extravagant ontology? Surprisingly enough, argues Balaguer, because this is the only defensible version of Platonism. Balaguer’s message is clear: if you want to advocate the existence of mathematical entities, don’t be shy, countenance all of them! Why should only Cantorian sets be accepted? What about the non-Cantorian ones? And why should we stop at sets, and not also introduce functions, numbers, and categories? The mathematical realm is a rich and multifarious domain, and that’s how it has to be if we are to solve the most pressing problem faced by Platonism: how can we have any knowledge of this realm if we have no contact with it, no form of access whatsoever? According to the FBP-ist, by extending the ontology of mathematics, we can answer this central question regarding mathematical epistemology. If all logically possible mathematical objects exist, we can explain the possibility of mathematical knowledge and the reliability of mathematical beliefs. To provide an account of the possibility of mathematical knowledge, the FBP-ist needs only to account for the fact that we can know that mathematical theories are consistent. Basically, FBP reduces the problem of knowing the truth of mathematical theories to that of knowing their consistency.⁶ After all, if FBP is true, the consistency of a mathematical theory M leads to its truth, since if M is consistent it will truly describe some part of the mathematical realm. And in this way, for the FBP-ist, knowledge of the consistency of M yields knowledge of M ’s truth. ⁵ There are worries about how to formalize properly the claim that characterizes FBP. Balaguer discusses some of these worries, without completely resolving them (see Balaguer 1998: 5–8). As Balaguer himself acknowledges, the formulation that comes closer to capturing FBP involves secondorder quantiﬁcation. Let ‘Y ’ be a second-order variable and let ‘Mx’ mean ‘x is a mathematical object’; in this case FBP can be (roughly) formulated as:

∃xMx ∧ ∀Y (♦∃x (Mx ∧ Yx) → ∃x (Mx ∧ Yx) ). For the sake of argument, I’m going to grant that there is a workable formulation—and a formalization—of FBP, although this is actually doubtful (see Restall 2003). ⁶ Since in this paper I’m focusing primarily on mathematical theories, following Balaguer, I’ll use the notions of consistency and logical possibility interchangeably. Nothing hinges on this.

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Moreover, to provide an account of the reliability of mathematical beliefs, all that needs to be explained is why it is the case that (as a general rule) if mathematicians accept p, then p.⁷ To indicate how this can be achieved, Balaguer provides the following argument: (i) FBP-ists can account for the fact that human beings can—without coming into contact with the mathematical realm—formulate purely mathematical theories. (ii) FBP-ists can account for the fact that human beings can—without coming into contact with the mathematical realm—know of many of these purely mathematical theories that they are consistent. (iii) If (ii) is true, then FBP-ists can account for the fact that (as a general rule) if mathematicians accept a purely mathematical theory T, then T is consistent. Therefore, (iv) FBP-ists can account for the fact that (as a general rule) if mathematicians accept a purely mathematical theory T, then T is consistent. (v) If FBP is true, then every consistent purely mathematical theory truly describes part of the mathematical realm, that is, truly describes some collection of mathematical objects. Therefore, (vi) FBP-ists can account for the fact that (as a general rule) if mathematicians accept a purely mathematical theory T, then T truly describes part of the mathematical realm. (Balaguer 1998: 51–2) After putting forward this argument, Balaguer indicates why its premises should be accepted, and, as a result, concludes that FBP leads to a new Platonist epistemology. For the purposes of this paper, there is no need to review all of Balaguer’s considerations (for details, see Balaguer 1998: 52–3). I’ll focus on the key premises of his argument—particularly premise (ii). Note that (vi) follows from (iv) and (v) only if FBP is true. Otherwise, we have at best the conditional: If FBP is true, then (vi) is true—which is, of course, substantially weaker than the claim that (vi) is true. In other words, FBP-ists can provide an account of mathematical knowledge and the reliability of mathematical beliefs only by asserting the truth of FBP. Without this assertion, the account doesn’t get off the ground. But does Balaguer’s overall strategy succeed? To support premise (ii), Balaguer develops an anti-Platonist—in fact, a factionalist—epistemology for Platonism. The epistemology is ﬁctionalist in ⁷ As Balaguer acknowledges, in explaining the possibility and reliability of mathematical knowledge, he follows related moves made by Hartry Field’s mathematical ﬁctionalism (see Field 1989). I’ll discuss Field’s proposal shortly.

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the sense that it doesn’t rely on the existence of mathematical entities. These entities play no role in the epistemological story, and they might just as well be taken as useful ﬁctions. The point of the ﬁctionalist epistemology is to establish how we can have knowledge of mathematical entities ‘‘without coming into contact with the mathematical realm.’’ The key idea is to advance a primitive notion of consistency (or logical possibility), and argue that this notion provides all that is needed for our knowledge of mathematical objects. After all, given FBP, if a mathematical theory is logically consistent, then it is true of part of the mathematical realm. Thus, given FBP and the principle of closure for the knowledge operator, if we know that a mathematical theory is consistent, we know that it is true (of some part of the mathematical realm). In other words, as noted above, for the FBP-ist, knowledge of the consistency of a mathematical theory immediately yields knowledge of the theory’s truth (regarding some part of the mathematical realm).⁸ So, the FBP-ist only has to explain how we come to know that mathematical theories are consistent; knowledge of their truth will follow straightway. The strategy is ingenious, of course, in that it uses what is perhaps the most attractive feature of anti-Platonism—the fact that it doesn’t seem to make mathematical knowledge a mystery⁹—in order to overcome the weakest aspect of Platonism, the difﬁculty of accommodating mathematical knowledge given the postulation of abstract entities. It’s precisely at this point that Fitch’s paradox comes in. First, note that, given the epistemological strategy that underlies FBP, the proposal is committed to the knowability principle (KP) in the context of mathematics; that is, every true mathematical claim can be known. After all, suppose that M is a true mathematical claim. Then, as the FBP-ist acknowledges, M is consistent (for the FBP-ist doesn’t think that there are true contradictions). But if M is consistent, it’s possible to know that M is consistent. (After all, only M ’s inconsistency would prevent the logical possibility of knowing that M is consistent, and so, if it were logically impossible to know that M is consistent, then M would be inconsistent. By contraposition, we get the intended conditional.¹⁰) Thus, if it’s possible to know that M is consistent, then for the FBP-ist, it’s possible to know M , given that, on the FBP-ist picture, M ’s consistency entails M ’s truth. In other words, every true mathematical claim can be known. ⁸ Note that the FBP-ist is not committed to the existence of true contradictions. Given two independently consistent, but jointly inconsistent, mathematical theories, their conjunction, being inconsistent, is not going to be true of any part of the mathematical realm. However, since separately each mathematical theory is (by hypothesis) consistent, each theory is going to be true of some part of the mathematical world. These parts, however, on the FBP-ist picture, do not overlap. ⁹ Typically, for the anti-Platonist, mathematical knowledge is conceptualized as some form of logical knowledge (knowledge of what follows from what). Using a primitive notion of consistency (or possibility), we can express the consequence relation—B is a consequence of A—as it’s not possible that A is true and B is false. ¹⁰ Recall that the FBP-ist is operating with the notion of logical possibility as the underlying notion of modality.

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It might be objected that the FBP-ist is not really committed to (KP). After all, there might be consistent mathematical statements that are so intractable that it’s not logically possible to determine their consistency. It may initially seem plausible that there are such statements. After all, nothing seems to rule out their existence. However, on further reﬂection, it’s quite unclear that we have any reason to believe that such statements exist. At stake is the existence of consistent mathematical statements such that any attempt to show that they are consistent leads to a contradiction. Of course, inconsistent statements would have this property (assuming classical logic, any attempt to show that they are consistent entails a contradiction). But inconsistent statements are not the relevant candidates here, since the assumption made involves consistent statements. As an alternative, it might be thought that Gödel’s incompleteness theorems provide such statements. But that doesn’t seem right either. First, a Gödel sentence is not intractable; in fact, it can easily be seen to be true. More importantly, from Gödel’s second incompleteness theorem, we have it that, under certain assumptions—namely, that T is a formal recursively enumerable theory that includes basic arithmetic and certain provability conditions—if T is consistent, T cannot prove its own consistency. The trouble, however, is that Gödel’s result is not relevant here. The assumption made in the objection above is not that the consistency of a certain mathematical theory M cannot be proved in M, but rather that M’s consistency cannot be proved at all—and just as a matter of logic. Moreover, the only assumption made in the objection is that the mathematical statements in question are consistent. But in order for us to apply Gödel’s theorem, we would also need the assumption that the mathematical statements in question constitute a formal recursively enumerable theory including basic arithmetic and certain provability conditions. In other words, the objection requires a much stronger claim than anything that can be reasonably supported by Gödel’s theorem, and the assumption made in the objection is much broader than those needed for the theorem to apply. Here is an illustration of the central point. Although we cannot show the consistency of arithmetic in arithmetic (assuming that arithmetic is consistent), we can show the consistency of arithmetic in set theory. However, the situation considered in the objection above is one in which the consistency of the mathematical statements in question cannot be established under any circumstance, for purely logical reasons—a much stronger claim! The burden is now on the FBP-ist to show that there are statements of this sort. I don’t see how this can be done. And unless the FBP-ist can show that there are such statements—and it’s not clear that this is the case—the argument given above to the effect that the FBP-ist is committed to (KP) still stands. Now, given the FBP-ist’s commitment to (KP), does it follow that the FBP-ist is thereby committed to (FP)? Well, recall that the argument for (FP), given in Section 1 above, established an inconsistency between two principles: (KP) and (NonO). According to the latter, we are non-omniscient; that is, there is a truth

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that nobody ever knows; in symbols: ∃p (p ∧ ¬Kp). So, the question now is whether the FBP-ist is also committed to (NonO). However the FBP-ist answers this question, problems emerge. In fact, we have a dilemma. Either the FBP-ist is committed to (NonO), or she isn’t. If she is committed, then the FBP-ist is committed to an inconsistency, given that (KP) and (NonO) are inconsistent. Alternatively, if the FBP-ist is not committed to (NonO), then there are three options: (i) The ﬁrst is that the FBP-ist is not committed to (NonO) in virtue of being committed to the negation of (NonO), that is to (O)—the claim that all truths are actually known; in symbols: ∀p (p → Kp). But this means that the FBP-ist would then be committed to (FP), given her commitment to (KP) and to the truth of mathematical statements p.¹¹ (ii) Alternatively, the FBP-ist is not committed to (NonO) because (NonO) lacks truth-value. But, for this move to the plausible, the FBP-ist needs to motivate why (NonO) is truth-valueless. Prima facie, (NonO) seems to be a perfectly intelligible and meaningful statement to the effect that there is a truth that nobody knows, and it doesn’t seem to have any of the usual semantic defective features that motivate lack of a truth-value (such as vagueness or indeterminacy). Moreover, even if this move were somehow well motivated, to make sense of truth-value gaps, the FBP-ist would need to introduce a non-standard semantics. But this conﬂicts with the FBP-ist commitment to a standard semantics. In fact, the use of a standard semantics is taken to be a signiﬁcant advantage of Platonism over various nominalist views (see Balaguer [1998]). Finally, (iii) the FBP-ist is not committed to (NonO) because she doesn’t know whether (NonO) is true or false. But, once again, the plausibility of this move would need to be established. It’s unclear what justiﬁes or even motivates, from a FBP-ist perspective, the limitation of such a knowledge claim. What is the reason why the FBP-ist doesn’t know whether there is a truth that nobody knows? Furthermore, if (NonO) is true—as it is typically taken to be, since we are not omniscient after all!—then this FBP-ist move is incoherent. After all, there would be something true, namely (NonO) itself, that nobody knows (according to the FBP-ist). But in this case, the FBP-ist would know that (NonO) is true! In the end, any of the four options just outlined (the ﬁrst horn of the dilemma and the three options that emerge from the second horn) are unacceptable for the FBP-ist. The ﬁrst makes the view inconsistent. The second generates a paradox for FBP. The third requires a major revision, and a substantial weakening, of the view. Finally, the fourth option makes FBP implausible, unmotivated, and possibly incoherent. But why does a commitment to (FP) engender a paradox for FBP? Given the strategy developed by the FBP-ist to obtain knowledge, it looks as though ¹¹ In other words, all of the conditions required to establish (FP) would then be met (see the argument for (FP), p. 253 above).

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from the fact that a certain mathematical theory M is possible (that is, logically consistent), and from the fact that we know that M is possible, we can infer, on the FBP-ist picture, that we know M . This is just what Fitch’s paradox (FP) allows us to infer! After all, for the FBP-ist, if we know that M is possible (i.e., logically consistent), then it’s possible to know M . (For, on this picture, all it takes to know M is to know that M is possible: M ’s logical consistency is sufﬁcient for M ’s truth. And if you know M , then it’s possible to know M .) Moreover, given FBP, if M is logically consistent, then M is true. But this means that, given Fitch’s paradox (FP), M is actually known. In other words, we conclude that M is actually known based only on M ’s truth (in fact, its logical consistency) and the possibility of knowing M . What should we make of the move to the effect that if it’s possible to know a true mathematical theory M , then M is actually known? We can perhaps take this conclusion as an important feature of full-blooded Platonism, and wonder what exactly was paradoxical about Fitch’s result in the ﬁrst place. Prima facie, it does seem counterintuitive to claim that the sheer possibility of knowing a true result is sufﬁcient for actually knowing it. However, as the FBP-ist insists, this is counterintuitive just because we don’t take the ontology of the mathematical world to be a plenitude of abstract entities and relations. Suppose the FBP-ist is right in claiming that every logically consistent mathematical theory is true of some part of the mathematical realm. In this case, on the FBP-ist picture, if it’s possible to know that mathematical theory is true, then we actually know the theory’s truth. This leaves open, of course, the issue of how we can know whether FBP itself is true; that is, whether the mathematical world is indeed the rich plenitude that the FBP-ist takes it to be. However, it’s unclear how the FBP-ist—or anyone else for that matter—could have that knowledge. The sheer consistency of FBP (assuming, for the sake of argument, that it is a consistent view) cannot be taken as a sufﬁcient condition for knowing that FBP is true. Making this move seems to assume a principle to the effect that consistency is sufﬁcient from truth. But this principle is not true in general, and clearly, it isn’t true of contingent domains, such as the actual world. However, perhaps the principle would be true of non-contingent domains, such as mathematics (assuming that mathematics describes non-contingent states of affairs). But even in this case, it’s not clear that the principle would hold, unless the FBP-ist picture of the mathematical universe is true, and hence the plenitude of the mathematical universe would entitle the FBP to move from the possibility of mathematical theories to their truth. The problem is that whether this move is acceptable is precisely the point in question. So, without begging the question, it’s unclear how the FBP-ist could invoke the modal principle under consideration in support of FBP. This suggests that the fact that FBP seems to lead to Fitch’s paradox should be taken not as a virtue, but as a difﬁculty for the proposal. After all, it’s only if

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FBP is true that the inference underlying Fitch’s paradox would seem acceptable. But it’s unclear how one could establish the truth of FBP in the ﬁrst place. One obvious way of establishing FBP’s truth is to invoke the inference that underlies Fitch’s paradox (see (FP), above). But this move, as we saw, begs the question, and it’s unclear how else the truth of FBP could be established. Alternatively, perhaps the FBP-ist could deny the commitment to the knowability principle (KP), by resisting the claim to the effect that if a mathematical theory M is consistent, then it’s logically possible to know that M is consistent. (Let’s call the latter italicized conditional: (C).) As we saw above, (C) is central to the argument that establishes the FBP-ist’s commitment to (KP). If (C) is rejected, the FBP-ist can avoid being committed to (KP), and hence can resist Fitch’s paradox. This move, however, won’t work. If the FBP-ist denies (C), he or she no longer can claim to have solved the epistemological problem of mathematics. After all, if it were logically impossible to know that a consistent mathematical theory M is indeed consistent, it would be logically impossible to know, on the FBP-ist picture, that M is true. And so, it would be logically impossible to know M . As a result, without (C) in place, it is unclear that the FBP-ist would have any epistemological advantage over standard Platonism. Thus, the cost incurred by FBP of positing a luxuriant ontology would come without the accompanying beneﬁt at the epistemological level. In response, the FBP-ist could insist that FBP was only meant to provide an epistemological story to explain the mathematical knowledge that we do in fact have, not to explain how we can know all mathematical truths. Perhaps there are unknowable mathematical truths—that is, truths whose consistency cannot be known (and, hence, which are not actually known). But this doesn’t challenge the success of the FBP-ist response to the epistemological problem of mathematics. After all, what the FBP-ist was after was only to explain how we have knowledge of mathematical objects and relations without having access to them. In particular, the project was only to explain how we have the mathematical knowledge that we have, not to guarantee that we are in a position to know every mathematical truth. Explaining the former is troublesome enough. This response makes perfect sense. It’s always advisable not to overextend one’s goals. However, by making this move, the FBP-ist would be acknowledging an intrinsic limitation in the view—a limitation that emerges in response to Fitch’s paradox. In order to avoid the latter, the FBP-ist tries to reject (C). Assuming for the sake of argument that this can be done, the FBP-ist is now committed to the existence of a consistent mathematical theory for which it’s logically impossible to know that it is consistent. The FBP-ist has here a commitment to something we don’t seem to have reason to believe exists. On the other hand, given the FBP-ist epistemological account, the mathematical theory in question cannot be known (assuming there is such a theory). This is an intrinsic limitation in the epistemological story offered by the FBP-ist.

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As a result, it seems that the FBP-ist faces a troublesome dilemma regarding the epistemology of mathematics in the context of Fitch’s paradox. If the FBP-ist rejects (C), she is committed to the existence of something we have no reason to believe exists. Moreover, it’s unclear then that FBP can offer a complete account of mathematical knowledge, since, by the FBP-ist’s own light, some consistent mathematical theories cannot be known. Alternatively, if the FBP-ist endorses (C), the FBP-ist is committed to (KP), and hence, to Fitch’s paradox. But, as we saw, this is similarly problematic. In light of these considerations, the option of taking Fitch’s paradox as raising trouble for FBP and articulating a different epistemological account for mathematics is a natural move, particularly for those who adopt a standard form of Platonism.

Fi t c h’s Pa r a d o x a n d St a n d a rd Pl a t o n i s m According to standard (or traditional) Platonism, mathematical objects, their properties and relations exist independently of our linguistic practices and psychological processes, and they are abstract (that is, they are not causally active nor are they located in space or time). However, as opposed to what goes on with FBP, the mere consistency of a mathematical theory is not sufﬁcient to guarantee the theory’s truth. After all, to be true, a mathematical theory needs to be more than consistent: it must correctly describe the mathematical objects in question, their properties, and the relations that these objects bear to each other.¹² How can we have knowledge of such objects, properties, and relations according to the traditional Platonist? Mathematical knowledge emerges from (i) the formulation of appropriate comprehension principles about mathematical objects (such as sets, numbers, functions, groups, graphs, topologies, and categories), and (ii) the exploration of consequences that these principles have for the corresponding objects.¹³ With the introduction of such comprehension principles, ¹² I’m assuming here for the sake of argument, together with Platonists, that the mathematical world is ‘‘consistent,’’ in the sense that no inconsistent theory correctly describes that world. Given that Platonists typically adopt classical logic as the underlying logic of mathematical theorizing, this consistency requirement seems prima facie reasonable. The requirement, however, can be dropped if the underlying logic is taken to be paraconsistent. As a result, even inconsistent but non-trivial mathematical theories can be studied. In this case, instead of assuming consistency as the minimal desideratum for a mathematical theory, non-triviality would play that role. That is, as long as not every sentence in the theory’s language is a theorem, the theory can be fruitfully pursued (see da Costa, Krause and Bueno 2007; and Bueno 2002). Of course, non-triviality is only a necessary requirement, since the mathematical theories in question need also to be mathematically interesting: they need to be mathematically tractable and allow for sophisticated, unexpected results (see Azzouni 2004). ¹³ A very elegant and insightful framework in which these comprehension principles can be formulated is provided by Ed Zalta’s object theory (see, e.g., Zalta 1983 and 2000). It should be noted, however, that object theory itself doesn’t commit one to Platonism, since the theory also allows for a nominalist interpretation (see Bueno and Zalta 2005).

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and by drawing consequences from such principles, the standard Platonist has all that is needed to articulate an account of mathematical knowledge. In a nutshell, mathematical knowledge is obtained by identifying the consequences of the comprehension principles that are postulated. Of course, this still leaves it open how we can have knowledge of the comprehension principles themselves. One move that traditional Platonists have explored at this point is to introduce a special notion of intuition, which is meant to work similarly to perception and should explain our knowledge of the comprehension principles directly in terms of this special access to the truth of the corresponding principles.¹⁴ It’s unclear, however, that such a move succeeds, given that presumably one would need to provide an account of how intuition itself functions, and in virtue of which features it is able to yield reliable access to the truth of substantive comprehension principles. Given that mathematical objects are abstract, the intuition in question would have to be this very peculiar cognitive faculty that is somehow akin to perception, but which allows us to apprehend abstract objects without any (empirical) access to them. In the end, explaining how such an intuition supposedly operates is likely to be as problematic as explaining directly the knowledge of the comprehension principles themselves. Whatever is the fate of traditional Platonism on the epistemological front, it’s important to highlight that, in contrast with FBP, modal notions do not play any role in the standard Platonist’s account of mathematical knowledge. Besides needing to provide a reasonable account of how we have knowledge of comprehension principles, the other component of the standard Platonist’s epistemology is to invoke the notion of logical consequence in order to draw consequences from such principles. However, the notion of consequence is typically spelled out by Platonists in model-theoretic, rather than modal, terms. And so, modality does not enter the picture even at this point. It might be objected, however, that unless the comprehension principles are true, the standard Platonist is in no position to claim that he or she does have knowledge of the objects, properties and relations in question. After all, even granting a suitable faculty of intuition, how can the standard Platonist know that the comprehension principles correctly describe the objects under investigation? Presumably, on the standard Platonist’s picture, the existence of mathematical objects and the properties they have are independent of any theorizing about these objects, or any apprehension of such objects via intuition. For the Platonist, mathematical objects are independent of any theorizing about these objects, or of any intuitive access to them, in the sense that these objects would exist and have the properties they have even if no comprehension principles have ever been formulated. But in this case, it’s unclear how the Platonist can guarantee that suitable comprehension principles correctly describe the mathematical objects ¹⁴ This move is often attributed to Gödel. I don’t claim here that this attribution is warranted.

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under study. This seems to require that we have some access to these objects independently of any theorizing, so that the correctness of our judgments about them can be assessed. But how can one apprehend a set, say, without specifying which kind of set it is (for instance, without determining whether it is a Zermelo–Frankel set or a von Neumann–Bernays–Gödel set)? In response, the Platonist could insist that the comprehension principles in question provide the meaning of the mathematical terms they introduce, and so these principles will be true in virtue of their form alone. Let’s grant that this is indeed the case. But the Platonist still owes us an account that establishes the existence of abstract objects independently of the framework determined by the comprehension principles in question. Why should we believe that by simply asserting certain analytically true mathematical claims, we will hence be able to pick out objects, properties and relations that exist independently of such claims? How can we guarantee that the objects that are referred to in these comprehension principles are precisely those that the Platonist takes to exist independently of the principles themselves? The issue here is not how we can know that a principle of the form ‘‘ ‘sets’ refers to sets’’ is true. Principles of this form are obviously true given the meaning of ‘‘refers.’’ The problem is how can we know when we assert that ‘‘there are inﬁnitely many sets’’ which sets we are referring to (say, Zermelo–Frankel sets or von Neumann–Bernays–Gödel sets)? And if we specify which sets we have in mind, how can we know that such sets exist independently of our speciﬁcation of them? This is the sort of independence that is central to the standard Platonist’s picture. It was precisely to overcome this sort of problem that the full-blooded Platonist postulated a dramatic increase in the size of mathematical ontology. If every mathematical object that logically could exist actually does exist, then the truth of a mathematical theory—and, in particular, the truth of appropriate comprehension principles—is guaranteed by the mere consistency of that theory. As discussed above, this move does seem to provide some epistemological help for the FBP-ist. However, as we also saw, this particular use of modal notions in mathematical epistemology opens the FBP-ist to the charge of engendering Fitch’s paradox. And, clearly, this is a difﬁculty that the standard Platonist would gladly like to avoid. As it turns out, however, the standard Platonist does seem to have the resources to resist Fitch’s paradox. On the standard Platonist’s perspective, the possibility of knowing a true mathematical theory is not sufﬁcient for knowing that such a theory is true, given that the standard Platonist doesn’t make the additional assumption (made by the FBP-ist) that the mathematical universe is a plenitude. As a result, the standard Platonist can clearly block Fitch’s paradox, since in general on the standard Platonist view, it’s not the case that: p → (♦Kp → Kp); that is, the sheer possibility of knowing a true mathematical result p doesn’t imply that p is actually known. After all, on the standard Platonist’s view, to know p one would need to show how to derive p from suitable comprehension

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principles. And in order to do this, the sheer possibility of knowing p—that is, the possibility that there is a derivation of p from appropriate comprehension principles—is not enough. An actual derivation of p needs to be provided so that one can claim that p is known. This means that the standard Platonist can easily reject the knowability principle (KP) that underlies Fitch’s paradox; according to this principle, every truth can be known. For the Platonist, however, there might be truths about mathematical objects that we have no means of knowing—that is, proving in a particular formal system. Gödel sentences provide a clear example of this phenomenon: they are true sentences that cannot be proved within a certain formal system. If we can know that such sentences are true, it’s not by proving them in the system. (This is another place where intuition comes in on the Platonist’s picture.) Moreover, there might be truths that are mathematically intractable, and so we can never obtain a proof that establishes them. Of course, in such cases, we cannot generally know that the results in question are true. But, for the Platonist, given the independence of the mathematical objects and their properties from our theorizing about them, nothing precludes the existence of intractable mathematical truths that cannot be known. As a result, if the standard Platonist can reject (KP), Fitch’s paradox cannot get off the ground.¹⁵ (I’ll return, in Section 6 below, to the discussion of the plausibility of rejecting (KP).) So, standard Platonism seems to be in a better position than FBP with respect to Fitch’s paradox. However, for the reasons discussed above, the proposal doesn’t seem to be able to provide a complete account of the epistemology of mathematics, since knowledge of the comprehension principles themselves is ultimately left open. Is there a better alternative?

Fi t c h’s Pa r a d o x a n d Ma t h e m a t i c a l Fi c t i o n a l i s m According to mathematical ﬁctionalism (or nominalism), mathematical objects do not exist (see Field 1980 and 1989). After all, the mathematical ﬁctionalist can resist the only argument for Platonism that doesn’t beg the question against the nominalist: the indispensability argument. This latter argument is meant ¹⁵ Similar considerations also apply to structuralist views about mathematics, such as those articulated by Shapiro (1997) and Resnik (1997). Note that on both views modal notions are not invoked to account for mathematical knowledge. In Shapiro’s case, given his commitment to second-order logic and a general theory of structure, he requires that the relevant mathematical structures be coherent. But just as the supposed consistency of Zermelo–Frankel set theory is not sufﬁcient to guarantee the theory’s truth, the supposed coherence of Shapiro’s theory of structure is not sufﬁcient to guarantee its truth either. In the end, given the additional requirements they introduce for mathematical knowledge, these views also seem to be safe from Fitch’s paradox. Now, it’s of course a much more contentious issue whether structuralism provides a complete epistemology of mathematics. In particular, it’s far less clear whether Platonism sits well with a structuralist epistemology. But this is a point to be explored elsewhere.

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to establish that we ought to be ontologically committed to mathematical entities given that they are indispensable to our best theories of the world.¹⁶ Hartry Field, however, provided a strategy to resist the argument in a particular, but important, case by showing that it’s possible to reformulate a signiﬁcant scientiﬁc theory, Newtonian gravitational theory, without quantiﬁcation over mathematical entities (see Field 1980).¹⁷ After providing detailed support for this view, Field has explored a major consequence of his position for the nature of mathematical knowledge (see Field 1984). One of the advantages of Field’s ﬁctionalism is that it accommodates straightforwardly the difﬁculties that plagued the standard Platonist proposal. As we noted above, it’s unclear how standard Platonists can explain our knowledge of a realm of causally inert and inaccessible mathematical entities. However, if there are no mathematical objects, this puzzle does not even arise, since we would not expect to have access to such non-existent entities (see Field 1989: 252). But another issue about the nature of mathematical knowledge still remains. How should we distinguish a person who has lots of mathematical knowledge (a mathematician, say) from another who hasn’t (a lay person)? Certainly, the ﬁctionalist has to come to terms with this issue. Field acknowledges the point, and he has advanced an interesting proposal. Mathematical knowledge is ultimately either empirical or logical knowledge (Field 1984). The idea is that what distinguishes a person who has lots of mathematical knowledge from another who hasn’t is the knowledge that mathematicians accept certain principles (which is empirical knowledge), and the knowledge that certain mathematical statements follow from others (logical knowledge). This move can be traced back to the suggestions made by early logical-empiricists who also spelled out (mathematical) knowledge in either empirical or logical terms. The distinctive feature of Field’s view is the way in which logic enters into the picture. The standard characterization of logical consequence (namely, Tarski’s 1936), which is crucial for an account of logical knowledge, quantiﬁes over mathematical entities, and thus is not nominalistically acceptable.¹⁸ What is required here is a nominalist account of the notion of consequence, and, more generally, of the applicability of mathematics to metalogic. And it is at this stage ¹⁶ For a thorough discussion of the argument, which was ﬁrst formulated by Quine and Putnam, see Colyvan (2001). ¹⁷ Of course, to refute completely the indispensability argument, the nominalist would have to show that quantiﬁcation over mathematical entities is dispensable in all of our best theories of the world. Field is, of course, aware of that, and what he provided is a program of nominalization rather than a complete nominalization of science. There are serious worries, however, as to how far the program can go, and whether it can successfully provide a nominalization of quantum mechanics (see, e.g., Bueno 2003). ¹⁸ A sentence α is a logical consequence of a set of sentences if and only if in all models in which all elements of are true, α is true as well. Moreover, a set of sentences is logically consistent if there is a model in which all the elements of are true. Models are, of course, abstract objects, typically formulated in set theory.

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that Field’s program leads him to introduce modal notions. Roughly speaking, a sentence B is a logical consequence of A if (A ⊃ B), where ‘’ is a primitive modal operator of logical truth. So, where we were expecting to ﬁnd an account of a purely logical notion, Field puts forward a primitive modal operator. Instead of logic, we have here modality. But how is the proposal spelled out? Field’s idea (1984 and 1991) is to reduce the logical side of mathematical knowledge to two forms of claims:¹⁹ (i) we know that a body of mathematical claims whose conjunction is A is logically consistent, i.e., we know that ♦A; (ii) we know that a claim B follows from a body of claims whose conjunction is A, i.e., we know that (A ⊃ B) (Field 1984: 84–5). In this way, mathematical knowledge is ultimately logical knowledge (see also Field 1991: 5–6 and 11–17). But what is this primitive modal operator? According to Field, it can either be an operator of logical implication or of logical truth—any of them will do the job, and they are interdeﬁnable. For instance, if we take the logical implication operator (‘→’) as primitive, it can be used to deﬁne a one-place operator of logical truth (‘L ’) in the following way: ‘L A’ is deﬁned as ‘(A ∨ ¬A) → A’ (Field 1989: 34; see also Field 1991: 8).²⁰ Moreover, Field stipulates that ‘L ’ obeys the laws ‘L A ⊃ A’ and ‘L (A ⊃ B) ⊃ (L A ⊃ L B)’, and also suggests that ‘L A’ should be taken as a logical axiom whenever A is a logical axiom, as well as taking ‘L A ⊃ L L A’ as a logical axiom (ibid.). Having said that, he then observes: The laws we have just found for ‘L ’ have, of course, a recognizably modal character: they are the characteristic S4 laws for a necessity operator. (Field 1991: 8; italics added; see also Field 1989: 34)

This passage suggests that, instead of having been imposed, the laws for ‘L ’ were discovered by some sort of investigation. But, as we saw, the nominalist has not exactly provided grounds for taking these laws as basic. Stipulating that the modal operator obeys the laws above does not answer the obvious epistemological question about this operator: how do we know that these laws are true? Of course, the fact that they are characteristic features of S4 cannot provide any warrant for the nominalist. After all, S4 has a semantics that is typically formulated in set theory, and thus is ultimately unacceptable for the nominalist, given the commitment to sets. Moreover, as a purely proof-theoretic construction, S4 is still abstract, since it relies on a notion of proof that is not tied to any actual inscription (a sentence is provable in S4 even if no one has ever written down that proof ). Furthermore, as the nominalist would of course acknowledge, the fact that we are all familiar with S4 laws clearly does not provide grounds for the claim that these are the laws that an operator of logical truth should obey. If the ¹⁹ I’ll focus the discussion on the logical aspect of mathematical knowledge, which, in any case, is the key one. ²⁰ As Field points out, ‘A 1 , . . . , An → B’ is deﬁnable from ‘L ’ as ‘L (A 1 ∧ . . . ∧ An ) ⊃ B’. Therefore, it does not matter which primitive one assumes (ibid.).

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nature of mathematical knowledge is to be explained,²¹ the nominalist should tell us why mathematical knowledge has the features it has: why should these S4 laws be taken as basic? In other words, the nominalist cannot simply assume those properties of knowledge claims that should be in fact the result of an account of knowledge. Of course, every theory has its primitive concepts. The problem I am raising is that, philosophically, it is not enough to ground mathematical knowledge on logical knowledge and then, in order to avoid commitment to mathematical objects, spell out logical knowledge in terms of a primitive modal operator—unless we already have an appropriate epistemology of modal notions. After all, the epistemology of modality is by no means less problematic than the Platonist epistemology of mathematics (see Shapiro 1993). In this sense, it is unclear what is gained epistemologically with the move to a primitive modal operator.²² In response, one can argue that since Field is trying to provide an account of mathematical knowledge (at least in the restricted sense of explaining the difference between the knowledgeable in mathematics and the ignorant), he is surely trying to obtain an epistemological advantage over the Platonist. And, prima facie, it does seem easier to explain how we know that certain things are possible, than to explain how we know that abstract objects exist. However, with further reﬂection, it becomes clear that this impression is precipitate. Given that the possibilities to be known are the consistency of certain mathematical theories, a substantial amount of mathematical information is required. Typically, in a model-theoretic approach, the consistency of a mathematical theory T is established by the construction of a model for T , in a suitably stronger theory T . And we know that, if T incorporates arithmetic, to avoid inconsistencies, T has to be stronger than T . Again this is due to a mathematical result (Gödel’s theorem). However, given Field’s rejection of the model-theoretic account of consistency, it is unclear (i) what exactly is meant by the consistency of a mathematical theory, and (ii) how Field’s primitive notion of consistency is related to the model-theoretic one. To be fair to Field, he does indicate one kind of relationship between these two notions of consistency. This is done through the following two principles (Field 1989: 108): (MTP # ) If (NBG ⊃ there is a model for ‘A’), then ♦ A. (ME # ) If (NBG ⊃ there is no model for ‘A’), then ¬♦ A.²³ ²¹ And, of course, providing such an explanation is a central feature of an account of mathematical knowledge. ²² The ontological gain with the introduction of such an operator is clear: it allows the nominalist to avoid commitment to abstract objects when doing metalogic. But the issue we are considering here is epistemological. ²³ ‘NBG’ refers to von Neumann-Bernays-Gödel (ﬁnitely axiomatized) set theory, and ‘♦’ is Field’s primitive notion of logical possibility.

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In Field’s terminology, ‘MTP # ’ stands for model-theoretic possibility, and ‘ME # ’ for model existence. The symbol ‘ # ’ indicates that, according to Field, these principles are nominalistically acceptable. But this is exactly what is unclear. It could be argued that (MTP # ) and (ME # ) are acceptable to the nominalist, since they are modal surrogates for the corresponding Platonistic principles: (MTP) If there is a model for ‘A’, then ♦ A (ME) If there is no model for ‘A’, then ¬♦ A.²⁴ But what is the difference between these two groups of principles? First, the ‘nominalistic’ versions make explicit the particular set-theoretical context where the models for ‘A’ are formulated (namely, in NBG). Second, the antecedents of the ‘nominalistic’ conditionals are ‘necessitated’. But why are these features enough to make the #-formulations nominalistic? Note that in (MTP) and (ME) a particular set-theoretic context is assumed, even if this is not made explicit. After all, the models for ‘A’ referred to in the antecedents are typically formulated in set theory, which provides a broad framework for model construction.²⁵ However, even if we grant that set theory is not assumed, a convenient mathematical framework surely is—for such a framework is required for the formulation of models. In other words, the intelligibility of (MTP) and (ME) depends on the assumption that there is an underlying mathematical framework from the start. So, whether it is set theory or another setting, the difference between the two groups of principles does not lie in the mathematical framework. With regard to the second point, simply by necessitating a Platonistic claim, and putting it in the antecedent of a conditional, one does not make the resulting sentence nominalistically acceptable in general. For example, if (MTP) is not acceptable to a nominalist, why should (MTP∗ ) If necessarily there is a model for ‘A’, then ♦ A be any different? It may be argued that (MTP∗ ) does not entail the existence of a model for ‘A’, since it can be true, even if it is possible that there is no such model. In reply, the same conclusion holds for (MTP), for it can also be vacuously true. However, as opposed to (MTP∗ ), (MTP) is taken to be nominalistically unacceptable. The general point is that simply by reformulating each mathematical statement A in terms of a conditional of the form (MTP∗ ) is not enough to provide a nominalization strategy for mathematics. This strategy resembles the so-called ‘if-thenism’, and, as Field knows, this proposal is quite problematic. For, if nothing satisﬁes the antecedent of such a conditional (which will be the case if nominalism is right), the conditional will be true even if we replace the conditional’s consequent by its negation (see, for instance, Parsons ²⁴ For a discussion, see Field (1989: 103–9). ²⁵ Of course, category theory may also be used at this point.

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1990). In other words, as a translation scheme for mathematics, this strategy is hopelessly inadequate, since for a mathematical statement A, both A and its negation can be translated into a (vacuously) true nominalistic claim—unless we make sure that the antecedent is satisﬁed.²⁶ But perhaps this complaint is uncharitable. Field’s proposal is not to assume a reformulation of mathematical claims via the use of conditionals plus the necessitation of the antecedents. The point is to express that a given result follows from a particular mathematical theory, and this should be done in the object language, without recourse to the model-theoretic notion of consequence. In the case in question, (MTP # ) states, in nominalistic terms, that if it follows in NBG that there is a model for ‘A’, then ♦A. And this formulation is nominalistically acceptable, since whether a particular result follows or not from a given mathematical theory is (i) an ‘‘empirical’’ question, which (ii) does not commit one to the truth of this theory. In this sense, the reference to NBG in (MTP # ) is not essentially different from a claim about a ﬁction, and what holds in it. The problem with this suggestion is that, in a context where we are dealing with abstract objects, we don’t expect ﬁctions to tell us about what is possible and what isn’t. So, on what grounds can the nominalist claim that, because there is a model for ‘A’ in (the ﬁction) NBG, ‘A’ is logically possible? In other words, can the nominalist believe in (MTP # ) without violating nominalism? I think the answer is no. For either the existence of certain models in NBG tells us something about logical possibility (Field’s ‘♦’)—but then it is unclear that the nominalist can bracket away the resulting ontological commitments, since these models are mathematical entities—or these models are not tied to logical possibility, but then (MTP # ) is groundless. In a nutshell, a story has to be told as to why our knowledge that a mathematical theory T is logically consistent is less problematic than our knowledge of T . The problem, as we saw, is twofold. On the one hand, knowledge of consistency seems to depend upon a substantial amount of mathematical knowledge. However, not only is this move closed to the nominalist (since mathematical knowledge then relies on the existence of mathematical objects), but it would also make Field’s proposal circular, since his epistemological account depends on the claim that mathematical knowledge is ultimately knowledge of consistency (and, as we saw, knowledge of consistency depends, in turn, on mathematical knowledge). On the other hand, if knowledge of consistency is not tied to mathematical knowledge, one loses the grip on the notion of consistency used in this context. For how can we recapture the information about the consistency of certain mathematical theories (on nominalistic grounds)? Thus, it is not clear in which respect the ²⁶ Note that Field does assume that the antecedent of (MTP # ) is satisﬁed; or more explicitly, he assumes that ♦NBG (Field 1989: 109). But what I am considering here is (MTP∗ ), and its relationship to (MTP). (MTP # ) will be discussed shortly.

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introduction of the primitive modal operator is helpful for illuminating the epistemological problem of mathematics.²⁷ Note that, according to several authors, primitive modalities are not always problematic. On the modalist’s view, for instance, primitive modal notions are inevitable if we are trying to provide an account of modality itself (see Shalkowski 1994). My point here is that, if we are considering mathematical knowledge, it is not obvious in what respect the move to a primitive modal operator can be of help in the absence of an appropriate modal epistemology. In order for Field to make room for the difference between those who know lots of mathematics and those who know very little, he has to provide an account of modal knowledge. After all, in his picture, that is ultimately what mathematical knowledge amounts to. The problem, however, is that usual accounts of modal knowledge often presuppose mathematical knowledge. For example, David Lewis’s main argument for the existence of modal knowledge rests crucially on the existence of knowledge of mathematical objects. After all, in Lewis’s view, just as we do have mathematical knowledge, despite the fact that we never causally interact with mathematical entities, we also do have modal knowledge, even though we never causally interact with possible worlds (see Lewis 1986: 108–15). Of course, on pain of circularity, this is not a route open to Field. Otherwise, he would be explaining mathematical knowledge in terms of modal knowledge and modal knowledge in terms of mathematical knowledge. Furthermore, a characterization of modal knowledge in terms of conceivability won’t work in this context either. After all, in the case of mathematics, we are often guided by mathematical theories to conceive what is mathematically possible or impossible. But this presupposes that such theories are reliable. And this is exactly what we expect to obtain from an account of mathematical knowledge, rather than something we could simply assume that we have from the outset. Despite the suggestive title of Field’s 1984 paper, ‘‘Is Mathematical Knowledge Just Logical Knowledge?,’’ the strategy he has actually taken (ultimately ‘‘reducing’’ logical knowledge to modal knowledge) only leads to another question ‘‘Is Mathematical Knowledge Just Modal Knowledge?’’ But then, the answer to this question is still left open. Let’s grant, however, for the sake of argument, that the mathematical ﬁctionalist can somehow overcome these worries and provide an account of mathematical knowledge. How would that account bear on Fitch’s paradox? As we saw above, for the mathematical ﬁctionalist, mathematical knowledge emerges from two features: (a) knowing that certain mathematical claims are consistent, and (b) knowing what follows from such claims. This imposes very little constraints on what it takes for us to have mathematical knowledge, and it’s hard to see ²⁷ Note that it won’t help Field’s case to claim that we have a priori knowledge of the properties of the modal operator ‘L ’, since this kind of knowledge is also modal, and we are back to the difﬁculty.

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how the mathematical ﬁctionalist could avoid Fitch’s paradox. After all, since the ﬁctionalist denies the existence of mathematical objects, it’s unclear how he or she could deny (KP), the knowability principle, according to which every truth is knowable. The standard Platonist, as we saw, can deny that principle given the additional constraint he or she imposes on mathematical knowledge: to be taken as knowledge, a mathematical result has to be proven from suitable comprehension principles. But this route doesn’t seem to be open to the mathematical ﬁctionalist, given the latter’s skepticism about the existence of mathematical entities. How can one motivate that certain claims may not be knowable if one denies the existence of objects that such claims are supposed to be about? As we saw, the standard Platonist can insist that, since the mathematical facts do not depend on us, there might be some that, due to their complexity and mathematical intractability, we won’t be able to track. However, no such move seems available to the mathematical ﬁctionalist. But perhaps the mathematical ﬁctionalist can use a different strategy of defense from Fitch’s paradox, and motivate the rejection of (KP). Once a certain body of mathematical principles and a particular logic are adopted, it doesn’t depend on us what follows (or not) from such a body of claims. In particular, if we adopt, say, NBG as the set theory we work with and ﬁrst-order logic as the underlying logic, there will be an inﬁnitude of results for which we won’t know whether they follow or not from the theory. In fact, due to (simple extensions of) Gödel’s ﬁrst incompleteness theorem, inﬁnitely many true arithmetical statements cannot be derived from such a theory. Moreover, consistency claims about mathematics are notoriously hard to settle in general. And since even on the ﬁctionalist view the logical facts are independent of us, it’s perhaps not surprising that it’s quite difﬁcult to decide such consistency claims. As a result, it seems that the mathematical ﬁctionalist could deny (KP) after all, since several truths cannot be known, and avoid Fitch’s paradox. But there’s still a worry here. The ﬁctionalist might be able to motivate the rejection of (KP) in this way, but the risk is to end up embracing skepticism about mathematical knowledge. Recall that, on the ﬁctionalist view, one of the necessary conditions for us to know a certain mathematical theory, say NBG, is for us to know that such a theory is consistent. However, due to Gödel’s second incompleteness theorem, we cannot know that NBG itself is consistent—unless we assume the consistency of a more powerful set theory, whose consistency, in turn, is more questionable than NBG’s. Thus, on the mathematical ﬁctionalist’s account of mathematical knowledge, we cannot claim to have knowledge of the results that follow from NBG, since one of the conditions for mathematical knowledge (namely, knowledge of the consistency of the theory in question) cannot be met. In other words, claiming that we know that NBG is consistent given that, say, we proved its consistency in a stronger theory won’t do. After all, if we have worries about NBG’s consistency, we would have still more worries about the consistency of a theory stronger than NBG that would need to be used

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in order to run such a proof. Since it’s unclear how the mathematical ﬁctionalist could be in a position to know the consistency of signiﬁcant mathematical theories, such as NBG, and given that this is taken to be a requirement for knowledge in mathematics, the position seems to lead to skepticism about mathematics. This would be the case for all substantial mathematical theories, such as set theory, but also arithmetic and analysis, whose consistency cannot be proved in general due to Gödel’s second incompleteness theorem.²⁸ It might be argued that this worry isn’t justiﬁed. Consider a similar move against reliabilism. If knowledge requires reliable cognitive faculties, do we need to know that our faculties are reliable? The reliabilist would insist that the answer is negative, of course. So, why can’t we make a similar claim in the case of NBG? That is, we could claim that there’s no need to know that the theory in which we prove NBG’s consistency is itself consistent. Having the consistency proof for NBG in some theory or other is enough. But this move doesn’t work: it’s not clear that it offers a reliable strategy in the ﬁrst place. Proving the consistency of NBG in a stronger theory would settle the issue regarding NBG’s consistency only if we knew that the stronger theory is itself consistent. After all, it won’t be of much help to prove the consistency of NBG in an inconsistent theory! And if we don’t know that the stronger theory in which we prove the consistency of NBG is itself consistent, we won’t be in a position to assert that NBG is indeed consistent. So, the analogy with reliabilism is not quite apt, given that we are not offered a reliable strategy to begin with. It might be objected that this response doesn’t push the analogy with reliabilism far enough. The point of the reliabilist move is to insist that, in order to have knowledge, there’s no need to have any positive evidence in favor of the reliability of a given process—or, perhaps, all that is needed is the absence of any salient negative evidence for the unreliability of the process. Similarly, in the case of NBG, all that is needed is that NBG be in fact consistent, independently of whether we have any evidence for its consistency. The problem here is that if NBG turns out to be inconsistent, we wouldn’t be able to determine that by simply working with the theory—unless we happen to stumble into an inconsistency. Frege worked for a long time with an inconsistent system without realizing it and if not for Russell, it is unlikely that he would have found the inconsistency. So, not even something as weak as the absence of any salient negative evidence for the unreliability of a process—e.g. no sign that we ²⁸ The mathematical ﬁctionalist could claim that in the case of such substantive mathematical theories, such as various forms of set theory, our knowledge of their consistency is obtained inductively: by not being able (so far) to show that such theories are inconsistent. Of course, as the ﬁctionalist would certainly acknowledge, this doesn’t prove that such theories are consistent, since someone may show that the theories in question, despite the appearances to the contrary, turn out to be inconsistent. Moreover, this move requires the ﬁctionalist to give up the claim that knowledge of the consistency of mathematics is part of mathematical knowledge, since on the inductivist conception, that knowledge is ultimately an inductive matter that cannot be established by mathematical techniques.

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are working with an inconsistent system—is sufﬁcient to support the reliability of that process. The reliabilist may, of course, simply run the risk: if the system in question (NBG in this case) turns out to be inconsistent, despite the appearance to the contrary, so be it. The important point, the reliabilist insists, is that if the system is consistent, it will yield knowledge—whether we know the system’s consistency or not. The problem is that, short of ﬁnding an inconsistency, there’s no way of telling that the system in question is indeed inconsistent. And if the system turns out to be inconsistent, the results obtained with it won’t be reliable. After all, assuming classical logic (which is the logic adopted by the mathematical ﬁctionalist), the negation of all the results obtained would also be derivable. It’s not by chance that the version of mathematical knowledge provided by the mathematical ﬁctionalist explicitly required that we know the consistency of the system we use. Moreover, this reliabilist move puts mathematical ﬁctionalists in the awkward position of having to assert the consistency of NBG without having grounds for that—in fact, without even requiring that they have such grounds! This move violates a basic norm of assertion, according to which you should assert only that for which you have some defeasible justiﬁcation. Of course, there are those who argue that the norm of assertion needs to be even stronger, since the norm ultimately requires knowledge: you’re entitled to assert only what you know (see Williamson 2000a). On this conception, the reliabilist move would be incoherent. But perhaps the reliabilist would claim that the mathematical ﬁctionalist doesn’t need to assert that NBG is reliable. It’s enough that NBG be reliable. But this would be, again, inadequate. Without being able to assert the consistency of NBG and other mathematical theories, the mathematical ﬁctionalist wouldn’t be in a position to make sense of signiﬁcant aspects of mathematical practice, where consistency claims are widespread. Even to make sense of Gödel’s theorems, it’s crucial to be able to assert the consistency of certain mathematical theories. A view that is unable to accommodate this basic feature of mathematical practice is entirely implausible. For these reasons, it’s unclear that the mathematical ﬁctionalist can completely resist Fitch’s paradox. Or, if the proposal manages to resist the paradox, by rejecting (KP), it seems to engender skepticism about mathematical knowledge. This raises the issue as to whether there is some version of ﬁctionalism that can resist the paradox in a well-motivated way. I think there is. Fi t c h’s Pa r a d o x a n d A g n o s t i c Fi c t i o n a l i s m Mathematical ﬁctionalism is a fairly traditional form of nominalism. It denies the existence of mathematical entities, and tries to explain the usefulness of mathematics by highlighting the fact that mathematical theories help to shorten

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derivations—even though in principle quantiﬁcation over mathematical entities could be dispensed with. Now, if mathematical entities don’t exist, then existential mathematical claims are false. So, claims such as ‘‘there are inﬁnitely many prime numbers,’’ which are taken to be true by mathematicians, turn out to be false. The Platonist (whether standard or full-blooded), however, has no difﬁculty in taking such claims literally: since prime numbers exist independently of us, and there are inﬁnitely many of them, the claim above is literally true. This is a signiﬁcant advantage of Platonism. In order to get verbal agreement with the Platonist, the mathematical ﬁctionalist has to introduce a ﬁction operator: ‘‘According to theory M , . . .’’, where M is a mathematical theory suitable to the context in question (see Field 1989). The ﬁction operator will then turn false operator-free existential statements into true mathematical claims: ‘‘According to arithmetic, there are inﬁnitely many prime numbers.’’ However, this means that the syntax of mathematical statements has to be changed, and as a result, as opposed to what happens in the Platonist’s view, mathematical discourse is not taken literally. This is a problem particularly if we want to make sense of mathematical practice, rather than simply construct a philosophical discourse parallel to that practice. Is there a way of preserving the central advantage offered by Platonism—of taking mathematical discourse literally—without the drawback of being committed to the existence of mathematical objects? I think there is. But this means developing a more robust form of ﬁctionalism about mathematics. This version of ﬁctionalism, which I call agnostic ﬁctionalism (for reasons that will emerge in a moment), is meant to provide the advantages of Platonism without its corresponding costs—or, equivalently, the advantages of nominalism without its accompanying troubles. The central idea of agnostic ﬁctionalism is that mathematical practice is always tied to the formulation of certain mathematical principles that characterize and deﬁne the meaning of the terms used in that domain of mathematics. For instance, in order for us to consider whether sets exist, we need ﬁrst to specify which sets we are considering. The term ‘set’ is referentially indeterminate, and we need to disambiguate between various different extensions of this term: each set theory speciﬁes a particular extension, a possible way of specifying the meaning of the term ‘set’, and the corresponding properties that such sets have. The speciﬁcation is done, just as in the case of the standard Platonist view, by providing suitable comprehension principles for sets. Once again, different set theories do that differently: consider, for example, the differences between the axioms for Zermelo–Frankel and von Neumann–Bernays–Gödel set theories, and the fact that the latter, but not the former, quantiﬁes over proper classes besides sets. Each set theory introduces its respective comprehension principles as constitutive for sets, and assumes a particular underlying logic for the theory. (Usually, that logic is left tacit, and it’s typically, but not always, taken to be classical.) Set theorists then draw consequences from the principles that have

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been introduced, and in light of the theorems that have been established, they determine the properties of the sets under investigation. The description so far, although focused on set theory, could be just as easily applied to different branches of mathematics, such as analysis, geometry, or arithmetic. In all of these cases, one needs to determine the objects of investigation by specifying suitable comprehension principles. In the case of analysis, there are many different options, ranging from classical, standard analysis (the usual analysis done in second-order logic) through classical non-standard analysis (such as the systems articulated by Abraham Robinson 1974) to non-classical standard analysis (when standard analysis is developed using some non-classical logic). In each case, suitable comprehension principles need to be speciﬁed and explored, and without such principles, it’s simply not determined which objects we are talking about. Similarly, there are various different geometries, from classical Euclidean geometry to various non-Euclidean geometries. Furthermore, we obtain additional geometrical systems by applying non-classical logics to the previously mentioned systems. Even in the case of arithmetic, we still need to specify the appropriate comprehension principles. If we want to determine the nature of natural numbers (that is, what kind of objects they are), we obtain different answers: one can adopt a neo-Fregean construction of such numbers, or one of many possible reformulations of such numbers in set theory, or the traditional formulation of arithmetic following the Peano axioms. In each case, we obtain a different system, and a different answer to the question of the nature of natural numbers. For convenience, we can simply stipulate that all such formulations are adequate, and show that they are equivalent for certain purposes (basically, the same results about numbers can be obtained in each system). But this doesn’t change the fact that, strictly speaking, each particular formulation of arithmetic provides a different characterization of the nature of numbers. For these reasons, it’s crucial to be explicit about the comprehension principles that are introduced in a particular branch of mathematics. And this may all look very Platonistic. But it isn’t. First, the truth of mathematical statements is now tied to the particular comprehension principles that specify the meaning of the mathematical terms that are employed in a given context. (The context here is determined by the speciﬁcation of the comprehension principles in question.) So, although the agnostic ﬁctionalist can follow the Platonist in taking mathematical statements as literally true, the truth of such statements is always dependent on the comprehension principles in question. Such principles determine a particular, internal context in terms of which the relevant mathematical statements are assessed. Second, no claim is made about the existence of mathematical objects beyond the context determined by the comprehension principles in question. Within the context of such principles, it’s trivially true (constitutively true) that there are mathematical objects of the appropriate sort. But nothing is claimed (or can be claimed) beyond such contexts. After all, beyond the contexts of

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such principles, it’s not clear what is meant by the mathematical terms under consideration: one needs comprehension principles to specify their meaning. It might be objected that the existence of mathematical objects cannot be warranted by the simple introduction of comprehension principles. After all, the truth of such principles cannot be guaranteed simply by the implicit deﬁnitions they offer of the relevant terms. Given that mathematical objects exist independently of any description we may have of them, the introduction of an implicit deﬁnition is not sufﬁcient to guarantee the existence of the corresponding objects. In response, note that the proposal here is not Platonist, and has no intention of guaranteeing the existence of mathematical objects independently of comprehension principles that systematize the discourse about them. Once a comprehension principle is given, we have the resources to talk about certain objects in exactly the same way as once a story is written we can talk about certain ﬁctional characters. That there are objects to talk about in each of these cases emerges from the systematization provided by the principles involved (comprehension principles in the case of mathematics, descriptions of ﬁctional characters in the case of ﬁction). However, from the fact that there are objects to talk about it doesn’t follow that the objects in question exist independently of the context provided by the principles in question. We can certainly talk about Sherlock Holmes, but we don’t take that object to exist. So, the view here allows us to quantify over mathematical objects without the assumption that they exist. This is accomplished by drawing a distinction between the existential quantiﬁer and the existence predicate. As will emerge shortly, this is a perfectly natural distinction, and one that provides a helpful device to avoid overextending one’s commitments (see McGinn 2000, and Azzouni 2004). The usual understanding of the existential quantiﬁer ends up mixing two very different functions of this quantiﬁer (McGinn 2000). One function (let’s call it the quantiﬁcational role) is to indicate that, in a certain domain of discourse, we are considering only some objects, rather than all objects, in that domain. The other function (let’s call it the existential role) is to assert that the objects in question exist. These are, of course, very different functions, and are better kept apart. Otherwise, we would be unable to say things such as: (∗ ) There are objects (such as ﬁctional entities, or frictionless inclined planes) that don’t exist. By restricting the existential quantiﬁer to its quantiﬁcational role, and introducing an existence predicate to play the appropriate existential role, we avoid having a quantiﬁer with these functions mixed. We can also express very easily sentences that have the form of (∗ ): ∃x (Ox ∧ ¬Ex), where ‘O’ stands for the predicate: is an object, and ‘E’ is the existence predicate. The conditions that the existence predicate should meet vary depending on one’s views about existence. This is, of course, not the place to take on such a

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large issue. It will sufﬁce to say that, in order to avoid begging the question against the Platonist, it’s enough to provide sufﬁcient conditions for us to know whether certain objects exist; namely, that we can track the objects in question, that we can interact with them, and reﬁne our access to them. Now, mathematical objects don’t seem to satisfy these conditions. But since these are only sufﬁcient (and not necessary) conditions, this doesn’t mean that the objects in question don’t exist. However, we need not be committed to the existence of mathematical just because we quantify over them either. As a result, the view proposed here is indeed nominalist, at least in the minimal sense that the existence of mathematical objects independently of suitable comprehension principles is never asserted. Whether there are mathematical objects outside such contexts is a question that is not well speciﬁed enough to be answerable. The view is agnostic about this matter. Does agnostic ﬁctionalism engender Fitch’s paradox? I don’t think so. After all, similarly to what happens with Platonism, to have mathematical knowledge, the agnostic ﬁctionalist requires that one produces a suitable proof of the results under investigation from the relevant comprehension principles. So, the mere possibility of knowing a true result is not enough for one to know the result: an actual proof needs to be produced. The agnostic ﬁctionalist can then motivate very naturally the rejection of (KP), the knowability principle. Given some framework for mathematics, there are truths that cannot be known (they cannot be derived in the system). As we saw above, Gödel’s results illustrate such situations very clearly. And without a commitment to (KP), Fitch’s paradox doesn’t get off the ground. Does agnostic ﬁctionalism engender skepticism about mathematical knowledge? That is, is it impossible to have mathematical knowledge on the agnostic ﬁctionalist view? The answer, once again, is negative. As we saw, Field’s mathematical ﬁctionalism seems to engender skepticism about mathematical knowledge given the requirement that we know that mathematical theories are consistent, and the difﬁculty of satisfying this requirement in general. The agnostic ﬁctionalist, however, doesn’t impose such a requirement. After all, we can have knowledge of even inconsistent mathematical theories. In a paraconsistent set theory, for example, we can show that the Russell set, {x : x ∈ / x}, has certain properties and lack others (see, e.g., da Costa and Bueno 2001). The Russell set is obviously an inconsistent object, but it’s not a trivial object—in the sense that it satisﬁes every property. On the agnostic ﬁctionalist view, we can have knowledge of such an object in the same way as we can have knowledge of other mathematical objects: by specifying suitable comprehension principles and a suitable logic, and determining what follows from such principles. Knowledge of consistency is not a requirement. Rather, the agnostic ﬁctionalist avoids triviality: that everything follows from the comprehension principles introduced. But, in an inconsistent context, this can be done by adopting a suitable paraconsistent logic (see da Costa, Krause, and Bueno 2007). In this way, knowledge of inconsistent mathematical objects is not essentially different from knowledge of consistent ones.

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A Di a g n o s i s As we saw, the responses to Fitch’s paradox from traditional Platonism and agnostic ﬁctionalism both rely on the rejection of the knowability principle (KP). But is it really plausible to reject this principle? Prima facie, aiming to know all the truths in a given domain seems to be a perfectly reasonable goal of inquiry. Clearly, such a goal may have informed substantial parts of scientiﬁc and mathematical research, particularly on a realist construal of these activities. In fact, for the realist—either about science or about mathematics—to establish the truth, or the approximate truth, about a certain domain is taken to be the aim of inquiry. This seems to presuppose, at least in principle, the possibility of knowing such truths. So, (KP) may be integral to the realist’s enterprise.²⁹ However, it’s unclear to me whether the goal of knowing all the truths in a given domain is sensible, and so whether the presupposition that it’s possible to know all such truths should be accepted. First, as opposed to the realist’s claim, ²⁹ It might be argued that the aim of inquiry for the realist is not to establish all truths, but to establish all knowable truths. I’ll call the view that takes as the goal of inquiry the establishment of all knowable truths epistemic realism. It turns out, however, that thinking of realism in epistemic terms faces several difﬁculties. First, it turns, by ﬁat, a radical skeptic who denies that we can know any truths into the most successful epistemic realist. For the skeptic could insist that we have established all knowable truths: it just turns out that there are none! Second, leaving the skeptic aside, the restriction to knowable truths makes the epistemic realist view extremely hard to implement. How could the epistemic realist know when he or she has achieved the goal of establishing all knowable truths? Well, when he or she knows that all knowable truths have been established. To know that, the epistemic realist would need to know what are all the knowable truths. But this is precisely what needs to be established in the ﬁrst place. Finally, presumably the epistemic realist would need to provide some principled account to distinguish knowable and unknowable truths. However this is spelled out, it would be hard to distinguish the practice of this epistemic type of realist from that of an anti-realist who denies that there are evidence-transcendent truths. The difference between the two views seems to be only verbal. Both views would agree on the knowable truths (those truths for which we have evidence), and both would dismiss the other truths (the unknowable ones). Although the epistemic realist would claim that there are such unknowable truths and the anti-realist would deny that, nothing in their practice would distinguish what they do when they conduct their research. It should be noted that none of these worries apply to a realist who characterizes her view in terms of searching for the truth (rather than knowable truths), since the view is not formulated in epistemic terms. First, the radical skeptic who claims that no truths can be known wouldn’t be a successful realist, since, as the realist would certainly point out, the fact that we can’t know whether something is true doesn’t entail that there is no independent truth to be found. Whether such independent truths can be known or not, the realist is after them. Second, the realist view can be implemented, by articulating better mechanisms of access to the truth. Since the realist doesn’t know the truth, it’s fallible which of these mechanisms will actually work. But progress is made by devising and testing such mechanisms. (As will become clear shortly, this doesn’t mean that the realist is problem-free by invoking truth as an aim of inquiry. I’ll return to this point below.) Third, the difference between realism and anti-realism is not purely verbal. The realist insists that there are truths that transcend our evidence, but this doesn’t preclude us from trying to forge better evidential mechanisms to ﬁnd the truth. I conclude that the realist is better off characterizing her view in terms of truth, as is typically done, rather than via knowable truths.

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it’s not so obvious that that goal has informed particular research programs. After all, in some cases the guiding principle may not be truth, but something weaker and indistinguishable from truth in the context of observable entities, such as empirical adequacy (see van Fraassen 1980). The same point applies to mathematics, where goals other than truth for mathematical activity have been entertained, such as conservativeness (see Field 1989). Moreover, there are several limits to what can be known, given the boundaries of human cognitive abilities. This ranges from clear limitations to what we can perceive, given the sensory faculties we have, the amount of information we can process, and how reliably we can process that information, even when we extend our capacities using various sorts of instruments (from microscopes to computers). There are also intrinsic limitations to what certain formal systems can yield, and Gödel’s theorems provide a clear example of that. In retrospect, from the point of view of realism, given the impossibility of establishing that we have reached the goal of knowing all the truths about a certain domain, it’s hard to see why (KP) should have been taken even as a plausible, regulative presupposition to begin with.³⁰ It is a reasonable requirement for a goal of inquiry that the participants should be able to know when that goal has been reached. But this is precisely what cannot be established in the case of knowing all the truths about a domain. Now, this raises a difﬁculty for the presupposition of such a realist goal, namely, that it’s possible to know all such truths, as (KP) stresses. After all, what grounds do we have to accept such a possibility, given the well-known limitations to human knowledge? One of the signiﬁcant lessons from Fitch’s paradox is to make this point transparently clear, by showing that (KP) is incompatible, as it should be, with the claim that we are non-omniscient. And so, a natural interpretation of the paradox is to take it as providing a reductio of (KP): there are truths that we cannot know. In the end, we can know many things, but there’s a lot that we can’t. The excursion above in recent philosophy of mathematics drives this point back home. Conclusion Although Fitch’s paradox poses an unexpected problem for accounts of knowledge, it’s possible to resist the paradox’s conclusion depending on the details of ³⁰ Of course, those in the intuitionist tradition have much to say in support of (KP). And, as a result, they try to block Fitch’s paradox by other means. But as I noted in the introduction to this paper, I’m exploring here the implications of Fitch’s paradox to the philosophy of mathematics. To do that, I have assumed, for the sake of argument, that Fitch’s paradox is a genuine paradox, as it seems to be in the context of classical logic. If it turns out that the inference leading to the paradox can be legitimately blocked, for example by changing the underlying logic, then, according to those who advocate such a change, there is no paradox. The issue then becomes how independently wellmotivated are the moves to block the relevant steps in the derivation of Fitch’s result. But this issue, although of course important, is entirely different from the one I’m concerned with in this paper.

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the epistemology one adopts. In particular, the paradox doesn’t seem to threaten a traditional form of Platonist epistemology that is based on the introduction of suitable comprehension principles. Since these principles introduce additional demands on what is required from mathematical knowledge, the traditional Platonist doesn’t face the paradox. However, as we saw, it’s still not clear that the traditional Platonist can successfully account for mathematical knowledge, at least if Platonism is true and mathematical objects do exist independently of our ways of describing them. After all, it’s then unclear how the traditional Platonist can account for knowledge of the comprehension principles themselves. In response to this epistemological worry, a more robust form of Platonism, full-blooded Platonism (FBP), was developed. The view may in principle be better situated to provide an epistemology of mathematics than traditional Platonism, given the increase in the ontology and the corresponding use of modal notions. But, as it turns out, the proposal is particularly susceptible to Fitch’s paradox. In the end, this poses a dilemma for Platonist epistemologies: If these epistemologies provide a full account of mathematical knowledge (such as FBP), they seem to be open to Fitch’s paradox. If they don’t provide a full account of mathematical knowledge (such as standard Platonism), they don’t seem to be open to Fitch’s troublesome conclusion, since they can then reject (KP). But then they would hardly be adequate, given their failure to yield a comprehensive account of mathematical knowledge. In either case, additional work is needed on the Platonist’s front. The situation seems more promising on the nominalist’s side. Although mathematical ﬁctionalism still faces some worries on the epistemological front, it might be able to resist Fitch’s paradox. However, as we saw, the response provided doesn’t seem to be as well motivated as it should be. As an alternative, agnostic ﬁctionalism has been proposed as a way of having the beneﬁts of Platonism (taking mathematical discourse literally) without the costs associated with nominalism (having to yield a parallel discourse for mathematics). In particular, agnostic ﬁctionalism provided a well-motivated response to Fitch’s paradox, by indicating why the knowability principle (KP) shouldn’t be accepted. In the end, we seem to have here a view that seems to offer the best of both worlds.

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17 Performance and Paradox Michael Hand The knowability paradox, or Fitch’s paradox, is thought to threaten semantical (Dummettian) antirealism. Here I suggest that the lesson of the paradox concerns the theoretical location at which to impose the antirealist’s ‘‘epistemic’’ constraints on truth, i.e., on the ‘‘central notion’’ of the antirealist’s meaning theory.¹ In particular, the knowability principle —that every truth is knowable—is not a successful way of capturing the antirealistic insight that truth is epistemically conditioned. I try to sharpen the intuitive feeling that the ‘‘paradoxical’’ Fitch conjunction ϕ & ∼ Kϕ is subject to a sort of ‘‘recognition-transcendent’’ truth that is of no interest for the realism/antirealism dispute, being instead an instance of a tame and well-understood pragmatic (not semantic) phenomenon. I suggest that antirealism’s epistemic constraints on the notion of truth are only mistakenly believed to motivate a global pragmatic constraint on ‘‘veriﬁcation procedures’’ and performances of them—which is the effect of the knowability principle—and are rightly taken to be local semantic constraints on the structures of such procedures. At the end (almost), in view of a not uncommon conviction among antirealists that the appropriate response to the paradox is to seek a restricted knowability principle to enforce the desired epistemic constraints on truth while avoiding paradox, I argue that no restricted knowability principle can serve the crucial meaning-theoretical purpose that antirealists reserve for their notion of epistemic truth. T h e Pa r a d o x The argument of the paradox is well known. The knowability principle (KP), (KP) ∀ϕ (ϕ → ♦Kϕ ), I am much indebted to Jon Kvanvig and Joe Salerno for comments on earlier drafts. ¹ In fact, the central notion of antirealistic meaning theory is not truth (canonical veriﬁability) but canonical veriﬁcation. To keep the current presentation simple, I sometimes talk as if truth itself is the central meaning-theoretical notion. The points I make are easily converted into a stricter statement in terms of canonical veriﬁcation procedures; the sections below on truth’s ‘‘recognition-transcendence’’ are especially pertinent.

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is incompatible with the common-sense claim that some truths remain always unknown, (CS) ∃ϕ (ϕ & ∼ Kϕ ). Thus what might appear to be a principle that is at least philosophically defensible, the thesis that for every truth it is possible that it be known at some time by someone, turns out to commit its proponents to a claim most semantic antirealists would reject (or at least, would deny being committed to simply in virtue of their antirealism), the claim that every truth does eventually become known.² The argument is simple. If ϕ is an always unknown truth, then this very fact, ϕ & ∼ Kϕ, is unknowable, for to know it requires knowing ϕ as well as knowing that ϕ is never known. Thus, if all truths are knowable, then there are no such truths as ϕ & ∼ Kϕ. For a true ϕ, then, we have ∼∼ Kϕ, and opponents of antirealism who countenance classical principles of inference rejoice in concluding Kϕ. Antirealists should have a difﬁcult time swallowing even the weaker conclusion, since it yields the unpalatable result ∀ϕ (∼ Kϕ →∼ ϕ ), i.e., only falsehoods remain forever unknown. Here is a formalization of the argument. The epistemic operator K is subject to two rules of inference: factivity and distributivity. (fact) Kϕ ϕ (dist) K(ϕ & ψ) Kϕ K(ϕ & ψ) Kψ First, note the inconsistency of K(ϕ & ∼ Kϕ ). K(j&∼Kj) dist K∼Kj ∼Kj

K(j&∼Kj) fact

dist Kj ⊥

Call this the key reductio. There are various ways to get from the key reductio to the result of the paradox. For example, ϕ & ∼ Kϕ, taken as an assumption for ∃-elimination on CS, together with ϕ & ∼ Kϕ → ♦K(ϕ & ∼ Kϕ ), from KP by ∀-elimination, yield ♦K(ϕ & ∼ Kϕ). The key reductio then permits the conclusion ♦ ⊥ from those two assumptions by means of obvious rules for ♦. This is as good as ⊥ itself for reductio purposes, which fact can be codiﬁed, for ² Some theorists have taken to calling ∃ϕ(ϕ & ∼ Kϕ) a nonomniscience claim, indicating that they think its denial, equivalent (classically) to ∀ϕ(ϕ → Kϕ), expresses omniscience. The claim that for each truth there is time when it is known—monotonicity of knowledge being assumed—does not entail that there is a time when all truths are known.

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instance, in a rule ♦ ⊥⊥. Now, by ∃-elimination, we have a reductio on the assumptions KP and CS. Fitch himself already observed that such a result is available for any factive distributive operator. Moreover, factive distributive operators are a dime a dozen, and a bit of familiarity with a few lessens considerably any sense of surprise at the result for K.

In d e x i c a l s a n d A s s e r t i ve Se l f - d e f e a t I follow Kaplanian orthodoxy in the semantics of indexicals. Some indexical sentences are such that although they express propositions (relative to contexts) that are not necessarily true, the sentences cannot be asserted falsely. For instance, ‘‘I am here now,’’ whenever asserted, is true. Indeed, ‘‘I am here now’’ is, following Kaplan’s use of the term, analytic. A sentence is analytically true if and only if for every context c, it expresses a true proposition relative to c. On the other hand, ‘‘I am speaking English’’ is true relative to any context in which the speaker says it (as well as any context in which the speaker—actually it’s better to avoid the term ‘‘speaker’’, since the ‘‘speaker’’ of a context need not be speaking; let us say the ‘‘agent’’—says anything in English at all), though it is not analytically true. It is false relative to contexts wherein the agent speaks in a different language, as well as those wherein the agent says nothing. An assertion of ‘‘I am speaking English’’ is self-fulﬁlling —it cannot be asserted falsely—though the sentence is not analytically true. ‘‘I am here now’’ is an extreme example of such a self-fulﬁller, for it manages to be true relative to any context, whether asserted therein or not, simply in virtue of the interaction of its indexicals. Analyticity is a semantical phenomenon. A sentence’s analyticity explains its self-fulﬁllment: because it is true at any context, it is true at any context in which it is said. More pertinent to present purposes are ‘‘I am speaking English’’ and similar examples, e.g. ‘‘I sometimes speak English,’’ which are merely self-fulﬁlling. (S is asserted in a context if and only if the context’s agent asserts S at the context’s time, etc.) From a purely semantical point of view, mere self-fulﬁllment is not interesting: although analyticity is a semantical phenomenon, mere self-fulﬁllment isn’t.³ Every contingent proposition requires for its truth that the world be (nonvacuously) a certain way. Some propositions require for their truth that the world be a certain way concerning what propositions are expressed therein, by whom, when, where, and how. Some propositions may require the world to be some way ³ This distinction between analyticity and mere self-fulﬁllment depends on the semantical principle of the ‘‘totality of character’’ in the semantics of indexicals: a sentence expresses a proposition relative to contexts wherein it is not uttered as well as those wherein it is. See Tsohatzidis (1992); Hand (1993); Tsohatzidis (1993); Zimmerman (1997); Kuppfer (2001, 2004); and Hand (2005: manuscript, still in preparation) for discussion.

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involving that very proposition’s being asserted or denied, or in some way used or not used, somehow or other—most generally, it may require something contingent of the used sentence itself (e.g., that it is eventually known). An asserted sentence may express, relative to its context, a proposition that is true in virtue of having been asserted: ‘‘I sometimes speak in English.’’ An asserted sentence may express, relative to the assertion’s context, a proposition incompatible with the assertion’s having been made in the context: ‘‘I never speak English.’’ The pragmatic interest of this phenomenon is the fact that a given use of a sentence may make the world a way that sufﬁces for the truth (or falsity) of its expressed proposition. A helpful example of this phenomenon can be seen in the relationship of (1a) and (1b). (1a) Je ne te tutoie jamais. (1b) Je ne vous tutoie jamais. Relative to any context, these say the same thing.⁴ Still, (1a) cannot be asserted truthfully, while (1b) can. Although they say precisely the same thing, what they say is about how they say it. (1a) says it one way; (1b) says it a different way. If I put to you what they say, and it is true in the context, then I have not used (1a) to do so. This does not mean that (1a) and (1b) say different things, but only that what they say, they say differently, and what they say concerns how I say things to you, and thus in particular how I say to you what (1a) and (1b) both say. An assertion of (1a) falsiﬁes itself, but at the same time it falsiﬁes (1b) as well (relative to the same context). Assertion of (1b) does not falsify itself, nor does it falsify (1a). They are equivalent, but the world is different if I use (1a) than if I use (1b), relevantly different as to whether members of the pair are true or false. Note that self-fulﬁllment and self-defeat are pragmatic properties of assertions. We may call a sentence itself self-fulﬁlling or self-defeating when, relative to any context in which it is asserted, its assertion self-fulﬁlls or self-defeats. Consideration of such derivative properties of sentences will shed light on the knowability paradox.

Pr a g m a t i c Se l f - d e f e a t i n Ge n e r a l Self-defeat of the above sort is a special case of a more general phenomenon of self-defeat. Consider ⁴ This pair is discussed repeatedly in the papers mentioned in the preceding note. (1a) and (1b) exploit the fact that French has two second-person singular pronouns, the informal ‘‘tu’’ and the formal ‘‘vous’’ (which also serves as the second-person plural, as does ‘‘you’’). Zimmerman 1997 prefers English examples, and introduces two English forms of singular ‘‘you’’: ‘‘youinf ’’ and ‘‘youform ’’. (1a) says, roughly, ‘‘I never address you informally,’’ using the informal pronoun to do so, and (1b) says the same, but uses the formal pronoun.

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(2) ϕ is true but I don’t know that ϕ is true. (2) can be true, yet I cannot know (2). That is, (3) is impossible. (3) I know that [ϕ is true and I don’t know that ϕ is true]. For (3) to be true, I must know that ϕ is true and at the same time know that I don’t know that ϕ is true. (3) is inconsistent in the manner of the paradox: the operator ‘‘I know that . . .’’ is factive and distributive. (2) is interestingly analogous to the earlier self-defeating sentences. Those can be true, but cannot be asserted truthfully. (2) can be true, but I cannot know it. Fitch’s ϕ & ∼ Kϕ can be true, but it cannot be known at all. We might naturally say that (1a) is a self-defeater with respect to assertion, that (2) is a self-defeater with respect to ﬁrst-person knowledge, and that ϕ & ∼ Kϕ is a self-defeater with respect to knowledge simpliciter. John Mackie (1980) gives another helpful example of a self-defeater. Surprisingly (in the way that the knowability paradox is surprising), the principle that every truth can be written truthfully in green ink entails that every truth is written in green ink.⁵ Let ϕ be a truth that is not written in green ink. Then there is the further, conjunctive truth (4), (4) ϕ is true but not written in green ink, which is incompatible not only with every true sentence’s being written in green ink, but with the assumption that every true sentence can be written truthfully in green ink. To write (4) in green ink is to write a false sentence in green ink, because to write it in green ink requires writing ϕ that way, thus falsifying the second conjunct. If, therefore, it is assumed that every truth is writable truthfully in green ink, then there is no truth not written, sooner or later, in green ink. No such difﬁculty arises for the unparadoxical principle that every sentence can be written in green ink, but only the claim that every true one can be written truthfully in green ink, i.e., can be written in green ink and still be true; i.e., that every truth is compatible with its being written in green ink. In light of the foregoing, it is hardly surprising that the principle fails. Some propositions are about (or entail something about) how they are used (in a very broad sense of ‘‘used,’’ meant to include such features as being known, or being written truthfully in green ink, or or indeed pretty much anything about the propositions and expressions of them) so as to ensure that they cannot be used that way and be true. The epistemic self-defeat seen in ϕ & ∼ Kϕ is a species of what we may call generally self-defeat with respect to (the operator) O. When O is factive and distributive, the claim that a non-O truth ϕ is a non-O truth, ⁵ Preserving the analogy with the knowability paradox, take the principle to assert that every true sentence can be written in green ink by someone at some time; similarly the conclusion is that every true sentence is sooner or later written in green ink.

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ϕ & ∼ Oϕ, cannot itself be an O-truth. Some truths are incompatible with being O. That a given truth is non-O is such a one. Since the key reductio for a given operator O hinges only on the factivity and distributivity of O, the fact that ϕ & ∼ Oϕ is subject to the reductio is not always a strictly pragmatic phenomenon. Truth itself is factive and distributive, but the deduction of ∼ Tr(ϕ & ∼ Trϕ ) indicates a semantic fact, that ϕ & ∼ Tr(ϕ ) cannot be true. Any factive distributive operator not equivalent to truth is stronger than truth (since it is factive). When the additional force of O involves pragmatic matters concerning what language-users do, such that ϕ and ∼ Oϕ are compatible, then the key reductio demonstrates the pragmatic self-defeat of ϕ & ∼ Oϕ. Thus we have a recipe for pragmatic self-defeat. Pick any distributive operator A involving pragmatic matters—expressed in English, written in green ink, known—such that ϕ and ∼ Aϕ are compatible, and deﬁne a new operator Oϕ =def ϕ & Aϕ. Then O is factive and distributive. So ϕ & ∼ Oϕ is pragmatically self-defeating with respect to O; thus Fitch self-defeat. The further conclusion ∼ ♦O(ϕ & ∼ Oϕ ) may surprise us when we attribute to all truths the possibility of possessing some stronger (factive, distributive) pragmatic property O that we expect is not possessed by all truths. We simply overlooked the fact that the operator affords a case of Fitch self-defeat. Reﬂection on such examples should minimize the surprise that often attends ﬁrst exposure to a speciﬁc case of Fitch self-defeat. I introduced the assertive self-defeat of ‘‘I am not speaking’’ and (1a) to show that there are non-Fitch forms of self-defeat that are simple and well understood, thereby (I hope) easing the lesson of Fitch self-defeat in particular: for any factive distributive operator O, ϕ & ∼ Oϕ is a Fitch self-defeater. Here is an especially vivid example due to Kvanvig (Chapter 13 of this volume). Let our factive distributive pragmatic operator O be ‘‘it is true and wished for . . . ’’. One might expect that for any truth ϕ, it is possible that ϕ be both true and wished for. Yet again O(ϕ & ∼ Oϕ ) is inconsistent, and ϕ & ∼ Oϕ a Fitch self-defeater: it cannot be true-and-wished-for that ϕ is true but not true-and-wished-for, for this requires both that ϕ is true-and-wished-for and that it is true-and-wished-for that ϕ is not true-and-wished-for, and the latter entails that ϕ is not true-andwished-for. Thus if there is a truth not wished for, then not all truths can be wished for. One more example. Say that knowledge that ϕ is shared when each of at least two individuals sooner or later knows that ϕ. The principle of shared knowability—that for every truth ϕ, it is possible that there are at least two individuals and two times such that one of the individuals knows ϕ at one of the times and the other at the other—is incompatible with the proposition that some truth is never known by more than one person. The principle entails that Wright’s (1987a) example ‘‘Thatcher is a master criminal’’ is false, for if it is true, it is known by at most a single person, the criminal Thatcher herself.

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Our response to the knowability paradox ought to be that this is hardly a big deal. There is nothing perplexing about it, nor, now, surprising. K is just another run-of-the-mill factive distributive pragmatic operator. If all truths are K-able, then there are no truths of the form ϕ & ∼ Kϕ, and so there are no truths ϕ such that ∼ Kϕ.⁶ The notion of pragmatic self-defeat stems immediately from reﬂections on the semantics of indexicals. It is hard to see how this semantic apparatus might be uncongenial to antirealists simply in virtue of their stance on the metaphysical issue separating them from realists. It is prima facie neutral as to whether the involved notion of truth is the realist’s epistemically unconstrained one or the antirealist’s epistemically constrained one. The whole account of self-defeat, and in particular the account of Fitch self-defeat for factive distributive operators, seems so remote from the philosophical issues at stake, and from that lofty perspective so mundane, that the real paradox of the knowability paradox consists in the idea that a single unremarkable instance of Fitch self-defeat would have such overpowering force as is sometimes attributed to it against a metaphysical view on the nature of truth. If antirealism has been previously formulated in such a way that this phenomenon does profoundly undermine it, then antirealists are at worst guilty of an oversight, a mistake in formulation that should be easily correctable in a transparent way. (Is it really the case that it would have been reasonable for Dummett, say, to abandon his project straightaway had the phenomenon of Fitch self-defeat come to his attention in semantic antirealism’s early days?)

A n t i re a l i s t i c Tr u t h a n d Re c o g n i t i o n -t ra n s ce n den ce Antirealism opposes any conception of truth that permits truth to obtain ‘‘recognition-transcendently.’’ Opposition to recognition-transcendence motivates the antirealist’s rejection of the principle of bivalence, because this principle, which accords to every (meaningful) sentence a determinate truth-value, embodies a notion of truth that outstrips any intelligible idealization of our epistemic capacities for determining truth-values. Presumably, not every sentence is ‘‘decidable’’ with respect to some ﬁnite extension of our abilities to recognize truth. Most notably, quantiﬁcation over inﬁnite domains introduces undecidability into our language even if all our epistemically basic sentences are decidable. On this model, epistemically nonbasic sentences are taken to be logical compounds built inductively from basic ones. ⁶ I am not claiming that K is a conjunctive operator obtainable by the recipe above. This would amount to the extremely contentious claim that knowledge can be analyzed as truth plus some further nonfactive property, e.g. belief+justiﬁcation, or belief+justiﬁcation plus an anti-Gettier feature. The plethora of proposals in recent epistemology concerning the last of these inspires no conﬁdence, and some theorists reject the very project of analyzing knowledge in anything like this way.

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Dummett’s famous arguments that provide the ‘‘philosophical basis of intuitionistic logic’’ are taken to show that a theory of meaning adequate for explaining our understanding of our sentences cannot employ an epistemically unconstrained ‘‘central notion.’’ Antirealism thus rejects the classical conception of truth, and denies that distinctively classical rules of inference are valid. The antirealistic idea of a meaning theory for a language is roughly as follows. Sentences that possess truth-values are associated with methods for ascertaining those truth-values, and among these methods are so-called canonical ones. A canonical method for recognizing ϕ’s truth-value, when ϕ is not epistemically basic, is something like an intuitionistic normal form deduction from literals (or at least the notion is thus inspired, and said to have certain attractive features also found in normal form proofs). It is an oversimpliﬁcation to say ϕ’s truth-value depends merely on the values of its immediate subformulas, which depend in turn on theirs, and so on down to epistemically basic sentences, but it will sufﬁce for present purposes. Oversimplifying again (and again because it sufﬁces for present purposes), we may simply posit that the canonical methods of recognizing the truth-values of epistemically basic sentences are null, thus we assume that the basic sentences are all epistemically decidable.⁷ Canonical methods for ascertaining truth-values are meaning-constituting. A language-user’s grasp of ϕ’s canonical truth-recognitional procedure is what her understanding of ϕ consists in.⁸ Other, noncanonical procedures for ϕ are justiﬁed by reference to ϕ’s canonical one, just as nonnormal natural deduction proofs in intuitionistic logic are reducible to normal form ones. A grasp of noncanonical procedures for ϕ is not required for understanding ϕ. In this way an antirealistic theory of meaning avoids truck with a notion of recognition-transcendent truth. When ϕ is true, it has a canonical procedure traversing at most a ﬁnite number of more basic sentences, terminating at a ﬁnite number of epistemically basic ones. Each step of traversal is epistemically constrained —this is the point of barring distinctively classical rules of inference and other objectionably inﬁnitary steps, e.g. allowing the procedure to take a quantiﬁed sentence to an inﬁnite number of instances all of which must be traversed. ⁷ The meaning-theoretical story gets especially complicated concerning conditionals; ϕ → ψ requires an effective method for converting any proof of ϕ into a proof of ψ. Such a method is paradigmatically a proof, from the open assumption ϕ, of ψ. Such a canonical procedure can thus be seen to proceed from the assumption toward the periphery where the basic sentences lie, then changing direction and moving away from the periphery toward the conclusion. Certain difﬁcult meaning-theoretical matters arise in this connection. Because a conditional’s truth seems to depend only on the structures of canonical procedures for its antecedent and consequent, it is hard to see how its truth-value depends differentially on the values of epistemically basic sentences. This complexity is not pertinent to our discussion of ϕ& ∼ Kϕ, so we ignore it. ⁸ Because these methods are determined inductively and parallel the logical structure of their sentences, such a theory of meaning is also able to account for a grasp of an undecidable ϕ as well as a decidable one, thus avoiding the old-fashioned veriﬁcationist’s conclusion that only epistemically decidable sentences are meaningful.

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Such a procedure has the structure of a ﬁnite tree whose undischarged leaves (assumptions) are epistemically basic sentences.⁹ Truth-values of the leaves, by hypothesis none of which requires introduction of a recognition-transcendent notion of truth, ﬁx the truth-values of (some, and in the present case the relevant) nonbasic sentences in the tree by means of antirealistically acceptable, epistemically constrained rules of stepwise traversal (e.g., intuitionistic rules of inference). The involved notion of truth in general is thus epistemically constrained. As we shall see, this need not entail that all true nonbasic sentences are thereby knowable. Let us simplify further. The operator K abbreviates a doubly quantiﬁed sentence, but this complication too is irrelevant to a successful antirealistic response to the knowability paradox. We shall avoid all issues pertaining to the semantical treatment of sentences with K dominant by treating them as basic sentences (!), with various conditions imposed on available distributions of truth-values among them; factivity requires that ϕ be assigned truth when Kϕ is, and distributivity requires that Kϕ and Kψ be assigned truth when K(ϕ & ψ) is. This is all that will matter in the following. The antirealistic meaning theory must involve an epistemically constrained ‘‘central notion.’’ Its associated epistemology of understanding, i.e., its account of what it is to understand a sentence, is a rather ﬁne-grained ‘‘truth-conditional’’ one. To know a sentence’s meaning is to know, in a very particular way, its truth-condition: ϕ is true when there is a canonical procedure for ϕ that issues in the value true. This procedure embodies a partial ordering such that ϕ’s daughters, the sentences immediately preceding ϕ in the ordering, are those a grasp of which are needed for a grasp of ϕ itself. There are only a few ways that ϕ’s daughters may relate to ϕ, and these are the epistemically constrained relationships amounting (in our idealization) to the rules of intuitionistic logic.¹⁰ Our very simpliﬁed picture of a meaning theory ignores the inconclusive, defeasible but nonetheless justiﬁed attributions of truth that are common in empirical discourse. Besides, the whole idea of conclusive reasons for judging an empirical claim true is problematic. The picture also abstracts entirely from the question of the meanings of epistemically basic sentences, not to mention the question of whether there are such sentences in empirical discourse. Moreover it ignores matters pertaining to the meanings of sentences with K dominant, consigning them to the class of basic sentences and capturing only K’s factivity and distributivity by the artiﬁcial device of imposing constraints on assignments of values to basic sentences. Still, this picture manages to embody ⁹ It is especially important to recognize that the complications arising in connection with conditionals and universal quantiﬁcations are irrelevant here, despite the classical appearance of this picture. ¹⁰ Some supplementation of this picture is needed in order to get from basic false sentences to their negations that are needed as open leaves in the tree, but this can be left aside. The proper treatment of falsity is a delicate matter: see my 1999 and Tennant’s 1999.

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everything in an antirealistic theory of meaning that we need in order to explore how the knowability paradox bears upon antirealism. That is, it sheds sufﬁcient light on how an antirealistic theory of meaning avoids objectionably recognition-transcendent truth for us to see why Fitch self-defeat does not threaten antirealism.

Ve r i ﬁ c a t i o n Pro c e d u re s a n d t h e Ac q u i s i t i o n o f K n ow l e d g e It is surely possible that although we all grasp ϕ’s tree, no one ever performs it. Indeed, we can be sure that there are plenty of epistemically decidable sentences that are true, and which we all understand, but whose canonical procedures no one has bothered to perform and never will, nor any indirect procedures for them. This is, on reasonable assumptions, nothing more than the antirealistic rendition of the common-sense claim CS: there are truths never known. Can an antirealist avail herself of this construal of CS? To do so commits her to a potentially troublesome attitude toward these trees. ϕ’s tree exists, in some sense appropriate for the existence claim, and the involved basics are true and false in a distribution of values with respect to which, according to the semantical relationships embodied by its tree, ϕ is true. In other words, these relationships sufﬁcient for ϕ’s truth obtain despite the failure of anyone ever to note that they obtain. Should antirealists grant to ϕ’s tree whatever sort of existence has just been asserted?¹¹ Note that I endorse what a paragraph ago seemed obvious: (i) ϕ is true: it stands in the right semantical relationships to its involved basics and the intervening sentences occurring in its tree, and so its tree exists, just as such abstract entities as, say, natural numbers and functions on them exist; and (ii) ϕ is never known to be true: no one ever performs ϕ’s canonical procedure (its tree) nor any noncanonical procedure for it. There are, then, two distinct issues, belonging to distinct levels of linguistic explanation. First, there is ϕ’s truth, to be characterized meaning-theoretically in terms of the structural features of ϕ’s tree, here conceived as an abstract entity akin to other abstractions that antirealism countenances. Second, and posterior, there are matters of performance of recognitional methods pertaining to ϕ’s truth. The ﬁrst level is a properly semantic one. The second, lower level is a pragmatic one, presupposing strictly semantical notions in its explanations of pragmatic phenomena. ¹¹ There is much to be said concerning this question. See the helpful discussion in Raatikainen. The main advocate of this attitude toward ‘‘veriﬁcation procedures’’ has been Prawitz; Dummett has typically opposed it. (See Raatikainen for references.) After this paper was written, Cesare Cozzo’s 1994 was brought to my attention, wherein he proposes a solution to the knowability paradox that rests, as does the present one, on this conception of these procedures. Comparison with my proposal must await a future opportunity.

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We have seen that an antirealistic theory of meaning avoids recognitiontranscendent truth by imposing epistemically motivated constraints on the structures of canonical procedures, rejecting distinctively classical rules of inference as well as certain other epistemically troublesome ones. Earlier I mentioned as an example of the latter a rule requiring ‘‘traversal’’ of an inﬁnite number of daughters in the determination of a sentence’s value. ‘‘Traversal,’’ however, invites a performance-level reading, while in fact the antirealistic rejection of the rule belongs to the higher, semantical level. To be sure, such a rule is rejected on epistemic grounds, but the antirealist’s meaning-theoretical constraints can be characterized abstractly and belong to the semantic level of the theory. In fact the entire realism/antirealism issue ought to be conceived as belonging to this semantical level.

Un p r o b l e m a t i c “ Re c o g n i t i o n - t r a n s c en d en c e” There are sorts of ‘‘recognition-transcendence’’ of truth to which antirealism need not be allergic. An important aspect of empirical truth is that opportunities to discover it come and go. A truth may have been discoverable at a past time but no longer. ‘‘Whip sneezed downstairs at 3:00 this morning’’ may be true, and had I been awake downstairs at that time I could easily have known it. But I was asleep upstairs, and he sneezes softly. No sneeze-traces remain. Shall we count the truth of the sneeze claim as exhibiting an antirealistically objectionable sort of recognition-transcendence? If so, we are reduced to Ayer’s view that the only true claims about past events are those that we can now establish as true. Antirealists should admit that some truths ‘‘transcend’’ our epistemic capacities merely in virtue of our temporal or spatial remoteness or other incidental, though now permanent, inabilities to perform the procedures needed for discovery. We should hold that our merely missing the opportunity to evaluate the sentence does not count against its truth. Our having missed our only opportunity to perform the tree is not a semantical fact about the tree at all, but only enters our theorizing at the performance level. By present lights, such ‘‘lost opportunity’’ cases are easily explainable in terms of the distinction between our two theoretical levels: decidability, truth, entailment, contingency, logical compatibility, etc., are semantical notions. Lost opportunities are matters of performance, not entering into an account of truth itself. The knowability paradox seems to introduce recognition-transcendence of a related sort. The truth of Fitch’s ϕ & ∼ Kϕ is in some way recognition-transcendent. If antirealism’s opposition to recognition-transcendent truth is correctly codiﬁed in the knowability principle, then the recognition-transcendent truth of ϕ & ∼ Kϕ is thereby assimilated to the realist’s distinctive recognition-transcendent truth

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as opposed to the antirealistically unobjectionable recognition-transcendent truth of lost-opportunity cases. ¹² T h e Re c o g n i t i o n -t ra n s ce n den t Tr u t h o f ϕ & ∼ Kϕ It is common sense that ϕ can be true but never known to be true, never investigated. It is a certainty that ϕ & ∼ Kϕ cannot be known. This is a third, distinctive sort of recognition-transcendent truth: ϕ & ∼ Kϕ can be true, but its truth ‘‘outstrips’’ our epistemic capacities. What sort of recognitiontranscendence is this? Recall (1a): ‘‘Je ne te tutoie jamais.’’ It can be true, but it cannot be asserted truthfully. The mere assertion of it makes the world such that the assertion is false. It can only be true when unasserted. At the heart of this assertive self-defeat is the fact that our language has resources for talking about what is asserted, and these can be slyly exploited to produce such self-defeaters. Fitch self-defeat is parallel: ϕ & ∼ Kϕ can be true, but it cannot be known. It cannot be investigated and found to be true. In particular, no performance of its tree can result in knowledge of its truth. By coming to know ϕ, we make the world be such that ∼ Kϕ is false. At the heart of this epistemic self-defeat is the fact that our language has resources for talking about what is known, and these can be slyly exploited to produce such cases of self-defeat. (1a)’s assertive self-defeat is a ‘‘pragmatic’’ matter posterior to an account of truth. The phenomenon of assertive self-defeat emerges straightforwardly from our semantics together with the fact that our language has the right expressive resources. Fitch self-defeat is likewise a ‘‘pragmatic’’ matter. The phenomenon of Fitch self-defeat emerges straightforwardly from our semantics together with the fact that our language has the right expressive resources. The parallel is close. In both cases, the explanation lies in the language’s possession of sufﬁcient linguistic resources to construct sentences that cannot have a certain ‘‘pragmatic’’ property. The parallel seems to me to tug rather strongly against the thought that antirealism must ﬁnd the Fitch self-defeat of ϕ & ∼ Kϕ troublesome. (And let us not forget Mackie’s claim that Fitch self-defeat ‘‘should be no more surprising than the fact that while I may be saying nothing at t 1 , I cannot say truly at t 1 that I am saying nothing at t 1 ’’. Indeed.) ¹² See Crispin Wright’s suggestive discussion in his 2003. He suggests that the Fitch problem is of the same species of innocuous ‘‘recognition-transcendence’’ as lost-opportunity cases, being matters of ‘‘contingencies of epistemic opportunity’’ as opposed to ‘‘necessities of limitation.’’ He does not explain the assimilation, however. The issue is not simple: in lost-opportunity cases, we take the sentences to be knowable now in virtue of having been knowable then. Fitch’s ϕ & ∼ Kϕ is not a lost opportunity case: there is no then. I agree that the recognition-transcendence of ϕ & ∼ Kϕ is innocuous, but for a reason very different from that of lost opportunities.

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T h e Pe r f o r m a n c e Pr i n c i p l e Rules of inference are the ‘‘building blocks’’ of canonical procedures. Antirealists hold that these must be epistemically constrained. That is, each must be performable in the appropriate sense. It does not follow from this that any procedure built from them is likewise performable under any distribution of values to basics. Due to the presence of linguistic resources permitting talk about these procedures themselves, a language may fail to meet the Performance Principle: if the steps (inference rules) that make up canonical procedures are performable with epistemically basic ingredient sentences—i.e., are performable when they constitute a sentence’s entire procedure—then every (nonbasic sentence’s) procedure built from them is performable too. In other words, if each step in a procedure is performable in the basic case, then the whole procedure is performable. The principle entails, for instance, that if ϕ and ψ have performable canonical procedures, then so does ϕ & ψ. Note that this principle is not a semantical principle. Epistemic constraints on procedural steps are imposed by the antirealist upon the very structures of these procedures—rules of inference embodying a nonepistemic notion of truth are barred. Whether the Performance Principle holds for a language is a further, performance-level issue posterior to imposition of the antirealist’s constraints on truth. The antirealist need not move to this postsemantical level and impose a new pragmatic constraint. Yet the knowability principle is just such a further constraint. The knowability principle says nothing explicit about performability of procedures, but it entails something about them. Given that no truth can be known without having been discovered, and that discovery requires the (abstract, nontemporal) existence of that truth’s canonical procedure, the knowability principle amounts to the Performance Principle. An antirealist’s ready adoption of the knowability principle amounts to acceptance of the Performance Principle without proper attention to the expressive power of the language at hand. This is the mistake underlying the antirealist’s attraction to the knowability principle. For a language containing K, the principle fails. The set of truths with performable procedures is not closed under procedural extension by means of procedural steps that are already antirealistically acceptable. Even conjunction does not always preserve performability. Nonetheless, conjunction preserves ‘‘epistemic decidability,’’ understood as a structural feature of procedural trees themselves. The present language of modal epistemic logic within which we phrase the knowability principle has sufﬁcient resources to formulate a Fitch selfdefeater, ϕ & ∼ Kϕ. Yet we have assumed that antirealism’s epistemic constraints are in force. The knowability paradox demonstrates that these constraints, imposed on the steps of canonical procedures, do not ensure performability

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of all truths’ procedures. Since these procedures already satisfy the antirealist’s epistemic constraints on truth, the Performance Principle’s failure is of no great metaphysical signiﬁcance. That principle is a substantive pragmatic principle not entailed by the antirealist’s epistemic constraints on truth. But is this weaker understanding of the antirealist’s slogan Truth is epistemic (or Truth is epistemically constrained or Truth cannot outstrip our epistemic capacities) really strong enough to be the lesson of the antirealist’s arguments? I must leave the issue for another time. To make the case would require a careful re-examination of versions of those arguments, with an eye toward whether their conclusion can, in light of the Performance Principle’s failure, be rephrased as a claim about the structures of canonical procedures, or whether the conclusion requires formulation in pragmatic terms like knowability and performability. Related antirealistic concerns, involving such ‘‘proof-theoretic’’ matters as harmonious rules of inference and whether a logical constant’s inclusion in a language yields a conservative extension of that language, are prima facie compatible with the spirit of the present proposal. In a nutshell, truth of ϕ consists in there being a properly constrained canonical procedure for ϕ which, under the circumstances, yields for ϕ the value true. Since the language at hand has sufﬁcient expressive resources for the expression of the Fitch self-defeater ϕ & ∼ Kϕ, the Performance Principle fails, and in particular the procedure for the truth ϕ & ∼ Kϕ is not performable.¹³ When the conjunction is true, it is not knowable. This sort of ‘‘truth-transcendence’’ is of an unobjectionable sort, being a pragmatic phenomenon independent of the antirealistic imposition of epistemic meaning-theoretical constraints operative at the semantic level, unlike the recognition-transcendence to which antirealism takes exception and in this way like that of lost-opportunity truths. The very term ‘‘veriﬁcation procedure’’ is a misnomer, when the language under discussion is one for which the Performance Principle fails. Epistemically constrained procedures for truths are not always performable. This is no reason to think that an epistemically constrained central notion of a theory of meaning for the language is unavailable. It is no reason to think that the attendant notion of truth is thereby antirealistically unacceptable for being insufﬁciently constrained epistemically.

A g a i n s t t h e Re s t r i c t i o n St r a t e g y I hold that antirealism’s epistemic constraints do not commit the antirealist to the knowability principle. The knowability paradox is a diverting but irrelevant sideshow in pragmatics. A different response to the paradox is to impose a ¹³ Note that ∼ Kϕ is not epistemically basic. In reality, of course, even the sentence of which Kϕ is an abbreviation is not a basic one.

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restriction on the knowability principle so as to avoid the key reductio by precluding instances of the principle that would otherwise give rise to it, while maintaining the principle’s status as the fundamental epistemic constraint on the antirealist’s theory of meaning. According to this restriction strategy, truth’s epistemicity consists in its knowability under the imposed restriction conditions. A good deal of attention has been focused on the pros and cons of restrictionstrategic proposals.¹⁴ I need not address details of particular ones, since my comments apply to any restricted principle and the role it can play in articulating the conceptual structure of antirealism, that is, whether it can serve as the antirealist’s fundamental epistemic constraint on the antirealistic meaning theory’s central notion. No such proposal can save antirealism, as familiarly understood, from the paradox, for no antirealist who takes knowability to be antirealism’s fundamental epistemic constraint on truth can abide a restriction on it. Rather, the antirealist must formulate an epistemic constraint on truth that is unrestricted, and presumably some restricted knowability principles would be derivative, pragmatic consequences. (The latter approach is not a restriction-strategic one as I am using the term.) It is already clear from the paradox that antirealists cannot hold the unrestricted knowability principle to be true, regardless of the theoretical role they might have liked it to play. There are of course restrictions that successfully prevent emergence of the paradox. On my own view, for instance, performability (and knowability) can be expected to be preserved in most cases, indeed in all cases except self-defeaters. If this is right, then Neil Tennant’s (1997) restricted knowability principle is one I will endorse. I will maintain it as a pragmatic thesis, however, derivative upon across-the-board epistemic constraints at the semantic level. The ‘‘restriction strategy,’’ on the other hand, seeks both to avoid the paradox and to maintain the meaning-theoretical centrality of the new, restricted principle. Both Tennant (1997) and Michael Dummett (2001) are restriction strategists in this sense. For example, in ‘‘The Philosophical Basis of Intuitionistic Logic,’’ Dummett writes, The argument told in favor of replacing, as the central notion for the theory of meaning, the condition under which a statement is true, whether we know or can know when that condition obtains, by the condition under which we acknowledge the statement as conclusively established, a condition which we must, by the nature of the case, be capable of effectively recognizing whenever it obtains. (Dummett 1978: 226–7)

That is, truth, as the realist conceives it, cannot serve as the ‘‘central’’ meaningtheoretical notion, and must be replaced by a property whose presence we ¹⁴ Tennant’s proposal is criticized by Jonathan Kvanvig and myself (1999), to which he responds in Tennant (2001b), and by Timothy Williamson (2000b), to which he responds in Tennant (2001a). Tennant (2002) criticizes Dummett’s proposal, and Berit Brogaard and Joseph Salerno (2002) helpfully investigate both Tennant’s and Dummett’s proposals.

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can always effectively discern. So truth, antirealistically conceived, is knowable—always. This conception of truth, unlike the realist’s, satisﬁes the requirements placed upon it by its role in a theory of meaning. Whatever plays this role in a theory of meaning must be epistemic. Tennant, in discussing Dummett’s disavowal of ‘‘global antirealism’’ in favor of a more piecemeal approach, considers the apparent generality of Dummett’s arguments and writes of the meaning-theoretical principles of Manifestationism, Molecularism, and Compositionality, These are principles of very wide scope, which are not to be thought of as standing or falling depending on the discourse in question. There is therefore certainly enough depth and substance in the antirealist’s initial thoughts about meaning for it to be quite in order to represent him as putting forward a global antirealism. In particular, before he even considers what is peculiar to any one discourse, the anti-realist will be committed to the tenet that truth is in principle knowable. (1997: 50)

He adds later in the same work, concerning the force of the knowability paradox against the antirealism in question (i.e., one that balks at concluding that no truths remain unknown), that the antirealist will still want to maintain that all truths are knowable, even if not actually known. That, after all, is what makes him an anti-realist. (1997: 265)

I read this as indicating not merely that some (restricted or unrestricted) knowability principle is a necessary commitment of antirealism, perhaps following from some deeper antirealistic thesis about truth, but rather that knowability is again the antirealist’s fundamental epistemic constraint on truth. Thus do our restriction strategists take the epistemic character of truth to be the constitutive claim of their antirealism, and to equate truth’s epistemicity with its knowability. For such antirealists, restricting the knowability principle is not an option. They escape the reductio, but forfeit the equation of truth’s epistemic nature with its now restricted knowability. Restriction strategists grant that the paradox demonstrates the untenability of the knowability principle. They fail to realize that a restricted knowability principle cannot serve as the key epistemic feature of truth that antirealism requires, no matter how the restriction strategist proposes to restrict it. If the fundamental issue separating antirealism from realism is whether truth must invariably be subject to some interesting epistemic constraint due to its meaningtheoretical role, then this feature is not knowability, restricted or otherwise. Whatever the relationship of truth to knowability, the latter must be at best an occasional consequence of truth’s being as antirealism distinctively requires. Antirealism’s fundamental epistemic constraint on truth cannot amount to a knowability principle, restricted or unrestricted, though it can be expected to entail some restricted ones. That truth is epistemically conditioned is a claim about the very nature of truth, motivated by its role in the antirealist’s meaning theory. It precludes truth’s

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ever obtaining nonepistemically. Epistemic truth is the notion in terms of which the antirealistic meaning theorist explains what it is to know the meaning of any (meaningful) sentence.¹⁵ If truth’s epistemic conditioning consists in nothing more than the knowability of some but not all truths, then the antirealist’s meaning theory is a lost cause. It endeavors to explain, strictly by means of its epistemic conception of truth with respect to some sentences, what it is to know the meaning of a sentence, even one that falls outside the restriction and whose truth need not be ascertainable. To assert that the epistemic nature of truth amounts to its knowability—that knowability is the fundamental antirealistic constraint on truth—and then respond to the paradox by restricting this knowability defeats the primary meaning-theoretical purpose of the notion of truth. Antirealism imposes epistemic constraints on truth generally, not merely in special cases. A restricted knowability principle has the form ∀ϕ (ϕ → (Rϕ → ♦Kϕ ) ), and if R is relevantly interesting then the realist and antirealist will disagree on the principle. As Tennant observes concerning his own proposed R, ‘‘To claim that every such truth is in principle knowable is still to forswear metaphysical realism’’ (1997: 275). But let condition R be the complement of R among sentences; the restricted principle says nothing at all about truth of R -sentences. It locates the epistemicity of truth in an epistemic feature of R-truth while the strategist hastens to add that not all R -sentences are false. (Indeed, the restriction strategy is designed precisely to permit its proponents this addendum, lest they succumb to the paradox, afﬁrm that no propositions outside the restriction are true, and conclude that no truths remain unknown.) This issue, whether R-truths in particular are all knowable, cannot be the fundamental meaning-theoretical one concerning truth that divides antirealism from realism. If it is, then the realism/antirealism issue is not a general one about truth. If antirealism is a distinctive meaning-theoretical view about the nature of truth, then it must inform us even about the truth of paradox-generating propositions. Antirealists may be unable to produce such a truth, but nonetheless their view must not require them to treat R -propositions so dismissively as to be helpless against the suggestion that these are subject to a nonepistemic notion of truth (or else are all false). To enshrine a restricted knowability principle as the fundamental epistemic constraint on truth reduces the issue to one concerned only with propositions satisfying the restriction, as if antirealism allows the metaphysical chips to fall where they may when it comes to the remainder, not all of which are false. Is nothing left of the dispute when realists and antirealists are asked to comment on the meanings of R -propositions? Are R -propositions ¹⁵ As mentioned earlier, the matter is made somewhat complicated by the fact that not all meaningful sentences possess a truth-value, on the antirealistic account. This does not mean that our understanding of such a sentence is not explained in terms of recognitional abilities regarding truth. Indeed, it is because antirealistic truth is conditioned by our epistemic resources that it is not subject to bivalence.

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simply irrelevant to the dispute? They are certainly not meaningless. A theory of meaning must bring its ‘‘central notion’’ to bear on them despite their peculiarity, yielding (if truth is not itself the central notion) an epistemically constrained notion of truth applicable to them as well as to well-behaved others. The antirealist cannot say that ‘‘Truth is epistemically conditioned’’ expresses only an insight into the nature of R-truth, while refusing to assert that R-truth is all the truth there is. This is not to deny that some restricted knowability principles hold, but such a principle cannot be the fundamental epistemic constraint on truth. The antirealist’s afﬁrmation of it cannot be, as Tennant says, ‘‘what makes him an antirealist,’’ for surely antirealists must deny that truth is ever nonepistemic. And yet restriction strategists do not hold that truth is always knowable. At this point, our antirealist must produce an account of truth’s epistemicity that applies across the board, and must explain by means of this account why failure of the unrestricted knowability principle has no bearing on the antirealist’s insistence that truth is epistemically constrained. To do this is to give up the restriction strategy. T h e Id e a l i s m Pr o b l e m Tennant has drawn attention to another self-defeater with respect to knowledge, here phrased as (I). (I) There are no epistemic agents. (By ‘‘epistemic agents’’ I mean individuals able to know things.) Even the antirealist would like to grant that (I) is possible, but (I)’s truth precludes its being known. It is thus unknowable. Is this ‘‘recognition-transcendence’’ a sort to which the antirealist need object? Must the antirealist judge (I) to be necessarily false? (I call this problem the ‘‘idealism problem’’ because the antirealist wishes to avoid this typically idealistic claim.) The antirealist is not required to hold that (I) is necessarily false, on my view, for truth of (I) would consist in the fact that the canonical procedure for (I) takes, under imagined circumstances, the value true. The fact that there is no one around to perform the procedure does not threaten the antirealist’s implementation of her epistemic constraints, but rather is just another case of pragmatic self-defeat. Just as ‘‘I never speak English’’ can be true only when unuttered, ‘‘There are no epistemic agents,’’ like ϕ & ∼ Kϕ, can be true only when unknown, i.e., when its procedure goes unperformed. Fi n a l C o m m e n t The proposal at hand has some worrisome parts. It assumes that a canonical procedure for a true sentence can be conceived by the antirealist as an abstract

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nontemporal formal object that may never be instantiated and may not even be instantiable. It imports a semantics/pragmatics distinction and applies it in a way that may turn out to be inappropriate. It endangers the Dummettian idea that we manifest our grasp of a truth by recognizing a canonical veriﬁcation of it when presented with one. In these ways and perhaps others it may involve unhappy concessions to realism. Even so, it strikes me (right now, at any rate) as a principled and indubitably antirealistic response to the knowability paradox, which can be seen as a bitter reminder of certain linguistic phenomena that antirealists must not ignore.

18 The Mystery of the Disappearing Diamond C. S. Jenkins

In t ro d u c t i o n : Tw o Pu z z l e s The proof now often known as the ‘paradox of knowability’ was originally formulated in print by Fitch in his 1963 (and previously suggested to him by Church in a referee’s report—see Salerno, Chapter 3 of this volume). It presents a challenge to a claim which is commonly associated with certain forms of global anti-realism. ‘Global anti-realists’, as I use the term, are those who are sympathetic to some version of the claim that reality, in its entirety, is dependent in some signiﬁcant way upon ourselves (usually upon our minds and our ways of thinking). This sort of world view often gives rise to the thought that, because of its minddependent nature, all of reality is epistemically accessible to us. And this thought in turn is often taken to amount to the claim that all true propositions are knowable. The Church–Fitch argument purports to show that, provided we accept only a couple of uncontroversial principles about knowledge, the claim that all true propositions are knowable commits one to the apparently much stronger claim that all true propositions are known. This (for all but the most extreme) is an obviously undesirable consequence. As is customary, in this discussion I shall use ‘Kp’ to mean ‘It is known by some being at some time that p’, hiding the two existential quantiﬁers. I shall begin by presenting what I think is the clearest exposition of the paradox argument, that found in Williamson (2000a) (except that, for the sake of simplicity of presentation, where Williamson uses quantiﬁcation over propositions I shall use schematic letters ‘p’, ‘q’ etc. to stand for arbitrary propositions). I am grateful for helpful comments and suggestions made by Jc Beall, Eline Busck, Lars Gundersen, Jon Kvanvig, Aidan McGlynn, Daniel Nolan, Joe Salerno, Kim Stebel and members of the NAMICONA research centre at the University of Århus who attended a seminar where a version of this paper was presented in August 2005. I would also like to repeat my thanks to those acknowledged in Jenkins (2007) (Nick Denyer, Dominic Gregory, Michael Potter and an anonymous referee), since the work represented there provides a foundation for the current paper in many respects.

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The argument relies on the factivity of knowledge: FACT: (Kp ⊃ p) and the claim that knowledge necessarily distributes over conjunction: DIST: (K(p & q) ⊃ (Kp & Kq) ) to show that what Williamson calls ‘weak veriﬁcationism’: WVER: (p ⊃ ♦Kp) entails what he calls ‘strong veriﬁcationism’: SVER: (p ⊃ Kp). The argument runs: (1) (2) (3) (4) (5) (6)

(K(¬Kp) ⊃ ¬Kp) (K(p & ¬Kp) ⊃ (Kp & K(¬Kp) ) ¬K(p & ¬Kp) ¬♦K(p & ¬ Kp) ( (p & ¬Kp) ⊃ ♦K(p & ¬Kp) ) ¬(p & ¬Kp)

by FACT by DIST from (1) and (2) equivalent to (3) by WVER from (4) and (5)

Note that the only things we rely upon in order to derive (6) are WVER and the uncontroversial principles FACT and DIST. Of course, (6) isn’t quite the same as SVER, but SVER is a trivial consequence of (6) in classical logic. And rejecting classical logic in favour of intuitionistic logic won’t help us much, because it is worrying enough for defenders of WVER if it commits you to (6), the claim that no true proposition is unknown. Other, more radical, departures from classical logic could be used to block the reasoning in one way or another.¹ For current purposes, however, I shall assume that more conservative approaches are to be preferred where available. Two interesting sets of questions have been raised in connection with the Church–Fitch argument. One familiar set, which I shall call ‘the Classic Puzzle’, concerns the relevance of the argument to the question of whether global antirealism is true. It might be tempting to regard the argument as a straightforward refutation of that doctrine. But even those who (like myself) are inimical to global anti-realism tend to feel that such an easy victory is a bit cheap. It seems unlikely that the deep issues which divide the realist and the anti-realist can be settled by a six-line proof. Questions which I take to form part of the Classic Puzzle, and which have received a good deal of attention in the literature, include: • •

Does the Church–Fitch argument really refute global anti-realism? If it does not, is this because the argument is fallacious, or because anti-realists are not in fact committed to WVER? ¹ Thanks to Jc Beall for pressing me to clarify this point here.

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If anti-realists are not committed to WVER, how should their doctrine of epistemic accessibility be expressed?

Although I think all these questions are important and interesting, in this paper I shall be primarily concerned to address a second set of questions, of which Jonathan Kvanvig (2006 and Chapter 13 of this volume) has recently stressed the importance. According to Kvanvig, the preoccupation with the relevance of the Church–Fitch proof to anti-realism has led philosophers to neglect the fact that, regardless of whether one is an anti-realist or not, there is something deeply surprising about the fact that SVER follows from the apparently weaker WVER. There is a sort of modal collapse: the diamond in WVER just disappears by the time we get to SVER. This is the ‘mystery of the disappearing diamond’ of my title. I take the following questions to belong to the second set, which I shall refer to as ‘the New Puzzle’: • • •

Does this apparent modal collapse really occur? If it does not, where does the Church–Fitch proof go wrong? If it does, what satisfying explanation can we give of this collapse?

(Note that, although there is some overlap between the questions in the ﬁrst set and those in the second, the focal points of the two enquiries are rather different.) I am inclined to think that the best response to the Classic Puzzle is that anti-realists are not (or at least should not be) committed to WVER. At worst, they might be committed in virtue of their epistemic accessibility thesis to a different modal claim, one which does not commit them to SVER. In the next section, I shall outline my take on the Classic Puzzle. In the remainder of the paper, I shall try to get clear about what exactly the New Puzzle is and how (if at all) we can solve it. Although I shall begin by adding a question to the set which comprises the New Puzzle, I shall eventually propose that my favoured response to the Classic Puzzle provides resources for addressing the New Puzzle too.

C h a n g e - t h e - c l a s s a n d C h a n g e - t h e - c l a i m² One classic response to the Classic Puzzle is to adopt what’s sometimes known as a ‘restriction strategy’. (The name must be used with some caution, for reasons to be described shortly.) That is, to somehow reformulate the epistemic accessibility claim of the anti-realist so that it does not share the undesirable consequences ² An alternative way of categorizing the various strategies described in this section can be found in Brogaard and Salerno (2004). They describe Edgington’s manoeuvre as a ‘semantic’ restriction strategy, and Tennant’s and Dummett’s as ‘syntactic’ restriction strategies. A semantic/syntactic distinction need not match up extensionally with the distinction I draw between change-the-claim and change-the-class manoeuvres. For instance, it could be proposed that we limit the class of relevant propositions to those which meet some semantic criterion.

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of WVER. A common way of doing this is to say that not all true propositions are supposed by the anti-realist to be knowable, but only some. (‘Restriction strategy’ is a good name for these views.) Tennant (1997, chapter 8), for instance, has suggested that what anti-realists should say is that for every true proposition p such that it is consistent to assume that p is known, p is knowable. This prevents the anti-realist having to accept line (5) of the above proof. Dummett (2001) argues that anti-realists should say that basic propositions are knowable if true, which also prevents acceptance of line (5). Both of these are what I shall call ‘change-the-class’ strategies: they work by changing the class of propositions p for which the anti-realist holds p ⊃ ♦Kp. Another way of thinking about them is as changing the antecedent of WVER from ‘p’ to ‘p & q’ for some q; for instance, Dummett’s ‘q’ is ‘p is basic’. There has been a good deal of debate concerning the acceptability of these change-the-class strategies, which I won’t go into here. I’ll just note a (commonly shared) intuitive response, which is that such strategies can seem rather ad hoc. They have a whiff of monster-barring: we want to adhere to the general principle WVER, but we don’t like the consequences of certain particular instances of it, so we explicitly rule out those instances. The hard work to be done if this is your preferred tack is to dispel this whiff. A different sort of response to the Classic Puzzle is to change, not the class of propositions p for which the anti-realist holds p ⊃ ♦Kp, but rather the claim that the anti-realist makes about all propositions. One can think of this type of strategy as changing the consequent of WVER from ‘♦Kp’ to something else. A ‘change-the-claim’ strategy may or may not amount to something which could sensibly be called a ‘restriction’ strategy. For the resulting anti-realist claim may be strictly weaker than WVER, in which case the term ‘restriction’ would seem appropriate. Alternatively, it may be weaker in some ways (enough to avoid the paradox) and stronger in others, in which case the word ‘restriction’ is potentially misleading. In any case, it is important to note that the species of restriction involved in change-the-claim strategies is logical weakening rather than (the more speciﬁc notion of) delimitation of the class of propositions to which the knowability claim applies. My own preferred strategy with respect to the Classic Puzzle, which I’ll describe in a moment, is a change-the-claim strategy. Another is due to Cozzo (1994), who suggests that an anti-realist understanding of truth need only lead one to accept that if p is true then there is an ideal argument for p. This is intended as a claim that is in some ways weaker than WVER since, on Cozzo’s conception, an ideal argument for a true proposition p may exist without p’s being knowable. Hence it is consistent to suppose that all true propositions have ideal arguments although some are unknowable. But Cozzo’s anti-realist position might also be thought to be stronger than WVER in some respects. For all Cozzo says, it may be that some true proposition is knowable although no ideal argument for it exists. If that is so, then it is

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consistent to suppose that all true propositions are knowable although some lack ideal arguments;³ in which case Cozzo’s anti-realist position has strength which WVER lacks. Edgington (1985) has proposed a change-the-class-and-the-claim strategy. She suggests that the anti-realist should say that for all p, if actually p is true then actually p is knowable, or in symbols: Ap ⊃ ♦KAp. I’ll follow Williamson in calling Edgington’s proposed anti-realist principle WAVER. Edgington’s approach might be described as a ‘restriction’ strategy in two senses, in that WAVER is strictly weaker than WVER (for it is simply WVER applied to truths of the form Ap) and the class of truths mentioned in its antecedent is restricted. (Notice that WAVER is importantly different, with respect to its modal status, from the pure change-the-claim strategy which adopts as its anti-realist principle p ⊃ ♦KAp. The latter is false at some worlds where WAVER, and indeed WVER, are true.⁴ Thus the pure change-the-claim strategy generates something which is in some ways stronger than WVER, while WAVER is strictly weaker.) Using WAVER instead of WVER we block the paradox argument by changing the consequent of (5) to ♦K(A(p & ¬Kp)), which does not contradict (4). I shall not go into the merits of Edgington’s approach, except to mention that two problems with it are that it does not seem quite true to the spirit of anti-realism, because it only claims that certain necessarily true propositions are knowable (actually p is necessary if p is true), and that it might be difﬁcult for beings in other possible worlds to know propositions of the form actually p, since reference from within non-actual worlds back to the actual world is problematic. (See Williamson 2000a, chapter 12, for a discussion of these objections. R¨uckert attempts to respond to them in his 2004, but I query the success of these responses in Jenkins 2007.) My own view concerning the nature of mind-dependence anti-realism suggests two ways in which the global anti-realist might try to avoid a commitment to WVER. Firstly, I argue in Jenkins (2005) that anti-realism should not be characterized in modal terms. I don’t think anti-realism is the view that it’s impossible for there to be a truth which is not appropriately related to our mental lives. Instead, I suggest that anti-realism is (some form of) the view that what it is for something to be true (or, more carefully, what it is for something to be the case) is for it to be appropriately related to our mental lives. You might think the latter implies the former, but there are some who would dispute that. Our assessments of other possible worlds take place within this ³ This supposition is not consistent with the further assumption that some truths are unknown, of course. But that is irrelevant to the question of relative strength which is at issue here. ⁴ Consider some contingent falsehood f and a world w at which f is true, such that there is an omniscient being at w. Everything which is true at w is known (and hence knowable) at w. At w, therefore, WVER is true (and hence so is WAVER). But f ⊃ ♦KAf is false at w. For f is true at w, but it is not possible at w (or anywhere else) for someone to know Af. Af is false—hence unknown—at every world because f is false at our actual world.

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world. So you might think that, although what it is for a sunset to be beautiful is for us to think of it as beautiful, there is a possible world where a sunset is beautiful although we don’t exist at that world and hence don’t think of it as beautiful at that world. The reason that sunset is beautiful, even though what it is for it to be beautiful is for us to think it is beautiful, is that we in our actual world think of it as beautiful when we are assessing this possible world. (See my 2005: 202–4 for further discussion of this point.) However, even if you’re not persuaded by this line of thought (and I myself am not sure how persuasive it should be taken to be), there are reasons to doubt whether anti-realists are committed to the particular modal claim represented by WVER. As I argue in Jenkins (2007), the most the anti-realist is committed to is the view that reality is epistemically accessible. And that, I claim, amounts to the view that: WVER∗ : For any true p, the state of affairs S which at the actual world makes p true is recognizable.⁵ This doesn’t imply that p is knowable, because it may be that at any world where the state of affairs S is recognized, S does not make p true.⁶ In order to motivate this claim, I require that there be a difference (at the actual world) between knowing p and recognizing the state of affairs which renders p true. For if there is not, then there are no possible worlds where people do the latter without doing the former. States of affairs must be, to some extent, extensionally individuated, so as to allow the requisite distance between recognizing states of affairs and knowing the corresponding propositions. That is, the identity of a state of affairs must be somewhat independent of the proposition by which it is picked out, such that, even if recognizing a state of affairs always amounts to knowing some proposition or other, it need not be that recognizing the state of affairs which actually makes p true necessarily amounts to knowing the proposition p. It may be that, at some non-actual worlds, knowledge of a different proposition which picks out the same state of affairs sufﬁces for recognition of that state of affairs. What I recommend in Jenkins (2007) is that, for these purposes, we think of the states of affairs corresponding to true propositions p as extensional as regards the objects referred to in p and the ranges of quantiﬁer phrases in p, but hyperintensional⁷ as regards the properties ascribed to those objects or collections ⁵ I assume a one–one correlation between true propositions and states of affairs which make them true. ⁶ Some claim that the existence of a truthmaker necessitates any proposition it makes true; see e.g. Armstrong (1997). I am not tempted by truthmaker necessitarianism myself. For current purposes, however, I am equally happy to deny that the states of affairs I’m interested in are ‘truthmakers’ in the sense Armstrong has in mind. The idea behind the envisaged notion of truthmaking is that for every true proposition p, there is a state of affairs the existence of which supplies a certain kind of explanation of p’s truth. Explanation does not require necessitation. ⁷ Jenkins (2007) says ‘intensional’, intending the (somewhat old-fashioned) usage on which this simply means non-extensional. Current usage prefers to reserve ‘intensional’ for that which

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of objects. (So that, for instance, the state of affairs which renders true the proposition All cats purr can be represented as an ordered pair consisting of the class of all cats and the property of purring. The range of the quantiﬁer phrase ‘all cats’ is then treated extensionally, since classes are extensionally individuated, but the state of affairs retains a hyperintensional aspect, since the property of purring is not deﬁned by its extension or intension.) As with Cozzo’s proposal, it is not perspicuous to describe my strategy as a ‘restriction’ strategy. Like Cozzo, I change the claim that the anti-realist makes concerning all true propositions, and there are some respects in which the new proposal is weaker and some respects in which it is stronger. It is weaker because recognition of the state of affairs which actually makes p true can occur (at non-actual worlds) without knowledge of p. Hence it may be that every true proposition is such that the state of affairs which actually makes it true is recognizable, even though some of those propositions are unknowable. Hence WVER∗ avoids a commitment to WVER and the Church–Fitch argument does not commit WVER∗ ’s defenders to SVER. But WVER∗ is also stronger in some respects than WVER (for reasons related to those described in note 4 above). Consider some contingent falsehood f and a world w at which f is true, such that there is an omniscient being at w. Everything which is true at w is known (and hence knowable) at w. At w, therefore, WVER is true. But WVER∗ is false. For it is not possible at w (or anywhere else) for someone to recognize the state of affairs which at the actual world makes f true: there is no such state of affairs at any world.⁸

W h a t E x a c t l y i s t h e Ne w Pu z z l e ? So much for the Classic Puzzle. The New Puzzle is supposed to make the Church–Fitch argument interesting regardless of our solution to the Classic Puzzle, and regardless of whether we are realists or anti-realists. So let us now turn our attention to it. Why should realists care about the Church–Fitch argument, given that they can just deny WVER? Well for one thing, realism as I deﬁne it in Jenkins (2005) is consistent with WVER. We can accept that all true propositions are knowable (WVER) while denying that what it is for a proposition to be true is for it to be knowable (i.e. while remaining realists by my lights). We might simply want to combine realism with optimism about our epistemic capabilities. So some is non-extensional and non-hyperintensional. In any case, hyperintensionality is more speciﬁcally what is intended. ⁸ It is for this kind of reason that anti-realists should not regard WVER∗ as necessary; see Jenkins (2007), note 2. (A related principle involving the appropriate relativization to worlds will, however, presumably be regarded as necessary.)

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realists may feel tempted to accept WVER. These realists are obliged to interest themselves in the paradox of knowability, at least to the extent that anti-realist defenders of WVER are so obliged. And even realists who are not tempted to accept WVER might ﬁnd it surprising that a commitment to WVER commits one to SVER, and want to know more about why this is so. Moreover, even if realists try to free themselves of the Church–Fitch problem by denying WVER, realism does not itself supply a good enough explanation of why WVER is false, as Douven (2005: 63–4) has pointed out. Certain kinds of realist view might give us reasons to doubt whether it is feasible, or physically possible, to know certain truths. But taking the diamond in WVER to indicate only metaphysical possibility, realist thinking typically gives us no good reason to reject WVER. In fact, however, the primary way in which I think the Church–Fitch argument is equally important to realists and anti-realists alike is that the argument is simply interesting in its own right and regardless of whether we accept WVER. WVER is a thought-provoking thesis and it is interesting to think about what sorts of consequences it has. Moreover, the appearance of modal collapse between WVER and SVER is somewhat surprising. To quell this surprise we need either an explanation of what’s wrong with the reasoning from WVER to SVER, or else some satisfying explanation of the collapse. It might sound even more impressive to put the explanans this way (as Kvanvig does in his 2006: 55 and elsewhere): WVER and SVER are shown by the Church–Fitch proof to be logically equivalent, so we seem to have lost a ‘logical distinction’ between the two. SVER obviously commits us to WVER, and now Church–Fitch shows us that WVER commits us to SVER too. Given that FACT and DIST are both true, WVER is true iff SVER is. And given that FACT and DIST are both necessary, WVER is true in exactly the same worlds as SVER. However, the claim of logical equivalence is potentially misleading. It is far from obvious that FACT and DIST are both logical truths, but the proof of SVER from WVER uses them⁹ (see Jenkins 2006). Kvanvig acknowledges the non-logical nature of FACT and DIST upfront in his paper for this volume (Chapter 13). However, many remnants of his 2006 formulation remain (sometimes, but not always, qualiﬁed as ‘loose’ or ‘careless’ statements of the problem). And, as we shall see on p. 314 below, the question of whether or not the equivalence is logical is potentially important when considering what is the correct approach to the New Puzzle. So it is worth stressing again, to avert any potential confusion, that logical equivalence is not on the cards. Even Kvanvig’s most ‘careful’ statement of the supposed problem says that there is a ‘lost logical distinction between actuality and possibility with respect ⁹ Williamson (1993) points out that some arguments against WVER and related principles might be constructed without relying on DIST (section IV). But, as he points out, DIST is still required for the formal proof of the equivalence.

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to what is known’ (p. 211). This is very misleading. Setting aside the point about non-logicality, there is clearly a distinction ‘between actuality and possibility with respect to what is known’ as that phrase is most naturally understood. For the known truths are a proper subset of the knowable truths. Formulations like this risk conﬂating something genuinely alarming but not established by Church–Fitch with something established by Church–Fitch but not genuinely alarming. Another alarming-sounding but, on a little reﬂection, misleading formulation of the supposed problem is also offered: that there is ‘no logical distinction between universally knowable truth and universally known truth’. This sounds impressive enough. Except that the only explication offered of this claim is that it means that there is an equivalence (given FACT and DIST) between WVER and SVER. But that is the familiar equivalence which Church–Fitch obviously brings to light. Kvanvig is supposed to be telling us what is so alarming about this equivalence, not just attaching a new label to it. He is claiming to have noticed a ‘lost logical distinction [which] is part of a ﬁrmly entrenched understanding of the nature of the modalities of necessity, possibility and actuality’ (p. 222). But it is far from ‘ﬁrmly entrenched’ that WVER is not equivalent to SVER in the presence of FACT and DIST. Many people think it is. This is a mere quibble, however. One thing it is much more important to note is that what we do not have on our hands here is a case of complete collapse of ‘♦Kp’ into ‘Kp’, where complete collapse would mean that we could replace ‘♦Kp’ with ‘Kp’ wherever we liked. It’s just that (in the presence of FACT and DIST), we can make such a substitution within the consequent of this one conditional: ‘p ⊃ ♦Kp’. It’s thus misleading to say that the Church–Fitch proof threatens us with the conclusion that there is ‘no . . . distinction between actuality and possibility in this way’ (this volume, p. 208), or to say, however ‘carelessly’, that it suggests that ‘possible knowledge implies actual knowledge’ (p. 208). The difference in strength between ♦Kp and Kp is not undermined by the Church–Fitch proof, since there are still plenty of contexts where the latter cannot be substituted for the former, even given FACT and DIST. This fact should already go some way towards lessening any surprise we may feel at learning that WVER commits one to SVER. For this sort of thing, i.e. modal ‘collapse’, or other similar strenthening, within the consequent of one particular conditional, happens in many other cases too—cases where it is clearly nothing to be concerned about. For instance, it’s not surprising that we can replace ♦p with p in the consequent of a material conditional that has a necessarily false antecedent and end up with something that is equivalent to what we started with. Similarly, p ⊃ ♦p is equivalent to p ⊃ p, but I take it this is not surprising either. Neither of these equivalences does anything to ‘threaten the logical distinction between possibility and actuality’ in this area, i.e. the distinction between ♦p and p.

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One might think that the reason these cases aren’t surprising is that we’re dealing in logical truths, and it’s never surprising that two logical truths are equivalent. I’m not sure how good this response is. Insofar as it is ‘surprising’ that you can replace ‘♦Kp’ with ‘Kp’ in WVER without changing the circumstances under which it is true, you might think that it should be equally ‘surprising’ that when you can replace an occurrence of ‘♦p’ with ‘p’ in one of the above contexts without changing the circumstances under which it is true—i.e. without transforming the initial logical truth into something which is not a logical truth. But in any case, there are other parallel cases that do not deal in logical truths. For instance, ¬p ⊃ (p v q) is logically equivalent to ¬p ⊃ q. I’ll call this the case of the disappearing disjunct. I take it that the fact that the consequent here can be strengthened from (p v q) to q without changing the circumstances under which the proposition is true isn’t especially alarming or paradoxical, provided we have a good grip on how the material conditional works. Certainly it does not do anything to threaten the logical distinction between (p v q) and q. Moreover, further examples are available where the equivalence is not even logical (for an even closer analogy with the Church–Fitch case). For instance, given that it is a necessary, but non-logical, truth that all jade is either nephrite or jadeite, ‘(¬X is jade) ⊃ (X is nephrite or X is jadeite)’ is equivalent to, i.e. true at all the same worlds as, ‘(¬X is jade) ⊃ (X is nephrite)’. Again, nothing paradoxical is going on here; the distinction between ‘X is nephrite or jadeite’ and the stronger ‘X is nephrite’ is not under threat just because the latter can be substituted for the former in the consequent of this one conditional without changing the circumstances under which the conditional is true. This is another disappearing disjunct case from which nothing alarming follows. So I think the New Puzzle is best understood slightly differently from the way Kvanvig suggests. He encourages us to be surprised that the strengthening from ♦Kp to Kp in the consequent of WVER makes no difference to the circumstances in which the conditional is true. But it can’t be the mere strengthening that’s surprising, because that sort of thing (strengthening within the consequent of one particular conditional) happens all the time.¹⁰ It must be something else. What is it? I think answering that question will be half the battle of solving the New Puzzle. In fact, I think it will probably be almost all the battle. I think one important question which properly belongs to the New Puzzle is: •

Why are we surprised by the Church–Fitch proof?

¹⁰ Kvanvig’s comments about ‘multiple contexts’ at pp. 213–19 (targeted on Mackie’s ‘syntactic’ approach to lessening the surprise of the Church–Fitch result) might be thought to address this kind of point. But to take them that way would be to say that what the analogies I am drawing here actually do is ﬂag up ‘a more general paradoxicality’. That is to say, it would suggest that one really does ﬁnd the case of the disappearing disjunct, and like cases, paradoxical. Of course, if it’s that easy to come up with paradoxes, paradoxicality is not very worrying. Certainly no one should be looking to revise any of their beliefs just because they throw up the case of the disappearing disjunct.

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Once we know the answer to this question, we will be able to see how to give an explanation of why the proof works which satisfactorily removes this surprise. So l v i n g t h e Ne w Pu z z l e Consider a simple explanation of why the Church–Fitch argument works, and hence of why WVER commits one to SVER: E: Nothing of the form (p & ¬Kp) is knowable. That’s obvious. But given WVER, if something of that form is true, then it is knowable. That’s why, if WVER is true, nothing of the form (p & ¬Kp) is true. And that’s why if WVER is true then it follows that SVER is true too. Kvanvig thinks there is something deeply surprising about the Church–Fitch proof; he must, therefore, think there is more to the surprise the proof engenders than the kind of surprise we can get over just by thinking carefully about how the proof works. The proof is paradoxical, on his view, because we somehow cannot bring ourselves to accept that it works, even after we have seen it and fully understood it. It would be inappropriate to acknowledge the truth of E and just get over it. If that’s right, there must be something inadequate about the simple explanation E. But what? One thing Kvanvig feels is in need of explanation is that the Church–Fitch proof shows WVER and SVER to have the same modal status (this volume, (p. 211). We are supposed to be surprised when the proof forces us to accept this, because according to Kvanvig we would previously have thought that WVER ‘if true, is . . . necessarily true [since] it is a purported implication of a proper understanding of the nature of truth’, whereas SVER is supposed to be contingent. However, on the assumption that WVER is false, there is no reason to suppose it is non-contingent. So realists, at least, might well believe, before thinking about Church–Fitch, that both WVER and SVER are contingently false (contingently because presumably they will think there is a possible world where everything is known, at which both WVER and SVER are true). And they can, of course, continue to hold this after thinking about Church–Fitch. So the proof tells them nothing new about the respective modal status of the two claims. Since Kvanvig wants us to focus on an explanatory challenge which faces everyone alike, realist or anti-realist, this can’t be part of it.¹¹ ¹¹ He may intend to make a similar point when he says (p. 213) that the challenge of Church–Fitch is that ‘we are told that what looks like a modal truth is logically equivalent to what looks like a non-modal truth’. If not, it is unclear what point this passage is making. Obviously, both WVER and SVER are non-modal in the sense that neither is governed by a modal operator. It’s true that one contains a diamond and the other doesn’t, but that is true of a great many obviously and unsurprisingly equivalent pairs (e.g. p → p and p → ♦p).

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So what can it be that is wrong with E? Do we need further explanation of one or more of the claims it draws upon? I don’t think so, but even if we do it could surely be given (see, for example, my comments in footnote 13 below). If my characterization of the New Puzzle is accurate, one thing that the simple explanation E leaves out is an explanation of why, before encountering the Church–Fitch proof, we feel that WVER shouldn’t commit us to SVER. But again, however, there is a simple explanation available. What’s behind this fact is that when someone innocent of Church–Fitch hears the claim All true propositions are knowable, she just doesn’t think about true propositions of the form (p & ¬Kp). These aren’t exactly the kinds of things that spring to mind when this kind of general claim about true propositions is made, if one hasn’t been exposed to the Church–Fitch proof. According to the simple explanation E, it is attention to these cases that reveals why 1 and 2 are equivalent (or rather, why the surprising direction of the equivalence holds). The explanation of why we feel strongly beforehand that WVER shouldn’t commit us to SVER is simply that we haven’t thought hard enough about the full implications of WVER—we haven’t thought about what it will mean for propositions of the form (p & ¬Kp). Even if that’s right, though, it may be thought that there is still something lacking in these simple explanations which prevents our just getting over it. It might be said that they don’t really explain the disappearance of the possibility operator; they just explain why SVER follows from WVER. This is an interesting kind of worry. What counts as a good explanation of a fact does plausibly depend on (among other things) the way the fact is presented. The thought here would be that presenting the implication of SVER by WVER as a case of apparent modal collapse makes the simple explanation offered above inadequate (even if it is a good-enough explanation of the same fact under a different description—e.g. when it is described as the fact that WVER commits us to SVER). I am not unsympathetic to those who think the simple explanation E is explanation enough of the fact in question under either guise, and who think a sufﬁcient explanation of why we are surprised when we ﬁrst encounter the Church–Fitch proof is that we just hadn’t thought hard enough about all the instances of WVER. I am sympathetic, that is to say, to those who think the best response to the New Puzzle is basically an injunction to get over it. At any rate, I don’t think someone who has this view can fairly be dismissed as ‘living in logical denial’ (Kvanvig, this volume, p. 313¹²). ¹² As an aside, I note that on p. 313 Kvanvig shifts between two very different responses to Church–Fitch: on the one hand, denying that it is a paradox, and on the other, accepting that SVER is a necessary truth. The latter response is a proper subspecies of the former. Since Kvanvig mentions myself and Williamson in a footnote during this passage as advocates of the view he is discussing, it is worth pointing out that neither Williamson nor I accepts SVER, let alone accepts that it is necessary. We both think the proof is non-paradoxical for other reasons, which Kvanvig does not describe.

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Be that as it may, what’s required of a satisfactory explanation of a fact can depend, not just on how the fact is presented, but also on who the explanation is for. So, while some may reasonably ﬁnd the simple explanations adequate, others may reasonably demand to hear more before their puzzlement is resolved. For such people, I think more can be said, and that is what I shall try to offer in the rest of this section. First, let me make a quick remark concerning Kvanvig’s suggestion as to what kind of explanation we should be looking for. Kvanvig’s paradigm of a satisfying explanation of modal collapse is the Kripke-style semantic explanation of why what is possibly necessary is not logically distinct from what is necessary (the characteristic commitment of modal logic S5). If we accept a possible worlds semantics for the modal operators (and if we make the required assumptions about accessibility) it becomes obvious why ‘possibly necessarily p’ is equivalent to ‘necessarily p’. By thinking about the semantics for the operators involved in these two propositions we can see why the two are equivalent in strength. Two things are noteworthy about this. One is that it is not clear that an explanation that appeals only (or primarily) to semantics is to be expected or desired in the Church–Fitch case, where the equivalence is, as I stressed above, not establishable through logic alone but only with the aid of substantive nonlogical principles about knowledge, namely FACT and DIST. Compare the jade example on p. 311 above. A big part of the explanation of this equivalence is presumably the empirical fact that there are exactly two kinds of jade: nephrite and jadeite.¹³ The second point to note is that there is another respect in which the modal collapse that characterizes S5 is very different from the modal ‘collapse’ revealed by the Church–Fitch proof. The former, but not the latter, is a complete collapse. That is to say, with S5 we get intersubstitutability in all contexts between ♦p and p, whereas with Church–Fitch we get intersubstituability between ♦Kp and p only within the consequent of a certain conditional. For these two reasons I think that anyone who thinks it appropriate to seek a semantic explanation of the disappearance of the diamond in WVER, along the lines of the possible-worlds explanation of why ♦p implies p, probably has the wrong sort of target in his sights. But what sort of thing should we be looking for? I am tempted to think that some light might be shed on this matter through comparison (not formal, but psychological) between what happens when we think about the Church–Fitch argument and what happens when we think ¹³ Of course, semantics may also be part of, or may help underwrite, the explanation; for instance, the rigidity of ‘jade’ is presumably part of the explanation of why it is necessary that all jade is nephrite or jadeite. But for that matter, semantics may be made part of, or may help underwrite, explanation E—some appeal to the meaning of ‘knows’, in particular, the fact that ‘knows’ is factive, might be included in an explanation of E’s initial claim that nothing of the form (p & ¬ Kp) is knowable.

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about Russell’s set-theoretic paradox. Russell’s paradox proves that a seemingly innocent universal claim (‘Every property determines a set of objects with that property’) leads to contradiction. The Church–Fitch argument proves that a seemingly innocent universal claim (‘Every true proposition can be known’) leads to contradiction of a patent truth (that some true propositions are unknown). Understood correctly, Russell’s ‘paradox’ is not really paradoxical. It’s just a proof that the seemingly innocent claim that every property determines a set is in fact false, and less innocent than it seemed. To see why, we are invited to consider a speciﬁc (and initially unobvious) case. The paradox argument shows that, although being non-self-membered is a perfectly good property for sets to have, there is no corresponding set of non-self-membered sets. It can be correctly called a paradox only insofar as one is tempted by the thought that there really should be such a set, or that the general claim should be true, or perhaps simply that it should not be possible to disprove the general claim by this sort of method. Similarly, the Church–Fitch proof is not really paradoxical. It’s just a proof that the seemingly innocent claim that every true proposition is knowable is in fact false (assuming, that is, that not all true propositions are known), and less innocent than it seemed. To see why, we are invited to consider its implications in a speciﬁc (and initially unobvious) case. The Church–Fitch proof shows that, although there are perfectly good true propositions of the form (p & ¬Kp), there is no possibility of knowing a proposition of this form. It can be called a paradox only insofar as one is tempted by the thought that it really should be possible to know things like this, or that the general claim should be true, or that it should not be possible to disprove the general claim by this sort of method. I think that, in both cases, what happens is that an innocent-sounding (but in fact far from innocent) universal claim gets put forward in a confused attempt to express a similar, genuinely innocent, claim. In the case of Russell’s paradox, what was intended is better captured by the axioms of standard iterative set theory.¹⁴ And, I believe, in the Church–Fitch case, what was intended is better captured by the claim that if p is true then the state of affairs which at the actual world makes p true is recognizable. ¹⁴ It has been suggested to me that, in fact, the notion of set characterized in the axioms of iterative set theory is very different from that which was supposed to be characterized by naïve comprehension. While there are of course some differences, I think there are also enough similarities to allow us to describe the axioms of (say) ZFC as an attempt to capture what was previously supposed to be characterized by naïve set theory, the most important of these being that both are theories of how a number of things can be collected together to form a new entity which is something over and above its members. ( Thanks to Aidan McGlynn for raising this interesting issue in online discussion at . McGlynn here also raises the interesting possibility that something akin to what I think is happening in the case of the Church–Fitch paradox may also be happening when we feel tempted to accept the major premise of the Sorites paradox.)

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In both cases, the paradox arguments seem bafﬂing, and cry out for substantive explanation, if we mistake the apparently innocent claim for the genuinely innocent one. For we can’t understand how the innocent claim we meant to express could have such objectionable consequences as are being attributed to the claim we have actually expressed. However, once we realize that the claim we intended to make is not the same as the one we actually made, it is no longer so bafﬂing that the claim we actually made has these undesirable consequences. In short, then, there is no mystery about the disappearance of the diamond in the Church–Fitch proof. The diamond disappears because WVER has (admittedly unobvious) strength, which it should not have if it is to serve as an expression of one’s commitment to the epistemic accessibility of reality. Note that nothing comparable is going on in the case of the disappearing disjunct (see p. 311 above), which is why we don’t ﬁnd ourselves experiencing any comparable bafﬂement about that case.

C o n c l u d i n g Re m a rk s A couple of further points are worth mentioning. One is that, as is well-rehearsed in the literature, Church–Fitch-style arguments can be constructed using other factive operators on propositions,¹⁵ not just for ‘K’. (Kvanvig, this volume, pp. 214–19, offers a discussion of this point.) For instance, if we assume that every true proposition can be truly believed, we will be able to derive that every true proposition is truly believed. Insofar as these proofs are surprising in the same way that the Church–Fitch proof is surprising, it would be nice to be able to offer the same diagnosis of that surprise. And in some cases this is no problem. For instance, we might propose that when we say that any true proposition can be truly believed, we don’t really intend to express p ⊃ ♦TBp, but rather something like: ‘Every true proposition is such that the state of affairs which makes it true can be correctly taken to obtain.’ However, it is important (and interesting) to note that some comparable claims involving other operators will not be amenable to treatment along quite these lines.¹⁶ Consider, for instance, the claim that any true proposition p is such that the state of affairs S which makes p true can be recognized by me now. For any true proposition q of the form (the state of affairs S which makes p true obtains now and is not now recognized by me), it is not possible that I recognize now the state of affairs T which makes q true. (Because, plausibly, recognizing T ¹⁵ And, indeed, some non-factive operators, such as ‘It is rationally believed that’ (see Mackie 1980). ¹⁶ I am indebted to Kim Stebel for online discussion of this point at .

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now will involve recognizing S now, but q must be false if I recognize S now.) Yet some proposition of the form of q is surely true. Therefore it is not the case that any true proposition p is such that the state of affairs S which makes p true can be recognized by me now. Moreover, the way we show that this is not so is Church–Fitch-like, and might therefore be expected to engender exactly the same kind of puzzlement as other Church–Fitch-style arguments. Yet we won’t be able to explain that puzzlement by saying that what we really meant to express was something else—something involving recognition of states of affairs rather than knowledge of propositions.¹⁷ One thing to note about this kind of case is that any surprise engendered by the new Church–Fitch-style argument can’t have much to do with the prior plausibility of the claim that any true proposition p is such that a proposition expressing the state of affairs S which makes p true can be recognized by me now. For that claim has very little prior plausibility. However, it might seem that there is still a salient similarity with the original Church–Fitch argument concerning the claim that all true propositions are knowable. Admittedly, in this case we are not at all surprised that the claim is false, but we are nonetheless surprised that it should be disprovable in this way. If it is true that this aspect of the new argument is surprising in the same way as the corresponding aspect of the original argument was surprising, then it is not clear how my explanation of the latter surprise could be correct. For in the new case we have exactly the same kind of surprise, but cannot give the same explanation of it. However, I am inclined to think that the surprise engendered in the new case is not entirely comparable to that engendered in the original case. In the new case, I would suggest, all the surprise is engendered by the fact that we just haven’t thought about the problem cases. When we do think about them, and when we understand how the Church–Fitch-like proof works, we understand why the apparently weaker claim actually commits one to the apparently stronger one. And our feelings of surprise should be thereby resolved. We should get over it. I am prepared to grant that, in the original case, some people do encounter a deeper and more robust surprise than this—a kind of surprise that is not fully resolved merely by thinking carefully about the problem cases and understanding the workings of the Church–Fitch proof. That deeper surprise, I think, is to be explained by showing that there is potential for confusion between the claim that all true propositions are knowable and the claim that reality is epistemically ¹⁷ Note that this is not just because what we actually expressed was something involving recognition of states of affairs rather than knowledge of propositions, but because the way I appeal to states of affairs to block the original Church–Fitch argument is by appealing to the ranges of the quantiﬁers appearing in ‘Kp’ (see Jenkins 2007, section IV). There are no quantiﬁers in the new operator, however, so this manoeuvre cannot be used here.

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accessible (the latter of which, we are right to think, does not commit us to thinking that all true propositions are knowable). Some closing comments are perhaps in order to make explicit the relationship between my favoured approaches to the Classic Puzzle and the New Puzzle. In addressing the Classic Puzzle, I offer mind-dependence anti-realists a defence against the charge that the Church–Fitch proof shows their view to be untenable. My response to the Classic Puzzle is to argue that anti-realism is best understood as commitment to a claim that is not prone to the Church–Fitch argument. Whereas my discussion (indeed, any discussion) of the New Puzzle is an exploration of the way we respond when thinking about propositions which are prone to the Church–Fitch argument. The latter, while interesting in its own right, is strictly speaking tangential to a discussion of whether anti-realism is true, if I am right about what that doctrine amounts to. Nevertheless, my discussion of the Classic Puzzle offers us resources with which to address the New Puzzle. For it enables us to argue that, insofar as any deep surprise is engendered by the Church–Fitch proof, that surprise is due to the confusion of WVER with a claim to the effect that reality is epistemically accessible to us. It seems likely that other approaches to the Classic Puzzle will also generate resources for addressing the New Puzzle. For instance, if Edgington is right that the anti-realist epistemic accessibility claim should really be WAVER, it could be argued that the reason we are especially surprised by Church–Fitch is that we mistook WVER for WAVER and hence were surprised when WVER turned out to have consequences which the anti-realist epistemic accessibility claim should not have. Other change-the-class or change-the-claim strategies could be put to similar use. Kvanvig (Chapter 13 of this volume) considers the application to the New Puzzle of Hand’s (2003) response to the Classic Puzzle. This involves pointing out a ‘structural interference’ between the operators and connectives in K(p & ¬Kp): knowing the ﬁrst conjunct entails that the second conjunct is false. Pace Kvanvig, I think something like this may well be (at least part of) a good explanation of the Church–Fitch result for some audiences. (For instance, it could be used to provide explanatory background for the ﬁrst two sentences of my simple explanation E above.) Kvanvig’s rejection of this manoeuvre seems to rest on his not ﬁnding it sufﬁciently analogous to one that can be made in defence of the ontological argument for the existence of God (p. 220–2). This is very puzzling. It might be that the ‘structual interference’ manoeuvre works quite differently, in a way that is not analogous to this—or any—defence of the ontological argument. (For instance, it may function simply as a source of explanatory background for part of E). Note also that Kvanvig thinks that in order for the manoeuvre to work in the same way as the relevant defence of the ontological argument (and hence, apparently, to work at all) it must establish that SVER is necessarily false (p. 221). But no defence of the Church–Fitch equivalence should have to establish that

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SVER is necessarily false. Even once the Church–Fitch reasoning is understood and accepted in its entirety, SVER seems to be contingently false, as I pointed out on p. 312 above. For there are worlds where both WVER and SVER are true. Thus Kvanvig’s understanding of the way the manoeuvre is supposed to work if it works at all seems questionable. Some responses to the Classic Puzzle involve no change-the-class or changethe-claim strategy. Some, for instance, believe that WVER is a fair interpretation of the anti-realist’s epistemic accessibility claim, but deny that the derivation of SVER from WVER goes through in their preferred logic (see, e.g., Beall, Chapter 8 of this volume, for a discussion of logics which block the inference). This view can also provide resources for addressing the New Puzzle: it offers us grounds for denying that the puzzling equivalence is genuine. Others respond to the Classic Puzzle by holding that WVER is a commitment of anti-realism and the equivalence is genuine enough, and that anti-realism is thus undermined. Such a person might choose to respond to the New Puzzle by saying that all we need do is think hard enough about how the Church–Fitch proof works to enable ourselves to get over any initial surprise it generates. We should not be dismissive of this view, even if we think there is in fact more to say in response to the New Puzzle. It is often important to take seriously the possibility that a purported puzzle is no puzzle at all.

19 Invincible Ignorance W. D. Hart

There are truths that cannot be known. For suppose that all truths can be known. Then all truths actually are known. Otherwise, we may suppose for some p that p but it is not known that p. Then it can be known that p but it is not known that p. But when it is known that thus and such, it is known that thus and it is known that such. So it could be known that p and known that it is not known that p. But what is known is true. So it could be known that p and not known that p. But that is a contradiction, and no contradiction can be true. So all truths are actually known. But it is not in fact known whether the number of hairs on Caesar’s body at the instant Brutus’s dagger ﬁrst penetrated was odd. Hence, even if that number could be known, still, as promised, not all truths can be known. Fred Fitch, who ﬁrst published the argument in 1963, credited it to an anonymous referee of a paper Fitch submitted to the JSL in 1945 but never published. The respect due to provenance attaches Fitch’s name to the argument, and we might call the result that there are truths that cannot be known Fitch’s Formula. To call it Fitch’s Paradox is tendentious. But many do, so let us explore the tendency. On ﬁrst acquaintance with the formula, some ask for an example of a truth that cannot be known. The request is a confusion. For to answer it, one should present the requester with a truth, and the requester should not be satisﬁed without a demonstration that it is a truth. But then the requester need only note his or her mastery of the demonstration in order to recognize that he or she now knows the truth presented. Since it is thus actually known, it is out of the question that it cannot be known. In the jargon of the philosophy of mathematics, there could not be a constructive proof of Fitch’s Formula, that is, a proof that displays (or gives an algorithm for displaying) an example of what it claims to exist. A taste for constructive existence proofs is associated with various irrealisms about the objects (like numbers, sets, or vector spaces) mathematics describes, and a hostility to Fitch’s Formula suggests a less than robust realism. A realist about, say, numbers thinks that numbers, and how they are, do not depend on

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what we or anyone else might say, think, prove or know about them. The mode of dependence here denied is modal. So the realist thinks numbers can be as they in fact are even if it cannot be known how they are. Fitch’s Formula is of a piece with a robust realism. How might irrealism’s neurasthenia show up? A simple symptom would be belief that all truths actually are known. The traditional form is belief in a know-it-all god. Religion is often a refuge for the fearful. Since the orthodox can be dangerous when roused, let us confer only with those who admit there is ignorance. Thus we agree amongst ourselves that it is an absurdum that all truths are known. Fitch’s argument reduces the supposition that all truths can be known to this absurdum. So let us look now at the premisses of that argument. These are of two sorts, modal and epistemic. The modal logic involved is pretty minimal. We use what is called Gödel’s axiom, namely, that modus ponens transmits necessity as well as truth. If one is going to take modality seriously, it is hard to see how one would deny Gödel’s axiom. We use what is called the rule of necessitation, namely, that necessitations of theorems are theorems. If we take care that our axioms are necessary and that our other rules transmit necessity, then the rule of necessitation should hold. Gödel’s axiom and the rule of necessitation are all the purely modal machinery we need. On the epistemic side we use the premiss that what is known is true. That premiss is about as close to analytic as anything remotely philosophically interesting gets; a belief does not count as knowledge unless it is true. It would be heroic, even quixotic, to deny it. The other purely epistemic premiss we use is that when a conjunction is known, so are its conjuncts. This premiss is an instance of the claim that all logical consequences of what is known are known, and even a little familiarity with sophisticated deductive theories is enough to confute the general claim (though an interesting sorites remains here). But the instance we use is so transparent. If an explicit conjunction is known, it should be understood and recognized as a conjunction, whence commitment to the conjuncts seems inevitable. It seems more rational to afﬁrm that conjuncts of the known are known and to deny that all truths can be known than the reverse. That what is known is true and that conjuncts of the known are known are all the purely epistemic machinery we need. Is there a refuge left for the irrealist? Typing is a perennial favorite in solving, or suppressing, paradoxes. As long ago as 1903, Russell considered typing in The Principles of Mathematics to deal with paradoxes like his own and the liar. Zermelo’s (1908) revision of set theory (Zermelo 1967), seen through the axiom of foundation as the iterative hierarchy,¹ pictures sets as arranged in ascending ranks. Soon after Ramsey in 1925 simpliﬁed Russell’s theory of types so layering ¹ Boolos, G. 1998. ‘‘The Iterative Conception of Set,’’ in Logic, Logic, and Logic. Cambridge, MA: Harvard University Press: 13–29.

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only sets in types is to treat paradoxes like Russell’s (Ramsey 1960: 1–61), Tarski in 1931 used levels of languages to handle the liar paradox (Tarski 1956: 152–278).² Tyler Burge (1984: 83–118)³ and Anil Gupta (1988–89: 227–46) continue the device of stratifying language to proscribe semantic paradoxes. Alonzo Church in his referee’s report on the paper Fitch submitted to the Journal of Symbolic Logic in 1945 suggests typing knowledge as a way around Fitch’s Formula.⁴ Bernard Linsky proposes a type-theoretic reply to Fitch in his contribution to this volume (Chapter 11). In this spirit we might try typing knowledge. The basic idea is of a bottom type 0 of propositions (or sentences) in which knowledge (or the verb ‘‘to know’’) is absent, and then for each type n of propositions (or sentences) to be followed by a type n + 1 of propositions (or sentences) expressing knowledge (or predicating the verb ‘‘to know’’) of propositions (or sentences) of type n. Knowledge of any type would still be true, and knowledge of any type of a conjunction would be knowledge of that type of its conjuncts. The crucial difference would be that a proposition (or sentence) of type n could always be known but only of type n + 1. In private correspondence Hans Kamp and Charles Parsons showed independently that propositions (or sentences) not in fact known are consistent with this system. We have just imagined a system in which knowledge fractures into knowledge of type 1, knowledge of type 2, and so on. In this last sentence we used arabic numerals like proper names for types of knowledge. In the preceding paragraph we also used, in addition to these speciﬁc numerals, the predicate ‘‘is a type’’ and variables of quantiﬁcation ranging over types. If only because there is no clear end to the conﬂux of types, it is at least very hard to see how we could describe them without that predicate and those variables. So a forthright theory of types of knowledge should include, in addition to names for types, a predicate speciﬁc to types and variables of quantiﬁcation ranging over them. In such a forthright system, Fitch’s Formula recurs. Suppose that any truth of type n can be known at type n + 1. We will show that any truth actually is known at some type. For otherwise we may assume that p but it is known at no type that p. The assumption that p but it is known at no type that p must be of some type t; that is the point of type theory. Then it can be known at type t + 1 that p but it is known at no type that p. Thus, as before, it is possible that it is known at type t + 1 that p and it is known at type t + 1 that it is known at no type that p. So, again as before, it is possible that it is known at type t + 1 that p and it is known at no type that p. Hence, by universal instantiation, it is possible that it is known at type t + 1 that p and it is not known at type t + 1 that p. But ² I defend Russell’s and Poincaré’s assimilation of the set theoretic and semantic paradoxes which Ramsey separated in my 1984: 193–210. ³ In ‘‘Frege on Truth,’’ Burge says, ‘‘The notion of truth cannot be adequately represented in terms of a truth predicate that lacks some sort of stratiﬁcation.’’ See his 2005: 131. ⁴ Private communication from Joseph Salerno.

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contradictions still cannot be true. So any truth actually is known at some type. So, given the actuality of ignorance, some truths cannot be known at any type. Type theory that acknowledges itself violates itself. Suppose we try to say that all knowledge must be of some speciﬁc type or other, and that there can be no utterly general knowledge. We are claiming to know what we say, but our claim is a counter-example to what we said. Much the same sort of thing happens with the Liar Paradox; see my 1989–90: 161–5. It was pretty clear even from the early days of type theory that type theory self-destructs.⁵ If we say that every set is of some type (or rank), then no set can be the range of the variable in the universal quantiﬁer ‘‘every,’’ so sets are inadequate for semantics; they won’t let ‘‘all’’ mean all.⁶ The closest thing to a philosophical certainty may be that there are no philosophical certainties. But as it is a platitude of humility before a world beyond our control that not everything is known, so it might be a reasonable hypothesis of reason before a world independent of reason that not everything can be known. ⁵ At 3.332 in his 1961: 31, Wittgenstein wrote, ‘‘No proposition can make a statement about itself . . . (that is the whole of the theory of types).’’ 3.332 is obviously a counter-example to 3.332. I argue in my 1971: 271–88 that this paradox is the center of the distinction between saying and showing in the Tractatus. The Tractatus was published in 1919, but it would be impertinent to suppose that Russell had not seen long before that statements of the theory of types violate it. ⁶ If ZFC, Zermelo-Fränkel set theory with the axiom of choice, is true, then since it denies there is a set of all sets, there is no set of all sets. The model theory Tarski taught us says the domain of a model is a set, so no model is the world ZFC describes and ZFC is not true after all. If there is no set of everything, it is not there to be the domain of a model in which ‘‘Everything is self-identical’’ is true. Tarski taught us to say instead that it is valid, that is, that it comes out true in every non-empty domain (set) in which ‘‘is identical to’’ is interpreted as the identity relation restricted to that domain. But when we wrote ‘‘every set’’ just now, we used a quantiﬁer that, according to the doctrine we are expounding, does not have the interpretation we meant. The received solutions to the set theoretic and semantic paradoxes do not work. The late Raúl Orayen used to stress these points, but he died too early to publish them.

20 Two Deﬂationary Approaches to Fitch-Style Reasoning Christoph Kelp and Duncan Pritchard

0 . In t ro d u c t i o n Frederic Fitch (1963) famously argued that the thesis that all truths are knowable (henceforth, the knowability principle), in conjunction with a handful of apparently highly plausible logical and epistemic principles, entails the obviously absurd claim that all truths are known. This argument has become known as the paradox of knowability. Of course, it is only a paradox if one ﬁnds the knowability principle highly plausible in the ﬁrst place, since a basic prerequisite of an argument qualifying as a paradox is surely that it involves a highly contentious—indeed, unacceptable —conclusion which validly follows from highly plausible premises. Moreover, there is good independent reason to think that such a principle is not so plausible. It seems that the principle knowability is naturally understood as applying to cognizers like us; that is, to subjects with ﬁnite cognitive capacities and a ﬁnite lifespan. At the same time, it is plausible that there are some propositions that are too large to be grasped by such cognizers—for instance, some disjunctions with inﬁnitely many disjuncts. If these ﬁnite cognizers cannot grasp such propositions, however, then they cannot know them either. In consequence, for cognizers like us, some propositions must remain unknown. Provided that the principle of knowability is naturally understood as applying to cognizers like us, then it is not plausible that it is true. If so, however, the ‘paradox’¹ of knowability,² then, isn’t strictly speaking a paradox at all. We are grateful to Brit Brogaard, Tony Brueckner, Laurence Goldstein, Patrick Greenough, Allan Hazlett, Stephen Maitzen, Aidan McGlynn, Alan Millar, Paul O’Grady, David Papineau, Sven Rosenkranz, Peter Sullivan and to an anonymous referee from Oxford University Press. Special thanks also go to the editor of this volume, Joe Salerno, for all his (considerable) help. ¹ The thesis that the ‘paradox’ of knowability is not really a paradox has also been defended by Williamson (2000a, ch. 12). ² Of course, one might ﬁnd it independently puzzling that it is even possible to derive the conclusion that all truths are known from the premise that all truths are knowable. Fitch’s argument

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Nevertheless, there are (as we will see in a moment) substantive philosophical grounds in favour of the knowability principle, and thus even if Fitch’s argument does not point to a paradox as such it may still be thought to be a potential reductio of those philosophical views which feel the theoretical need to incorporate this principle. Accordingly, if there are such theoretical views then its defenders had better have something compelling to say in response to Fitch’s argument. In this paper, we will look at one—perhaps the only—theoretical view to which, on the face of it, the knowability principle is of central importance. We will then consider two deﬂationary responses to Fitch’s argument on behalf of defenders of this view. What we mean by a ‘deﬂationary’ response to the argument is a proposal which proceeds by weakening, on a principled basis, one of the principles essentially employed by that argument. The motivation for this strategy is this: ceteris paribus, if one can accommodate the considerations which prompt adoption of a certain principle by advancing a version of that principle which is (perhaps only slightly) logically weaker, then one ought to do so. If one can further show that the Fitch argument is blocked once the weaker ‘deﬂated’ version of the principle is adopted, then one will have succeeded in offering a deﬂationary response to the argument.³ The ﬁrst deﬂationary response that we will consider proceeds by weakening the factivity principle for knowledge. We will argue that this strategy does not stand up to closer inspection. Nevertheless, we claim that there are good grounds for holding that the second deﬂationary response that we consider—which rejects the principle of knowability in favour of a weaker principle—is effective at resolving the problem posed by Fitch’s argument.

1 . Se m a n t i c A n t i - re a l i s m The view to which the knowability principle is, on the face of it, of central importance, is often labelled ‘semantic anti-realism’. Semantic anti-realism is the rejection of realist theories of meaning (i.e., semantic realism). Indeed, semantic anti-realists often explicitly motivate their position by pointing to defects in realist theories of meaning. In this section, we will outline the problems which, according to the semantic anti-realist, beset realist theories of meaning and show how accepting the knowability principle can potentially avoid these problems. would then be philosophically interesting even if one did not ﬁnd the knowability principle plausible. Still, what is making the argument—taken in isolation—philosophically interesting is not that it poses a paradox. ³ This deﬂationary strategy—applied to epistemological issues—is explored and defended at greater length in Pritchard (2004). See also Greenough (2002).

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To begin with, let us look at the credentials of realist theories of meaning. Realist theories of meaning are commonly construed as having the following two features: (1) The meaning of a statement is identiﬁed with its truth-conditions. (2) There are evidence-transcendent truths.⁴ From these features of realist theories of meaning it follows that some statements have evidence-transcendent truth-conditions as their meanings. A general constraint on a theory of meaning is that it should at least be compatible with a theory of understanding—that is, for a theory of meaning to be satisfactory it must be compatible with an account of what a competent speakers’ linguistic understanding consists in.⁵ Semantic anti-realists suspect that realist theories of meaning will be unsatisfactory on just this score because they are incompatible with a satisfactory account of our understanding of statements with evidence-transcendent truth-conditions. Semantic anti-realists base their suspicion on a challenge to realist theories of meaning which arises from what they consider to be an important Wittgensteinian insight into the nature of understanding—namely, that understanding a concept consists in a set of practical abilities rather than in a state of mind. Certainly, if one is to be credited with a given practical ability then one must be able to manifest that ability in one’s behaviour. For instance, a child will be credited with the ability to swim only if she is able to manifest swimming behaviour in suitable circumstances. Hence, if the Wittgensteinian insight is to be taken seriously—that is, if understanding is to be conceived of as a set of practical abilities—then understanding must be manifestable in behaviour too. Presumably, the kind of behaviour in which understanding must be manifestable is linguistic behaviour (i.e., language use). According to the semantic anti-realist, however, what would count—at least minimally—as a manifestation by a speaker of her understanding of a statement in use is that the speaker is able to evaluate her own and other people’s use of the statement and, if circumstances render it appropriate, to adjust her use of it accordingly.⁶ Given that we understand what counts as manifestation of understanding in use in this way, however, it is hard to see how understanding of statements with evidence-transcendent truth-conditions could be manifested in use. After all, the truth-conditions of such statements are evidence-transcendent. As a result, there aren’t any circumstances that would provide the basis for an evaluation ⁴ Cf. Wright (1993a: 250). We take it that for present purposes this is an adequate representation of Wright’s statement of realism: ‘Realism about a given discourse, for the purposes of the Manifestation Challenge, is simply the combination of views (a) that the proper account of our understanding of its statements is evidence-transcendent truth-conditional, and (b) that the world on occasion exploits, so to speak, this understanding—does on occasion deliver undetectable truth-conferrers to such statements’ (ibid.). ⁵ Cf. Wright (1993a: 47). ⁶ Cf. Wright (1993a: 247).

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of one’s own or other people’s use of such statements. And, similarly, there aren’t any circumstances in the light of which one would adjust one’s use of such statements. If there aren’t any such circumstances, then understanding of statements with evidence-transcendent truth-conditions cannot be manifested. And, if understanding of such statements cannot be manifested, then it does not consist in a set of practical abilities after all—contrary to what the Wittgensteinian insight suggests. Accordingly, the challenge that semantic anti-realists pose to their realist opponents is to provide an account of understanding of statements with evidence-transcendent truth-conditions that is both faithful to the two core realist theses and respects the Wittgensteinian insight. Their suspicion is that this cannot be done.⁷ A related challenge that semantic anti-realists pose to semantic realists focuses on the acquisition of our understanding of statements with evidence-transcendent truth-conditions. Since if we accept a truth-conditional theory of meaning we acquire our understanding of a type of statement by bringing to bear evidence on the truth-values of instances of it, semantic anti-realists argue that it is hard to see how we could so much as acquire an understanding of statements with evidencetranscendent truth-conditions. Accordingly, semantic anti-realists challenge their opponents to provide an account of how we acquire our understanding of statements with evidence-transcendent truth-conditions.⁸ These two challenges were ﬁrst advanced by Michael Dummett (1978) and have become known as the manifestation and the acquisition challenge, respectively.⁹ Although there are further anti-realist arguments, these challenges—and the manifestation challenge in particular—appear to be the most common reason offered by semantic anti-realists as to why they ﬁnd realist theories of meaning problematic.¹⁰ Accordingly, semantic anti-realists have proceeded to deny at least one of the two core theses of realist theories of meaning. Initially, semantic anti-realists were tempted to deny the realist’s ﬁrst core claim—i.e., the commitment to a truth-conditional theory of meaning—and replace it with a theory that identiﬁes the meanings of statements with their assertibility conditions. The rationale for this is obvious, since by tying the meaning of a statement to its assertibility conditions (which are held not to be evidence-transcendent) rather than its truth-conditions, the anti-realist avoids the problems posed for a theory of meaning by allowing evidence-transcendent truths. More recently, however, this option appears to have become less appealing to semantic anti-realists. Instead, they have tended to reject the realists’ second core ⁷ Cf. Wright (1993a: 247–8). ⁸ Cf. Wright (1993a: 87). ⁹ See also Dummett (1993a). ¹⁰ For two further anti-realist arguments, see Wright (1993a), who outlines a challenge that proceeds from the normativity of meaning, and Putnam (1981), who adduces the so-called ‘modeltheoretic’ argument. Most contemporary anti-realists appear to accept the manifestation challenge. For examples, see Dummett (1978, 1993), Wright (1993a) and Tennant (1997).

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claim—i.e., that there are evidence-transcendent truths.¹¹ It ought to be clear that accepting the possibility of evidence-transcendent truths entails accepting the existence of unknowable truths, at least if one accepts the further (highly plausible) claim that in order to know a proposition one must have evidence in favour of it. Accordingly, if one holds, with the knowability principle, that there cannot be any unknowable truths, then it follows that one must reject the idea that there are evidence-transcendent truths as well. Given the foregoing, there is clearly a large theoretical pay-off in rejecting this key realist claim, since it avoids the worries just noted regarding our understanding of such truths. If there aren’t any such truths, then the fact that it is doubtful whether an understanding of them can be manifested—or acquired for that matter—won’t be a problem for the semantic anti-realist. Given that our primary interest is the Fitch argument, it is this second strand of semantic anti-realist thought—which, like the Fitch argument, has the knowability principle at its heart—that is our concern here. Henceforth, when we talk of ‘semantic anti-realism’ we will have this speciﬁc variety of semantic anti-realism in mind.

2 . Fi t c h’s A r g u m e n t Fitch’s argument clearly poses a fundamental challenge to semantic anti-realism. Indeed, given that it is an undeniable truth that we are not omniscient, unless the semantic anti-realist can ﬁnd some way to block this argument then she is faced with a reductio of her position. Since, in order to be able to discuss some options the semantic anti-realist may have to block Fitch’s argument, it will be a good idea to look at how the argument proceeds in a bit more detail. First, we will formalize the knowability principle in the following way:¹² (KP)

(∀P ) (P → ♦(∃s, t ) (Ks, tP ) )

Now we assume, for reductio, that one is not omniscient—i.e., that there is some truth (we’ll call it ‘P 1 ’) which is unknown: (1) P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) Given (KP), however, one can straightforwardly derive (2): (2) ♦(∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) ) ¹¹ For a good contrast of the two anti-realist approaches, compare the early and later essays collected in Wright (1993a). ¹² We will take quantiﬁcation over propositions (P , P 1 etc.), subjects (s, s 1 etc.), and times (t , t 1 etc.,) for granted. For a statement of the argument that does not rely on substitutional quantiﬁcation, see Kvanvig (1995).

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An essential feature of Fitch’s argument at this point is a sub-argument to the effect that (2) is false. This proceeds by ﬁrst assuming, for reductio, that the statement within the scope of the possibility operator at line (2) is true: (3) (∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) )) Plausibly, knowledge distributes across conjunctions, such that if a conjunction is known, then so are both of the conjuncts: (4) (∃s 2, t 2 ) (Ks 2, t 2 P 1 ) & (∃s 2, t 2 ) (Ks 2, t 2 ¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) Most will also agree that knowledge is factive, such that if one knows a proposition, then that proposition must be true. We can thus conclude (5): (5) (∃s 2, t 2 ) (Ks 2, t 2 P 1 ) &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) This is, of course, a contradiction. Since the assumption of this sub-argument leads to contradiction, we can therefore infer the negation of this assumption: (6) ¬(∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) ) Moreover, since this result has been derived based on no assumptions, we can also conclude that it is a necessary truth: (7) ♦¬(∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) ) Using standard modal logic, however, we can infer (8) from (7): (8) ¬♦(∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) ) Now (8) is obviously inconsistent with (2). It therefore follows that the original assumption—that we are non-omniscient—must be denied. The knowability principle, at least when combined with some very basic epistemic and modal logic, is therefore inconsistent with non-omniscience such that if we retain this principle then we must, it seems, accept the absurd conclusion that all truths are known.¹³ Fitch’s argument therefore poses a serious problem for semantic anti-realism. In the remainder of this paper we will explore two deﬂationary approaches that the semantic anti-realist could pursue in order to evade this argument. 3 . A De ﬂ a t i o n a r y Ap p r o a c h t o Fi t c h’s A r g u m e n t I : We a k e n i n g t h e Fa c t i v i t y Pr i n c i p l e The ﬁrst deﬂationary proposal that we will be exploring considers the prospects of offering an anti-realist response to Fitch’s argument which denies the factivity ¹³ See Williamson (1988b, 1992) for an argument to the effect that the conclusion just canvassed—that all truths are known—will not follow within intuitionistic logic from Fitch’s reasoning, even though it does follow that non-omniscience is false. For an excellent overview of the debate regarding Fitch’s puzzle, see Brogaard and Salerno (2004).

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of knowledge. It should be quite obvious that once the factivity of knowledge is denied, the argument that leads to the paradox, at least in its present form, will no longer go through since the step from (4) to (5) will no longer be valid. Of course, it is easy to say that one does not accept factivity and that, therefore, one isn’t impressed by Fitch’s argument. However, factivity seems to play an important—indeed, indispensable—role in any plausible theory of knowledge. In particular, it is one of the central guiding intuitions regarding knowledge that one cannot know falsehoods. That is, the very idea that there could be a case in which an agent knows a proposition and yet that proposition is false, just seems plain incoherent. Now of course one might claim that even the most deeply entrenched intuitions could be called into question on theoretical grounds. Even if this is so, however, it remains that any theory which denied factivity and thereby held that it was possible to know falsehoods would face a pretty severe uphill struggle when it came to gaining widespread acceptance. Nevertheless, there may be some room for manoeuvre here. After all, as we will now see, there is a potential logical gap—at least by semantic anti-realist lights—between the claim that there cannot exist any cases in which an agent knows a falsehood and the factivity claim that knowledge entails the truth of the proposition known. If this is right, then the semantic anti-realist can exploit this logical gap in order to motivate a weakened version of the factivity principle which can nevertheless retain the core guiding intuition behind factivity that there cannot exist cases of false knowledge. In order to see this, let us ﬁrst state factivity more formally: (FAC) (∀P ) (∀s) (∀t ) (Ks, tP → P ) Furthermore, let us state explicitly the intuition that is meant to drive adoption of (FAC)—viz., that there are no cases of false knowledge:¹⁴ (∗ ) ¬(∃P ) (∃s) (∃t ) (Ks, tP &¬P ) Now from (∗ ) we can derive (∗∗ ): (∗∗ )

(∀P ) (∀s) (∀t )¬(Ks, tP &¬P )

And from (∗∗ ) we can derive (∗∗∗ ): (∗∗∗ )

(∀P ) (∀s) (∀t ) (Ks, tP → ¬¬P )

From (∗∗∗ ) it might seem like a very small move indeed to get to (FAC), since all one needs to do is introduce the double negation equivalence rule (DNE) to eliminate the double negation in the embedded consequent. Crucially, ¹⁴ Note that to avoid unnecessary complications, we have here expressed the intuition in a slightly weaker form—i.e., that there are no cases of false knowledge, rather than that it is impossible for there to be cases of false knowledge. Nothing in what follows turns on this.

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however, intuitionistic logic does not contain (DNE), and yet it is precisely this logic that semantic anti-realists typically endorse. Accordingly, it follows that an anti-realist can accept the intuition guiding adoption of (FAC)—which we have expressed as (∗ )—without being compelled to endorse (FAC) itself. Instead, this guiding intuition merely entails the weaker claim which we have expressed as (∗∗∗ ), but which it is open to the semantic anti-realist to argue is itself a respectable version of factivity. We will call this weakened version of factivity, (FAC∗ ): (FAC∗ )

(∀P ) (∀s) (∀t ) (Ks, tP → ¬¬P )

This line of reasoning seems deﬂationary in just the right sort of way, since it shows that there is, at least by the lights of a particular theoretical outlook, a way of properly responding to the core intuition motivating (FAC) which results in a logically weaker principle. If this logically weaker principle can help the semantic anti-realist block Fitch’s argument, then this would thus be an extremely attractive way of resolving the situation. Unfortunately, however, closer inspection reveals that the present proposal is ultimately unsuccessful. True, on the face of it, (FAC∗ ) blocks the move from line (4) to line (5) in that it only gives us (5∗ ): (5∗ )

(∃s 2, t 2 ) (Ks 2, t 2 P 1 ) & ¬¬¬(∃sl, t 1 ) (Ks 1, t 1 P 1 )

Crucially, however, this triple negation collapses into a single negation, even within an intuitionistic logic, and thus one will be able to derive line (5) of the paradox of knowability anyway, even without having to appeal to (FAC). So one won’t solve the paradox of knowability by rejecting the factivity of knowledge and replacing it by the ever so slightly weaker (FAC∗ ). In order to get this line to work one would have to replace the factivity principle with something that is weaker even than (FAC∗ ). The difﬁculty facing such a proposal, however, is that it will not be able to do full justice to our intuition that one cannot know falsehoods. In this way, it is highly doubtful whether the present deﬂationary strategy can ultimately be successful.¹⁵ ¹⁵ Interestingly, in some recent (and unpublished) work Finn Spicer and Allan Hazlett have independently argued that there are good grounds for rejecting (FAC) outright. In particular, they argue for the plausibility of the claim that knowledge is best understood as reliable true belief. Thus, since one can have a reliably formed false belief, it follows that (FAC) must go. If such an argument could be made compelling, then it would hold out the prospect that the semantic anti-realist could exploit this proposal in order to block the Fitch argument on non-factivity grounds. Notice, however, that such a suggestion is not within a deﬂationary spirit. That is, unlike the deﬂationary proposal explored here, the move is not towards offering a logically weaker formulation of factivity which can nevertheless do justice (by the broader lights of that theory at any rate) to the intuitions which drive acceptance of factivity. Instead, this line of argument involves the straightforward rejection of those intuitions. This feature of the proposal makes this sort of manoeuvre dialectically problematic, and certainly ensures that it is not relevant for our purposes here.

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4 . A De ﬂ a t i o n a r y Ap p r o a c h t o Fi t c h’s A r g u m e n t I I : We a k e n i n g t h e K n ow a b i l i t y Pr i n c i p l e Although the proposal to deny factivity will not do the trick, there is a second proposal available that is in the same deﬂationary spirit and which is much more promising. The thought is that instead of rejecting one of the epistemic principles which are employed within Fitch’s argument, one instead rejects the very principle that is the target of that argument—i.e., the knowability principle itself. In its stead is then put forward a slightly weaker principle which can nevertheless accommodate the guiding motivation behind the knowability principle. Informally, the weakened principle that we have in mind is as follows: for all true propositions, it must be possible to justiﬁably believe them. More formally: (JP)

(∀P ) (P → ♦(∃s, t ) (JBs, tP ) )

In order to see why this principle of justiﬁed believability, as we will call it, suits the purposes of the semantic anti-realist it is important to ﬁrst notice that it accommodates the semantic anti-realists’ worries regarding realist theories of meaning. Recall that the semantic anti-realist argued that realist theories of meaning will have a problem explaining how we can acquire and manifest an understanding of the meanings of statements with evidence-transcendent truth-conditions. Recall, furthermore, that we saw that accepting the knowability principle will avoid this problem. Given the plausible additional assumption that one knows a proposition only if one also has evidence for it, it follows that if all truths are knowable then it must also be possible to have evidence for them. If it must be possible to have evidence for all true propositions, however, then there can be no evidence-transcendent truths. In this way, the semantic anti-realist can resist the realist’s second core claim that there are evidence-transcendent truths by accepting the knowability principle. Notice, however, that a parallel argument will show that accepting the justiﬁed believability principle will do the job just as well. After all, it is also plausible that one justiﬁably believes a proposition only if one has evidence for it. That means, however, that if, for all truths, it must be possible to justiﬁably believe them, then it must also be possible to have evidence for them. But if it must be possible to have evidence for all true propositions, then there can be no evidence-transcendent truths. In this way, the semantic anti-realist can resist the realist’s second core claim by accepting the justiﬁed believability principle. In short, this principle will do the job for the semantic anti-realist just as well as the knowability principle. Notice that while replacing the knowability principle with the justiﬁed believability principle will allow the semantic anti-realist to avoid the conclusion

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that we are omniscient—after all, justiﬁed belief is not knowledge¹⁶—but that does not mean that the semantic anti-realist is no longer susceptible to refutation by a Fitch-style argument. After all, a parallel argument for justiﬁed belief threatens to show that the justiﬁed believability principle entails that all statements are justiﬁably believed. And that, it would seem, is almost as bad for the semantic anti-realist as the original conclusion of Fitch’s argument. So there is still work to be done. One might think, however, that even if there is work to be done, it is not much work. After all, justiﬁed belief, as opposed to knowledge, is not factive. That is, one can justiﬁably believe a falsehood. For instance, one might reliably and conscientiously form the belief that there is a barn over there—and thereby have a justiﬁed belief in this proposition—even though this belief is nonetheless false because, unbeknownst to you, what you are in fact looking at is a barn facade. Accordingly, since Fitch’s argument relies on the factivity of knowledge, it follows that it will not go through if the knowledge operator is replaced by a justiﬁed belief operator. Hence it would seem as though all the semantic anti-realist has to do is to replace the knowability principle with the justiﬁed believability principle in order to avoid the conclusion of Fitch-style reasoning. 5 . Pro b l e m s w i t h t h e Se c o n d De ﬂ a t i o n a r y Ap p ro a c h t o Fi t c h - s t y l e Re a s o n i n g There are, however, problems on the horizon for this line of reasoning. In particular, one might think that the conclusion just canvassed is either false or uninteresting. To take the ﬁrst horn ﬁrst, one might think that it is false because even granted that justiﬁed belief is not factive, the following reﬂection principle does, nonetheless, hold: if, at a certain time, one justiﬁably believes that one does not at that time justiﬁably believe a proposition, then one does not at that time justiﬁably believe that proposition. More formally, we can express this principle as follows: (RP)

(∀P ) (∀s) (∀t ) (JBs, t (¬JBs, tP ) → ¬JBs, tP ) )

The signiﬁcance of this principle is that the relevant ‘factivity’ move in a Fitch-style argument employing the justiﬁed belief operator would be from (4 ) to (5 ): (4 ) (∃s 2, t 2 ) (JBs 2, t 2 P 1 ) & (∃s 2, t 2 ) (JBs 2, t 2 ¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) ) (5 ) (∃s 2, t 2 ) (JBs 2, t 2 P 1 ) &¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) ¹⁶ At least on standard views of justiﬁcation at any rate. If one held that justiﬁcation was factive, then there would be scope to contend that there is no logical gap between justiﬁed belief and knowledge. Such a theory of justiﬁcation would be highly revisionary, however, and so we can legitimately set this possibility to one side here.

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If the justiﬁed belief operator were factive, then that would straightforwardly license this inference. It ought to be clear, however, that even if the justiﬁed belief is not factive, then this inference will go through just so long as (RP) holds. So one might object that the mere fact that justiﬁed belief is not factive does not get the semantic anti-realist off the hook, since it is still plausible that justiﬁed belief satisﬁes the reﬂection principle which, it would seem, sufﬁces to generate the Fitch result. On the other hand, one might object that the result is uninteresting because it has long been established that the semantic anti-realist can resist the conclusion of Fitch-style argument by stating the epistemic constraint on truth in terms of justiﬁed believability. J. L. Mackie makes the point in the following passage: Suppose we read K [the knowledge operator in Fitch’s argument] as ‘It is justiﬁably believed at t 1 that . . .’ This will distribute over &, but we might expect the argument now to fail at step 4 [to 5 in the above statement of the argument], since this K is not truth-entailing. But step 4 [to 5] still goes through. If it is justiﬁably believed that p at t 1 that p is not justiﬁably believed at t 1 , then p is not justiﬁably believed at t 1 . On the other hand, if we read K as ‘It is justiﬁably believed at some time that . . .’, then step 4 does not go through. It does not follow that if it is justiﬁably believed at any time that p is not justiﬁably believed at any time, then p is not justiﬁably believed at any time. It might be justiﬁable at t 1 to think that p is false and never has been and never will be justiﬁably believed and yet there might be some other time t 2 at which p was, or will be justiﬁably believed. So the argument does not enable us to reject the principle that what is true can be justiﬁably believed at some time. (Mackie 1980: 91–2)

In this passage, Mackie distinguishes between two reﬂection principles for justiﬁed belief, one which he deems plausible and one which he deems implausible. The plausible reﬂection principle has it that if it is justiﬁably believed at t 1 that it is not justiﬁably believed at t 1 that p, then it is not justiﬁably believed at t 1 that p. This is, of course, the reﬂection principle—(RP)—that we formulated above. In contrast, according to the implausible reﬂection principle, if someone at some time justiﬁably believes that no one ever justiﬁably believes that p, then no one ever justiﬁably believes that p. This principle can be formalized in the following way: (RP∗ )

(∀P ) ( (∃s, t ) (JBs, t¬(∃sl, t 1 ) (JBs 1, t 1 P ) ) → ¬(∃sl, t 1 ) (JBs 1, t 1 P ) ) )

Mackie claims, correctly and for the right reasons, that (RP∗ ) is false. He goes on to claim, again correctly, that the conclusion of Fitch’s argument can be avoided if the epistemic constraint is construed in terms of justiﬁed believability at some time—i.e., what we have called the justiﬁed believability principle. Unfortunately, however, this last claim, while correct, is made for the wrong reasons. For, as we are about to show, Fitch’s conclusion can be derived from (RP), which Mackie deems plausible, and the justiﬁed believability principle.

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To begin with, we start with the relevant assumption for reductio—someone at some time justiﬁably believes that p and that no one ever justiﬁably believes that p: (3 ) (∃s 2, t 2 ) (JBs 2, t 2 (P 1 &¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) )) If (3 ) is true, then so is an instance of it. Or, in other words, if someone at some time justiﬁably believes that p and that no one ever justiﬁably believes that p, then there must be a particular epistemic subject who believes this conjunction at a particular time. Let the epistemic subject and time be s 3 and t 3, respectively. We then get: (4 ) JBs 3, t 3 (P 1 &¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) ) ) Since justiﬁed belief distributes across conjunctions, we get: (5 ) JBs 3, t 3 P 1 &JBs 3, t 3 ¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) Now, if one justiﬁably believes that there is no one at any time who justiﬁably believes that P 1 , then one also justiﬁably believes that, currently, one does not justiﬁably believe P 1 oneself.¹⁷ Accordingly, from (5 ) we can derive: (6 ) JBs 3, t 3 P 1 &JBs 3, t 3 (¬JBs 3, t 3 P 1 ) Given (RP), however, the second conjunct of (6 ) entails that s 3 does not justiﬁably believe P 1 at t 3 : (7 ) JBs 3, t 3 P 1 & ¬JBs 3, t 3 P 1 From here the Fitch-style argument proceeds as rehearsed. So we can argue to its conclusion without having to appeal to the implausible reﬂection principle, (RP∗ ). All that we need is (RP) which Mackie deems a plausible reﬂection principle. So Mackie’s distinction between the two reﬂection principles will not help the semantic anti-realist. If the semantic anti-realist is to get any mileage out of rejecting the principle of knowability and replacing it by the weaker principle of justiﬁed believability, then she must have some other way of resisting the Fitch-style conclusion. Fortunately for the semantic anti-realist, however, there is excellent reason to believe that (RP) does not hold. Consider the following case due to Saul Kripke: Pierre is a Frenchman who has lived most of his life in France. Having just ¹⁷ Some may object that the step from (5 ) to (6 ) will not go through because justiﬁed belief is not closed under known entailment (never mind under entailment simpliciter). However, the argument does not depend on such closure principles. It is plausible that if (RP) is valid—that is, if one justiﬁably believes at t 1 that one does not justiﬁably believe that p at t 1, then one does not justiﬁably believe that p at t 1—then so is the following reﬂection principle: if one justiﬁably believes at a given time that no one ever justiﬁably believes that p, then at that time one does not justiﬁably believe that p. Stated formally: (RP∗∗ ) (RP∗∗ )

(∀P )(∀s)(∀t )(JBs, t¬(∃sl, t 1 )(JBs 1, t 1 P ) → ¬JBs, tP ) will validate the inference to (7 ) without relying on any further closure principles.

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returned from a trip to London, one of Pierre’s best friends asserts ‘‘Londres est jolie.’’ Since Pierre knows his friend to be a man of exceptional taste he believes what his friend asserted and hence comes to believe that London is pretty. Now suppose that, by some unfortunate circumstance, Pierre ﬁnds himself stuck in a particularly unattractive part of London. Pierre is forced to take on a badly paid job that will just pay him enough to buy food and accommodation. At this time he learns English ‘directly’—that is, by direct interactions with other English speakers rather than referring to, say, translation manuals. Pierre uses the term ‘London’ as his neighbours do and learns everything his neighbours know about it which, let us suppose for the sake of argument, is not very much. On the basis of his experiences in the city he comes to believe that London is not pretty. At the same time, Pierre is still sometimes thinking about his nice life in France, and sometimes even of his friend who told him about the pretty city of London. In such moments Pierre thinks to himself: ‘‘Si seulement je serais en Londres . . .’’ Obviously, Pierre still believes that London is pretty and hence he has inconsistent beliefs. Moreover, his inconsistent beliefs are both justiﬁed. The testimony from a person with exceptional taste justiﬁes his belief that London is pretty while his direct experiences justify his belief that London isn’t pretty. It is plausible that whilst having inconsistent beliefs that are both justiﬁed, Pierre may also believe, justiﬁably, that he does not believe that London is pretty. Perhaps some psychologist analyses him and tells him that the source of his recent unhappiness is simply that he no longer believes himself to be living in a pretty city. Pierre thus comes to believe, and justiﬁably so (since on the basis of the reliable testimony from the psychologist), that he does not believe that London is pretty. But if one justiﬁably believes that one does not believe a proposition, then one also justiﬁably believes that one does not justiﬁably believe that proposition. Accordingly, Pierre also justiﬁably believes that he does not justiﬁably believe that London is a pretty city. Pierre’s case thus indicates that one can simultaneously justiﬁably believe all of the following: (a) a proposition, P; (b) its negation, not-P; and (c) the proposition that one does not justiﬁably believe P. Given that this is so, however, it can easily be seen that the reﬂection principle (RP) must fail. For, if (RP) held, it would follow that Pierre both does and does not justiﬁably believe that London is pretty. (RP) turns an inconsistency in Pierre’s belief-system (in conjunction with a second-order belief), into an inconsistency in the world. So it must be false.¹⁸ If (RP) is false, however, then the relevant Fitch-style argument no longer goes through. The semantic anti-realist is off the hook. There is, however, a further difﬁculty for the semantic anti-realist who endorses the justiﬁed believability principle. It remains true that since there are some statements that are true but will never be justiﬁably believed, it must, by ¹⁸ If one wants to run the argument by appeal to (RP∗∗ ) instead of (RP)—see the above footnote—one will need a slightly different case to make the present point. We provide such a case below.

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the justiﬁed believability principle, also be possible for someone at some time to justiﬁably believe an instance of this. Among other things that means that it must be possible for someone at some time to justiﬁably believe statements of the form ‘‘P but no one ever justiﬁably believes P’’ and, similarly, ‘‘P but I don’t justiﬁably believe P’’. And, as Dorothy Edgington (1985: 558) has pointed out, one might think that this is already bad enough for the semantic anti-realist. After all, it would seem that one just couldn’t have any evidence for statements of either form. If so, then it would seem that one also cannot justiﬁably believe such statements. Moreover, recall that the semantic anti-realist introduces the justiﬁed believability principle in order to ensure that meaning, construed truth-conditionally, can always be manifested in use. If there are truths of the form ‘‘P and no one ever justiﬁably believes P’’ and ‘‘P and I don’t justiﬁably believe P’’, then one must be able to manifest the meaning of those statements in understanding. Since justiﬁed believability is supposed to secure manifestability, it must be possible to justiﬁably believe statements of the form ‘‘P and no one ever justiﬁably believes P’’ and ‘‘P and I don’t justiﬁably believe P’’. But if it is impossible to have evidence that would support statements of this form, then one cannot justiﬁably believe such statements. So even if the semantic anti-realist can deny the reﬂection principle, (RP), Fitch’s argument shows that things are already bad enough for the semantic anti-realist even before the problematic principle comes into play. It would seem, however, that there are ways for the semantic anti-realist to respond to this difﬁculty. Let us begin with statements of the form ‘‘P and I don’t justiﬁably believe that P’’. In order to argue that statements of this form can be justiﬁably believed, the semantic anti-realist can simply point to Pierre’s case again and claim that Pierre might well come to believe that London is pretty (by believing that the proposition expressed by ‘‘Londres est jolie’’ is true) and that he does not believe that London is pretty (by believing that the proposition expressed by ‘‘I don’t believe that London is pretty’’ is also true). Since both of his beliefs are justiﬁed and since we typically acquire a justiﬁed belief in a conjunction by conjoining the justiﬁcation we have for the beliefs in the conjuncts,¹⁹ it would follow that the semantic anti-realist can comfortably allow that Pierre justiﬁably believes that London is pretty and that he does not justiﬁably believe that London is pretty.²⁰ Things are a bit more difﬁcult when it comes to statements of the form ‘‘P and no one ever justiﬁably believes that P’’. However, the situation is not hopeless for ¹⁹ Cf. Kvanvig (2006: 21). ²⁰ Of course, another consequence is that Pierre can come to justiﬁably believe a contradiction. This may initially seem an unwelcome consequence of the view. However, it is not clear why one could not justiﬁably believe a contradiction (at least so long as it is not an obvious one). A clever logician could easily tell me that what, in effect, is a complicated contradiction is true and I could come to believe this contradiction on that basis. Since the logician is an otherwise reliable informant on such issues, and since the testimony of reliable informants furnishes us with justiﬁed beliefs, my belief in the contradiction is surely justiﬁed.

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the semantic anti-realist. Consider, for instance, a case in which I am stranded on a lonely island. The only thing I have with me is a book about psychology which is written in English. I read about a brain lesion, named ‘X’, and learn that the only symptom of X, which occurs in 99 per cent of the cases, is a continuous strong belief on the part of the patient that she has X. Through introspection, I ﬁnd that I don’t believe that I have X. I therefore come to believe that I don’t believe that I have X. My belief, since based on reliable introspective capacities, is justiﬁed. I justiﬁably believe that I don’t believe that I have X. Since if one justiﬁably believes that one does not believe a proposition, P, then one also justiﬁably believes that one does not justiﬁably believe P, I justiﬁably believe that I don’t justiﬁably believe that I have X. Moreover, since I am on a lonely island, without drinkable water, and since I have every reason to believe that no one will come to my rescue and that I will die fairly soon, I also justiﬁably believe that no one will ever justiﬁably believe that I have X. To ﬁnish the story off, suppose that just before I left for my holiday, I called my doctor to get the results for some brain tests that they had done on me. My doctor told me that everything was ﬁne except that I have a brain condition called ‘Y’, which is completely harmless, and that I could go on the trip without any problem. On the basis of the testimony from my doctor I come to believe, justiﬁably, that I have Y. Now, since I am German and have talked to my German doctor our conversation was naturally in German. What I don’t know is that the German expression ‘Y’ and the English expression ‘X’ are names for the same brain lesion and that I am among the lucky 1 per cent of patients who don’t suffer from the symptoms. I am now in a condition in which I justiﬁably believe that I have X (on the basis of testimony from my doctor: ‘Sie haben Y’) and I also justiﬁably believe that no one ever justiﬁably believes that I have X (on the basis of introspection, what I have read about X in the psychology book and my unfortunate predicament of being on a lonely island about to die). If I were to conjoin my two beliefs, I would justiﬁably believe that I have X and that no one ever justiﬁably believes that I have X. There is thus a way for the semantic anti-realist to respond to Edgington’s worry. The semantic anti-realist may point out that, contrary to what Edgington claimed, one can justiﬁability believe statements of the form ‘‘P and I don’t justiﬁably believe P’’ as well as ‘‘P and no one ever justiﬁably believes P’’. The semantic anti-realist is, again, off the hook. In general, there are good grounds for holding that the deﬂationary strategy of replacing the knowability principle with the justiﬁed believing principle may well offer the semantic anti-realist a way of avoiding Fitch-style reasoning.

21 Not Every Truth Can Be Known (at least, not all at once) Greg Restall

Ab s t r a c t According to the ‘‘knowability thesis,’’ every truth is knowable. Fitch’s paradox refutes the knowability thesis by showing that if we are not omniscient, then not only are some truths not known, but there are some truths that are not knowable. In this paper, I propose a weakening of the knowability thesis (which I call the ‘‘conjunctive knowability thesis’’) to the effect that for every truth p there is a collection of truths such that (i) each of them is knowable and (ii) their conjunction is equivalent to p. I show that the conjunctive knowability thesis avoids triviality arguments against it, and that it fares very differently depending on another thesis connecting knowledge and possibility. If there are two propositions, inconsistent with one another, but both knowable, then the conjunctive knowability thesis is trivially true. On the other hand, if knowability entails truth, the conjunctive knowability thesis is coherent, but only if the logic of possibility is weak.

1 There are many things that we don’t know to be true. Ignorance is a fact of life. However, it is tempting to think that of the things that are true but not known to be true, each of them could be known. If the signiﬁcance of a proposition is to be explained in terms of its veriﬁcation conditions, for example, then if it is See http://consequently.org/writing/notevery/ for the latest version of the paper, to post comments and to read comments left by others. ¶ Thanks to Conrad Asmus, Allen Hazen, Lloyd Humberstone, Nick Smith and Timothy Williamson, to audiences at Monash University, the University of Melbourne, and Oxford University, and to commentators at http://consequently.org/writing/notevery/ for helpful discussions. Feedback from anonymous reviewers for this volume was useful in cleaning up the presentation.

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true, there must be some veriﬁcation conditions, and it is tempting to say that we could (at least potentially, in theory) have access to them. So, it is tempting to endorse the claim Every truth is knowable.

(♦)

which has come to be known as the knowability thesis, and its formalization (∀p) (p ⊃ ♦Kp)

(1)

for any truth, it is possible (♦) that it be known (K). This position is tempting to many, but Fitch has shown that the temptation comes at a very high cost. Using only inference principles that are very tough to reject, we can show that, given the knowability thesis, every truth is, indeed, known.¹

1.2 Here is Fitch’s proof that the knowability thesis fails if we are not omniscient. Suppose, for a reductio, that we are ignorant of some truth, so suppose that p is true but not known to be true. Then p ∧ ¬Kp is true. So, by the knowability thesis, this is possibly known: ♦K(p ∧ ¬Kp). Now, this is very hard to take. How could we know that p ∧ ¬Kp? If knowing a conjunction entails knowing the conjuncts, then K(p ∧ ¬Kp) entails Kp and K¬Kp.² Now knowledge entails truth, so K¬Kp entails ¬Kp, a contradiction. So, by a reductio, it is not possible that K(p ∧ ¬Kp), and we have (using the knowability thesis) refuted the hypothesis of ignorance. If the knowability thesis holds, a much stronger thesis holds too: every truth is not merely knowable, but known.

1.3 This phenomenon has come to be called Fitch’s Paradox, after F. B. Fitch, who ﬁrst formulated it (Fitch 1963). This paradox has generated a vast literature, including on the one hand ‘‘search and rescue’’ missions designed to ﬁnd the true principle underlying knowability thesis, and to save them from similar paradoxical fate, ¹ I indicated before that Kp should be read as ‘‘it is known that p.’’ But known by whom? Kp can be read either as ‘‘α knows that p’’ for some ﬁxed agent α, or ‘‘someone knows that p’’ without too much strain in what follows. The existential reading, which requires that p merely be known by someone, ensures that K is nothing like a normal modal operator. We do not have Kp, Kq K(p ∧ q), since it may well be that someone knows that p and someone (else) knows that q without anyone knowing that p ∧ q. In what follows, however, we soon move from reading Kp as the straightforward ‘‘p is known’’ to the idealization ‘‘p is a logical consequence of what is known,’’ and this does satisfy the principle that distributes knowledge over conjunctions: if p and q are consequences of what is known (by someone or other) then so is their conjunction. For any of these readings, the knowability thesis has some bite. It seems like a substantial claim that any truth is knowable by someone or other. It seems like a more substantial claim that any truth is knowable by you. ² Maybe we could know a conjunction without knowing the conjuncts. No problem: Just interpret Kp as ‘‘p is a logical consequence of what I know.’’ If the knowability thesis works for knowledge, it works for this K too. So, from now on, K will allow for deductive closure: if Kp and p q then Kq.

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and ‘‘seek and destroy’’ campaigns aimed at hammering more nails into the cofﬁn of any so-called principle of knowability that shows signs of life.³ This paper contains elements of both kinds of discussion. I shall present and motivate yet another revision of the knowability thesis, and then show that this revision is consistent and not subject to Fitch-paradoxical refutation—that is the search and rescue part of the story—and then I will show that this revision is not only consistent but either it is also almost trivially true and therefore, it is not likely to do the work that a veriﬁcationist or anti-realist might require of a principle of knowability, or it’s an interesting, controversial thesis about knowledge, which is coherent under certain conditions.

1.4 The knowability thesis, cast as the statement (1), is dead. Fitch’s paradox is a conclusive refutation, and even though many interesting moves are possible with the logic in which these principles are couched, defeating the inference from (1) to omniscience (Beall 2000; Wansing 2002), these answers do not address the question I take to be asked by Fitch’s paradox. I say this because upon reﬂection, the principles motivating a knowability thesis in fact undercut its application in a case such as p ∧ ¬Kp. Consider any truth p, of which we are ignorant. Given the knowability thesis we can, indeed, imagine coming to know that p. This is all well and good, but any way we can go about knowing that p makes it no longer the case that ¬Kp. But, ¬Kp was true, and so, maybe it too could be known. If we do not enquire as to the status of p (so we don’t come to know that p is true) but rather take ourselves to consider whether or not Kp, it seems plausible to suppose that we could conﬁrm that ¬Kp. In other words, it’s quite coherent to suppose that there is nothing that we can see that makes it impossible for us to know that ¬Kp. But the conjunction p ∧ ¬Kp is true, and none of the ways we have considered, of coming to know p, or coming to know ¬Kp will provide a way to know both p and ¬Kp. The conjunction p ∧ ¬Kp cannot be known ‘‘all at once.’’ Fitch’s proof, it seems, is not a trick to be avoided or to be explained away but a result to be understood.

2 This reasoning points the way to a possible answer: the Fitch-paradoxical conjunction p ∧ ¬Kp cannot be known ‘‘all at once’’ but it can be known ‘‘in pieces.’’ In particular, the ﬁrst conjunct can be known (or rather, there seems to be nothing preventing us knowing it), but it cannot be known if the second conjunct is known. Similarly, the second conjunct can be known, but ³ Brogaard and Salerno’s ‘‘Fitch’s Paradox of Knowability’’ (2004) is a ﬁne guide to this literature.

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it cannot be known if the ﬁrst conjunct is known. They cannot be known together. (1) is refuted, but it begs for a reformulation. Instead of saying that any truth could be known, let’s attempt to maintain instead that every truth can be known ‘‘in pieces.’’ That is, for any truth p, there is some collection of truths, each of which could be known, and when taken together, entail the original truth p. In other words, p can be factored into components, each of which is knowable.

2.2 If we could defend the knowability thesis in this weaker form, according to which unknowable truths could be factored into knowable pieces, then we may be able to provide some comfort to the anti-realist who takes meaningfulness to be a matter of knowability. For the fact that p ∧ ¬Kp is unknowable is no counterexample to its meaningfulness any more than the unknowability of p ∧ ¬p renders it meaningless. No, p ∧ ¬p is meaningful when p is meaningful, because we can understand p and its negation and its conjunction, even if to understand this is to come to see that it can never be known for it can never be true. The same kind of process can be seen in p ∧ ¬Kp, though now we have a conjunction which we can see that we will never know even though it may be true. It is meaningful because it is a conjunction of meaningful claims.

2.3 In fact, one could say that in the original naïve formulation (♦) didn’t mean what is expressed by (1), at least in its application to the statement p ∧ ¬Kp. For the p ∧ ¬Kp is not, in itself, one truth that is knowable, but two. There are two knowable truths here, not one. (This is altogether too tendentious a reading to take seriously, however. Nothing in this paper hangs on the idea that conjunctive knowability is what we really wanted in the ﬁrst place.)

2.4 Now consider what it is for a sentence to be a conjunction of knowable sentences. (In what follows, I will call these ‘knowables’ for short.) From the perspective of pure logic it matters not whether the original sentence is complex or atomic. For whatever may be expressed by a complex sentence may be expressed by an atomic sentence too. In whatever model theory we like, if we have a model in which a complex sentence is interpreted in some way, then as far as logic is concerned, any simple sentence may be interpreted in just that way too.⁴ But suppose that our original sentence was a complex sentence like p ∧ ¬Kp. This sentence is ⁴ This phenomenon underlies one of the substitutional properties of formal logics. If φ is a tautology containing the atomic sentence p, then φ , found by replacing p everywhere by another formula B is also a tautology.

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unknowable. If this is because it expresses an unknowable sentence, then if we interpret the atomic sentence q as ‘‘meaning the same thing’’ as p ∧ ¬Kp, then it seems that q will (relative to this interpretation, of course) be unknowable as well. But q has no conjuncts at all: it is a simple sentence. Have we sunk the ‘‘factoring’’ analysis before it could set sail?

2.5 This factoring analysis may survive if we are prepared to agree that while the sentence q from our example has no explicit conjuncts, it may have conjuncts implicitly. The sentence q is equivalent (relative to this model, again) to the conjunction p ∧ ¬Kp. As far as logic is concerned this will sufﬁce for a factorization. We will say that q is conjunctively knowable (relative to a model) if it is equivalent (relative to that model) to a conjunction, each of whose conjuncts are knowable (relative to that model).

2.6 This is my proposed revision of the knowability principle: Every truth is conjunctively knowable.

(♥)

In the rest of this paper, we will examine the fate of this principle.⁵

3 For the proposal to be formally evaluated, it must be stated more sharply. This thesis assumes a number of exotic elements of logical vocabulary, such as propositional equivalence, propositional quantiﬁcation, epistemic and modal operators. To properly state this thesis will require a great deal of machinery. The syntax of the claim is straightforward enough. We may formalize one version of this claim as follows: (∀p) (p ⊃ (∃q, r ) ( (p = q ∧ r ) ∧ ♦Kq ∧ ♦Kr ) )

(2)

This version posits a very strong version of conjunctive knowability: every proposition may be factored into a conjunction of two propositions, each of which is knowable.⁶ ⁵ After writing a draft of this paper, Joe Salerno brought to my attention Risto Hilpinen’s paper ‘‘On a Pragmatic Theory of Meaning and Knowledge’’ (2004), in which he argues that a Peircean pragmatism motivates a conjunctive knowability principle just like this. I must leave it to the reader to determine whether or not the results of the investigation below are congenial to the pragmatist project. ⁶ It could be that something could be factored into three knowable conjuncts but not two. As far as I can see, there is no natural upper limit to the number of conjuncts one could require in a formalization of (♥), so a perfectly general formulation would perforce be quite complex indeed, as

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3.2 The work comes in when we are required to characterize the logical properties of propositional quantiﬁcation, propositional identity and the modal and epistemic operators. Thankfully, for our purposes, we need not attempt to pin down the right principles governing ♦, K, (∀p) and =. Qua logician my job is to investigate the consistency of (♥) and its formalization (2). Fitch showed that (1) is inconsistent with deeply plausible modal and epistemic principles. I will show that (2) does not suffer that fate. (2) is consistent, and compatible with the strong principles of modal and epistemic reasoning. To do this, we need not ﬁnd the right principles of such reasoning. In doing this, it is acceptable to overshoot and require too much. I will provide a class of models that show that the revised knowability thesis (♥) and its formalization (2) can be absolutely unrestrictedly true at no cost to ignorance or to many other epistemic or modal principles. (There will, however, be an important caveat to be discussed in Section 5.)

3.3 Our logic will be the incredibly strong modal epistemic logic in which ♦ and K are both governed by the principles of the logic s5. This is unrealistic in the extreme, for it commits us to wild epistemic principles such as the claim that if p is true then we must know that we don’t know that ¬p (if p then K¬K¬p) and even that if we don’t know something we know that we don’t know it (if ¬Kp then K¬Kp). Neither of these principles is particularly plausible (even if we take Kp to mean that p is a consequence of what we know) but we will use such a strong logic nevertheless, since nearly every epistemic or modal principle endorsed by someone or other is valid in this logic: s5 ♦ ⊕ s5 K .

3.4 This logic has models of the usual kind for modal logics. Here a model is a quadruple W , R♦ , RK , [[ · ]] where W is a non-empty set of worlds, R♦ and R K are accessibility relations on W and [[ · ]] is a function assigning to each atomic sentence (for example, p) an interpretation—a set of worlds (in this case, [[p]]). In this modal logic we place no restrictions on which sets can be used to interpret sentences. All sets may be propositions in our model. The set [[p]] is the set of worlds in which p is true. In the usual way, the interpretation function is it would have to quantify over collections of propositions (there are some propositions which factorize p). This seems to be the right condition: (∀p)(p ⊃ (∃P)(( P = p) ∧ (∀q)(Pq ⊃ ♦Kq))) (2 ) where P is the second-order propositional variable used in the second-order propositional existential quantiﬁer ∃P ranging over classes of propositions (which is ‘‘really’’ a third-order quantiﬁer over objects, since propositions are ‘‘really’’ zero-place properties) and the ‘‘connective’’ sends a class of propositions to its conjunction.

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extended to assign sets to arbitrary sentences in the language. For conjunction, disjunction, the material conditional and negation we use the standard Boolean operations. A conjunction p ∧ q is true at the worlds where both p and q are true: [[p ∧ q]] = [[p]] ∩ [[q]], and similarly for the other Boolean connectives. The relations R♦ and R K are used to model the operators ♦ and K respectively. In our case R♦ and R K are both equivalence relations. R♦ is the equivalence relation governing ♦: #

♦φ is true at w iff φ is true at a world in w’s R♦ equivalence class.

We will call the worlds in w’s R♦ equivalence class the modal alternatives of w.⁷ Another way to understand the interpretation is as a function from propositions to propositions. [[♦φ]] is the union of all modal equivalence classes overlapping [[φ]]. Think of approximating the proposition [[φ]] by equivalence classes, counting in our approximation any equivalence class that at least partly overlaps the original proposition: [[φ]] is approximated by its closure. Similarly, an equivalence relation R K governs K: #

Kφ is true at w iff φ is true at all worlds in w’s R K equivalence class.

We will call the worlds in w’s R K equivalence class the epistemic alternatives of w. [[Kφ]] is the set containing all epistemic equivalence classes totally included in [[φ]]. Here, [[φ]] is approximated by its epistemic interior. The modal alternatives of w need not be the same worlds as the epistemic alternatives: a modal alternative need not be an epistemic alternative (we can know things that are not necessary) and an epistemic alternative need not be a modal alternative (we can be ignorant of some necessary truths). Now for the propositional quantiﬁer: # (∃p)φ is true at a world w if and only if for some set X of worlds, φ is true at w when we take the formula p occurring unbound in φ to be true at exactly the worlds in X. # For identity, we will say that φ = ψ is true at a world just when [[φ]] = [[ψ]]. This sufﬁces to ensure that we may, for example, infer from φ = ψ that θ(φ ) = θ(ψ). Any formula containing φ (for example, ♦Kφ) is true in the same worlds as the formula found by replacing those φs by ψs (in this case, ♦Kψ). ∃p,= An argument is s5 ♦ ⊕ s5 K valid if for every model, if the premises are true in a world in that model, the conclusion is true in that world too. A sentence is ∃p,= an s5 ♦ ⊕ s5 K tautology if and only if it is true in every world in every model.

3.5 Let me reiterate: This model theory is not to be endorsed as giving us the ‘‘true picture’’ of knowledge, possibility, propositional quantiﬁcation and propositional identity. It is intended as a grab-bag sizeable enough to catch all principles thought ⁷ This leads to a slight infelicity: w counts as a modal alternative of itself.

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to govern epistemic modal logic with propositional quantiﬁers. If we can ﬁnd models that both validate the conjunctive knowability principle (♥) and allow for ignorance, then they show that no principle true in these models collapses conjunctive knowability into omniscience.

3.6 Figure 21.1 shows a simple model in which (♥) holds. There are four worlds {a 1 , a 2 , b 1 , b 2 }. a1

a2

b1

b2

∃p,=

Figure 21.1. A simple s5 ♦ ⊕ s5 K

frame

The modal accessibility relation R♦ relates a 1 to b 1 and a 2 to b 2 ; the epistemic accessibility relation RK is orthogonal to the modal relation: it relates a 1 to a 2 and b 1 to b 2 . So, a world’s modal alternatives are those worlds sharing a number, and its epistemic alternatives are those sharing a letter. In Figure 21.1 (and in all other diagrams) solid lines join epistemic alternatives, and dashed lines join modal alternatives.

3.7 (♥) says that any proposition true at the world of evaluation is a conjunction of two propositions which, at the world of evaluation, are knowable. What are the propositions in our model that are knowable at any world? Any proposition true at both a 1 and a 2 is knowable at all worlds, since it is known at a 1 and a 2 (and hence, it is possibly known there), and at b 1 , the world a 1 is possible, and at b 2 , a 2 is possible, so at b 1 or at b 2 , this proposition is also possibly known. So, if {a 1 , a 2 } ⊆ [[φ]], then φ is knowable at any point in the model. Similarly, any proposition true at both b 1 and b 2 is knowable at every world. And these propositions are the only propositions knowable at any world. The propositions which can not be known are ∅, each singleton proposition {a 1 }, {b 1 }, etc., and the two diagonal propositions {a 1 , b 2 } and {a 2 , b 1 } and the modal alternative propositions {a 1 , b 1 } and {a 2 , b 2 }. All other propositions are knowable, from the point of view of every world.

3.8 It will be helpful to consider why for any interpretation of p, the proposition denoted by p ∧ ¬Kp is not knowable at any world. If [[p]] = X ⊆ W , then [[Kp]]

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consists of the interior epistemic approximation of X, and [[¬Kp]], then, is the union of all equivalence classes not totally inside X. So, its intersection with X (the set [[p ∧ ¬Kp]]) consists of the union of all X-overlapping parts of epistemic equivalence classes that overlap X but do not fall completely inside X.⁸ In the case where [[p]] = {a 1 , a 2 , b 1 }, [[Kp]] = {a 1 , a 2 } and so [[¬Kp]] = {b 1 , b 2 }, and [[p ∧ ¬Kp]] = {b 1 }. This proposition is not knowable, because it contains no epistemic equivalence classes as a subset.

3.9 In this model, every true proposition is a conjunction of two knowable propositions. The singleton {a 1 } is the conjunction of {a 1 , a 2 } and {a 1 , b 1 , b 2 }. The same goes for each other singleton. The pair {a 1 , b 1 } is the conjunction of {a 1 , a 2 , b 1 } and {a 1 , b 1 , b 2 }. The same goes for each other pair. It follows that (2) is true in our model (Figure 21.2). a1

a2

b1

b2

Figure 21.2. {a 1 } as a conjunction of knowables

3.10 So, we have shown that (2) survives consistently and coherently. In this model there is much ignorance (a proposition true at a 1 alone is true there but not known to be true), yet every proposition is a conjunction of two knowable propositions. The principle (♥) of conjunctive knowability is secure. Truth and knowability can be intimately connected, even if not every truth is knowable. What the friend of knowability cannot have whole she is allowed to have if she will accept it in two pieces.

3.11 At this point, the story takes a different turn. Conjunctive knowability is secure, but it either is almost certainly not what the veriﬁcationist wants, or it is too high a price for the veriﬁcationist to pay. In the rest of this paper I shall show that if knowability does not entail truth (so some falsehoods are knowable while not being known), then conjunctive knowability, in the form of (2) is ⁸ In topological terms it is the part of the (epistemic) boundary of X that is also inside X.

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not only true, but it’s very hard to refute in an epistemic modal logic. It puts precious few constraints on knowledge or necessity, and so, is not useful as criterion for favouring one theory over another. If principles acceptable to the realist lead them to accept (2) while maintaining their realism, then (2) will do no good as a principle designed to favour the anti-realist. On the other hand, if knowability entails truth, then any non-trivial account of conjunctive knowability is inconsistent with plausible modal principles. (In particular, with the modal principle of transitivity: ♦♦p ♦p.) 4 ∃p,=

So, consider what we have done so far. We have a model of s5♦ ⊕ s5 K in which conjunctive knowability is satisﬁed. It turns out that this is not a one-off affair. ∃p,= In an epistemic modal logic like s5♦ ⊕ s5 K , and its much weaker cousins in which the modal and epistemic accessibility relations satisfy fewer constraints, (2) turns out to be very easy to validate. Not only are there many models in which (2) is true, it turns out that (2) is a consequence of other, unproblematic modal and epistemic principles. In particular, (2) follows from the following thesis about possible knowledge, satisﬁed in the models we have seen: (∃q) (♦Kq ∧ ♦K¬q)

(3)

This is relatively uncontroversial, given one understanding of how possibility and knowledge (or the consequences of what is known) are connected. Provided that, for some q, both q and ¬q are possibly true (and this is not too difﬁcult to imagine) then it is not much more difﬁcult to conclude that for some q, both q and ¬q are possibly known. Of course, a circumstance in which one knows q is, perforce, one in which ¬q is not known, and vice versa, for there to be a q such that ♦Kq and ♦K¬q, there must be at least two distinct modal alternatives, one at which q is known, and the other at which ¬q is known. All that requires is that we have two modal alternatives whose epistemic closures do not intersect, like so (Figure 21.3).

Figure 21.3. Two inconsistent knowables

Given a q such that both it and its negation are knowable, we can prove that for any true p, there are knowable p 1 and p 2 where p = p 1 ∧ p 2 and

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both p 1 and p 2 are knowable. If q is such that (♦Kq ∧ ♦K¬q), then we may choose p 1 to be p ∨ q and p 2 to be p ∨ ¬q. Then by simple Boolean reasoning, p 1 ∧ p 2 = (p ∨ q) ∧ (p ∨ ¬q) = p. However, since ♦Kq, we have ♦K(p ∨ q).⁹ Similarly, since ♦K¬q, we have ♦K(p ∨ ¬q). Both p ∨ q and p ∨ ¬q are knowables, regardless of how unknowable p might be! (2) is a trivial consequence of the trivial truth (∃q) (♦Kq ∧ ♦K¬q). It looks as if (2) tells us little about the connection between truth and knowability.

4.2 Where can the fan of conjunctive knowability resist this analysis? It might be thought that a friend of relevance would quail at the identiﬁcation of p with (p ∨ q) ∧ (p ∨ ¬q), as well they should. The inference from p to (p ∨ q) ∧ (p ∨ ¬q) is valid in almost every logic you care to mention, as it is found by composing the inferences from p to p ∨ q, and from p to p ∨ ¬q and from these to their conjunction. All are simple lattice moves. The problem with relevance is in the other direction. To get from (p ∨ q) ∧ (p ∨ ¬q) we need q ∧ ¬q p, and this is relevantly invalid. Nonetheless, rejecting this identity¹⁰ is not going to stop this argument from getting off the ground. The crucial premise in the argument was that q and its negation were both knowable, and could be used in the factorization of p. There is no requirement that q and its negation be used for this purpose. Provided that we are given two incompatible propositions (say, q 1 and q 2 ) that are knowable—so q 1 , q 2 ⊥ for the trivial proposition ⊥, and ♦Kq 1 and ♦Kq 2 —then even in relevant logics, the sentences p and (p ∧ q 1 ) ∨ (p ∧ q 2 ) are true in exactly the same situations. Blocking the inference from (p ∧ q 1 ) ∨ (p ∧ q 2 ) (with the rule q 1 , q 2 ⊥) to p requires blocking the distribution of conjunction over disjunction, not any odd behaviour about negation or relevance. So, pleading relevance or paraconsistency will not give the fan of conjunctive knowability (or its enemy, for that matter) a straightforward way out of the problem.¹¹

4.3 Denying that one can infer K(p ∨ q) from Kp is not going to help, either, for as we have seen, we can replace talk of what is known by talk of what is a consequence of ⁹ By distribution of both K and ♦ over logical consequence: since q p ∨ q, then Kq K(p ∨ q) (remember, we read K(p ∨ q) as ‘‘p ∨ q is a consequence of what is known’’) and so, ♦Kq ♦K(p ∨ q): all are reasonable principles. ¹⁰ Which, it must be said, is not the same identity as that between p and (p ∧ q) ∨ (p ∧ ¬q), a factorization seen again and again in different kinds of reasoning. ¹¹ A not-quite-straightforward way out of the problem is to deny that there are any incompatible pairs of propositions. To be sure, in many relevant logics, there is no way to construct formulas φ and ψ such that φ, ψ ⊥. Nonetheless, in most models for such logics there are ways to interpret φ and ψ such that their conjunction is absolutely inconsistent. Merely take [[φ]] and [[ψ]] to have empty intersection, so their conjunction is true nowhere.

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what is known (at least in decidable logics), and clearly, if p is a consequence of what is known, so is p ∨ q. Furthermore, possibly being a logical consequence of what is known is not very far removed from being possibly known, so reading Kp throughout as ‘‘p is a consequence of what is known’’ does little violence to the principles in question, and it validates the inferences used in our deduction. So requiring high standards for knowledge, so high that logical consequence can lead you from what is known to what is not, is also not a way out of the problem.

4.4 So, conjunctive knowability is not only consistent, but it is trivially so, if possibility and knowledge are connected as given by (3), that is, if possible knowledge can sometimes outrun truth, just as Fitch’s paradox has shown us that truth can sometimes outrun possible knowledge.

5 Nonetheless, (3) is by no means uncontroversial.¹² What if we reject (3) and hold, instead, that only truths may be possibly known? So, let us embrace (4): ♦Kp p

(4)

5.2 Before proceeding, I wish to do away with a bad argument for (4). No-one should argue as follows: ‘‘What is possibly known must be true, because of necessity, what is known is true. It is, therefore, impossible for what is known to be false. It follows that if something is possibly known, it is true.’’ This contains a modal fallacy. We have attempted to infer from the innocuous ‘‘it is impossible for what is known to be false’’ (¬♦(Kp ∧ ¬p) ) to the much stronger ‘‘if something is possibly known, it is true’’ (which as a material implication is ¬♦Kp ∨ p). In the ﬁrst, the truth of p (or its not being false) is under the scope of the possibility operator, and in the second it is not. That is a bad argument for (4). If you contemplate (4), do not do so for that reason.

5.3 In the case of an epistemic modal logic modelled with an accessibility relation R♦ for possibility and R K for knowledge, (4) is straightforward to guarantee: we need simply that (∀x) (∀y) (xR♦ y ⊃ yR K x) ¹² Thanks to Nick Smith for pressing me on this point.

( 4 )

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for then, when we are at x and we have some modally accessible world (call it y) in which every epistemically accessible world has p true, p is true at x, since x is epistemically accessible from y. Conversely, if we have some x and y where xR♦ y but not yR K x, then if p is true at everywhere other than x (or, if you like, everywhere epistemically accessible from y, if everywhere other than x seems like overkill) then at y, Kp is true, and hence at x, ♦Kp is true. However, at x, p is false. So, if we are allowed to assign the extension of a proposition at whim in our models (and it is hard to see why not) then condition ( 4 ) corresponds precisely to the validity of (4).

5.4 Similarly, we can say, precisely, what condition on R K and R♦ corresponds to conjunctive knowability in its weakest possible form. First note that, in a given model, if p is conjuctively knowable when [[p]] is a singleton set (so p is true at one world only) then every proposition is conjunctively knowable. (If φ is true at x, and if p is true at x alone, then consider the propositions, each knowable, which jointly entail p. These jointly entail φ —relative to that model—too, which shows that φ is also conjunctively knowable.) So, what does it take for a proposition true at x alone to be conjunctively knowable? Well, we must ﬁnd for any world y distinct from x, a proposition which is knowable but not true at y. If that is not found, the conjunction of all knowable propositions will not entail p, since it will also be true at y, where p is not true. So, we require the following condition (∀x) (∀y) (x = y ⊃ ∃z(xR♦ z ∧ ¬zR K y) )

(5)

for if (5) does not hold, then any z modally accessible from x will include y as epistemically accessible, so no proposition false at y will be knowable from x, as it will not be known at any modally accessible worlds.

5.5 It follows that normal epistemic modal models for conjunctive knowability satisfy (5). Alas if ( 4 ) and (5) both hold, then if R♦ is transitive, it is trivial in the sense that xR♦ y only if x = y. Here is why: if ( 4 ) holds, then ¬zR K y means that ¬yR♦ z, which when substituted in ( 4 ) gives (∀x) (∀y) (x = y ⊃ ∃z(xR♦ z ∧ ¬yR♦ z) ) but if x and y are non-identical and yR♦ x, then whenever xR♦ z by transitivity yR♦ x, which contradicts what we have assumed.

5.6 We have a syntactic proof of this modal collapse as well. We can show that (4), ♦♦p ♦p and conjunctive knowability (in the most general form ( 2 )), ensure that ♦p p, in the presence of propositional quantiﬁcation.

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5.7 Here is the proof: Suppose ♦p. So there is some possible circumstance in which p is true. Consider one. In this circumstance p is true, so there are propositions r 1 , r 2 , . . ., which together entail p, and each of which are possibly known. So, we have ♦Kr 1 , ♦Kr 2 , etc., and r 1 , r 2 , . . . p. Now consider the actual circumstance in which ♦p is true. In this circumstance, each ♦Kri is possible: that is, ♦♦Kri . But possible possibility is (we assume) possibility, so we have ♦Kri for each i. But by (4), ♦Kri ri , so each ri is true. But r 1 , r 2 , . . . p, so p is true too. In other words, we have inferred p from ♦p.

5.8 So, we cannot have (4), (5), ♦♦p ♦p and the non-triviality of ♦. One, at least, must go. Which one is to go? I am tempted to do away with (4), but we have already seen what can be done without (4): it makes conjunctive knowability all too easy. Making R♦ trivial is unacceptable, for then the only possibilities will be truths, so if every proposition is a conjunction of knowables, it will be a conjunction of knowns, and hence, every truth will be a consequence of what is known, making all ignorance vanish. We avoid Fitch’s paradox and its heirs by denying the premise that we are not omniscient. To do away with (5) is to give up the task of exploring the consequences of conjunctive knowability. The only remaining option here (given the machinery of normal epistemic modal logics and their possible worlds models) is to explore the rejection of the transitivity of R♦ . As a result, we will examine what follows if we deny the inference from ♦♦p to ♦p.

5.9 Denying transitivity of R♦ is a severe price to pay to save conjunctive knowability. It turns out that it is enough. In the remaining paragraphs of this section I will show that we may maintain (4), making every knowable a truth, and (5), making every truth conjunctively knowable, without concluding that every truth is known. A model showing this is relatively simple (Figure 21.4). The worlds are the (positive and negative) integers Z. We have xR♦ y iff y = x or y = x + 1. (Notice that this is not transitive, since 0R♦ 1 and 1R♦ 2, but we don’t have 0R♦ 2. Nonetheless, it is reﬂexive, so at the very least, p ♦p.) We have xR K y iff y = x or y = x − 1. (Notice that this is not transitive either, so we do not have Kp KKp, but it is reﬂexive, so Kp p, as one would hope.)

x−2

Figure 21.4. The Model

x−1

x

x+1

x+2

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5.10 That is the model. Let’s see how it manages to satisfy (4 ) and (5). We have satisﬁed (4 ) by ﬁat: if xR♦ y, then y = x or y = x + 1, in which case either x = y or x = y − 1, ensuring that yR K x. So, (4 ) is satisﬁed, ensuring that ♦Kp p. Conjunctive knowability, in the form of (5), is satisﬁed too. If x = y, then there is always some z where xR♦ z but not zR K y. If we don’t have xR K y, then choosing x for z will sufﬁce (since xR♦ x always). If we do have xR K y, then since x = y, we have y = x − 1. Then choose x + 1 for z. We have xR♦ z (z is one step up from x) but we don’t have yR K z (y is two steps down from z, which is just too far).

5.11 How does this model work? At every point, x, knowledge is a little limited because x is epistemically indistinguishable from x − 1. Only propositions true at both x and x − 1 may be known at x. Nonetheless, the world x + 1 is modally accessible from x, and at this world, x − 1 is not epistemically accessible but x is. This means that any proposition true at x is a conjunction of two knowable propositions (Figure 21.5). If p is true at the set X (including x) then consider two propositions q 1 and q 2 , true at X ∪ {x − 1} and X ∪ {x + 1} respectively. q 1 , true at X ∪ {x − 1}, is known at x (and so is possibly known at x) and q 2 , true at X ∪ {x + 1}, is known at x + 1 (and so is also possibly known at x). In this case, as in our other models, every proposition true at a point is a conjunction of two knowable propositions. Nonetheless, not every proposition is known: at every point there is ignorance.

x−2

x−1

x

x+1

x+2

Figure 21.5. {x} as a conjunction of two propositions knowable at x

5.12 So, if knowability entails truth then we can maintain the conjunctive knowability thesis in the form of (2), but only at the cost of rejecting the s4 principle for possibility: ♦♦p ♦p.

6 Fitch’s paradox shows us that not every proposition is knowable: at least not all at once. Fitch’s paradoxical sentence is an example of a proposition that cannot be

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known, but which can nonetheless be split into pieces, each conjunct of which can be known. It turns out that this modest fallback position is coherent. We may coherently hold that every proposition can be factored into a conjunction, each of which are knowable. Thinking of this in terms of possible worlds, it comes quite close to one original consideration in motivating of knowability. Propositions divide possible worlds into those that are in and those that are out. Conjunctive knowability tells us that for any world that a proposition takes to be out, we can know something that would rule out that world. Think of the discriminations that a proposition makes as constituted by all of the worlds inconsistent with it. According to conjunctive knowability, no proposition makes a discriminiation essentially beyond our grasp. This is coherent. If two inconsistent propositions are knowable, then conjunctive knowability is coherent but trivial. If knowability entails truth, then conjunctive knowability is both coherent and substantial.

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Index adjunction 107 agnostic ﬁctionalism 273–7 algebra 25 n. 7 aliases, problem of 156, 157 Analyticity 285 Anderson, C. A. 170 n. 13 Anderson, A. R. 21 n. 2, 26 n. 7, 119 anonymity, problem of 24 n. 5, 156, 157 anti-realism 5–6, 65–6, 74, 183–204 and equivalence thesis for truth (ET) 62, 63 and equivalence thesis for warranted assertibility (EA) 63–4 global 298, 302, 303 intuitionism and 78 knowability and 209–11 about meaning 289–92, 297–8 and recognition-transcendence 289–92, 293–4 semantic 53–5, 77, 250–1, 325–8, 332–8 and truth 184–8, 289–92 Armour-Garb, B. 207 n. 6 Armstrong, D. 307 n. 6 Artemov, S. 131 assertibility of disjunctions 192–3 warranted 63–4 asserting 217–18, 273 assertions 285–6 self-defeating 286–7, 294 self-fulﬁlling 285 Ayer, A. J. 293 Azzouni, J. 276 Balaguer, M. 254, 255, 258 Baltag, A. 139 Barcan Marcus, R. 21 n. 3 Beall, Jc 7–8, 102 n. 9, 119 n. 23, 125 n. 32, 303 n. 1, 319, 341 belief 3–4, 39, 40, 164–6, 169–75, 176 disbelief principle 166, 171 false common 145 type theory and 169–70 believing 22, 31 Belnap, N. 26 n. 7, 119 van Benthem, J. 8, 33 n. 4, 130–46 Bigelow, J. 5 n. 4 Binkley, R. 166 bivalence 66–7

Boolos, G. 321 n. 1 Brock, Stuart 243 Brogaard, B. 5 n. 5, 9, 10, 47 n. 22, 130, 147, 158, 187 n. 5, 207 n. 3, 227 n. 4, 230 n. 5, 231 n. 6, 242 n. 3, 247, 249 n. 20, 250 n. 21 & n. 22, 252 Bueno, O. 9, 261 n. 12 & n. 13, 265 n. 17, 277 Burali-Forti paradox 80 Burge, Tyler 322 Burgess, J. 8, 35 n. 10 Burks, A. W., 23, 33 n. 5, 46 Burm´udez, J. 7 Cantorian sets 254 Cantor’s proof 80 Carnap, Rudolf 21 n. 3,173 causal necessity 23 causal possibility 23 causation, partial 25–6 Church, Alonzo 8, 67, 168, 179, 302, 322 ﬁrst referee report on Fitch’s ‘‘Deﬁnition of Value’’ 13–15, 36–40, 163 n. 1 identiﬁcation as anonymous referee 35–7 second referee report on Fitch’s ‘‘Deﬁnition of Value’’ 15–18, 41–5, 169 n. 11 theory of types 168 Clark, P. 82 closure 107, 111, 118 Cogburn, J. 5 n. 4, 8, 20 Colyvan, M. 265 n. 16 coming to know 131–3, 139 common knowledge 134, 135–6, 140, 145 and multi-agent learning 143–4 and public announcements 138, 141–3 compositionality 298 conditional fallacy 38, 46, 48, 250 n. 22 conjunctive knowability thesis 341–53 worlds theory and 344–6, 351, 352, 353, 354 consistency 117, 171, 254, 256–61, 267, 269, 271–3 contradictions 109–10, 112, 117 contraposition 107 in Fitch’s paradox 97–9 da Costa, n. C. A. 261 n. 12, 277 Costa-Leite, A. 9–10 Cozzo, Cesare 189 n. 9, 207, 292 n. 11, 305–6, 308

368

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de dicto knowledge 155–7 de re knowledge 155–7 decidability 66–7, 190–3, 195–6, 290 decidable sentences 59–61, 66–7, 209 deontic necessity 23 deontic possibility 23 desire 23, 27 knowledge and 40, 41–3 Devidi, D. 6 n. 7, 8, 189 n. 9, 207 n. 3, 249 dialetheism 97, 100–1, 102 disbelief axiom 173 disbelief principle 166, 171 discovery principle 147–57 de dicto knowledge 155–7 de re knowledge 155–7 formalization of 151–5 objections to 149–51 temporal paradox 147–9 distribution 106, 111, 117 distribution axiom 165 van Ditmarsch, H. 138, 139 Divers, John 244, 245, 246, 249, 250 doing 22, 26 Douven, I. 309 Dummett, Michael 54, 63, 64, 66, 129, 183, 192–3, 207, 237, 248 n. 18, 250, 289, 290, 292 n. 11 anti-realism 1, 5–7, 53–4 basic/atomic sentences 73 indeﬁnite extensibility 78, 79–81, 82–4 inductive speciﬁcation 89–90 intuitionistic logic 186–7 justiﬁcationism 51–2 knowability principle, restriction of 6–7, 187 n. 5 manifestation argument 250–1, 326 n. 4, 326–7 meaning theory 297, 327 natural numbers 80–1 negation 189 ordinal numbers 79–80, 83–4 quantiﬁcation 82–4 response to Fitch’s paradox 89–90 restricted knowability principle 6–7, 73, 305, 297–300, 304 n. 2, 305 Dunn, J. M. 114 n. 18, 119 n. 24 Edgington, Dorothy 9, 140, 158, 189 n. 10, 204 n. 28, 207, 248, 304 n. 2, 306, 337, 338 Egr´e, P. 146 Enderton, H. B. 35 n. 10, 80 ephemeral truths 150–1 epistemic logic dynamic 134–6, 138, 145 event updates 136–7, 139–40

and evidence 130–1 static 133–4 epistemic realism 278 n. euclidean geometry 275 evidence-transcendent truths 326–8, 332 excluded middle 97–8, 108 factiveness 225 and normalization 233–5 operators 30–2, 33–4 restriction on 230–7 factivity principle: weakening of 329–31 Fagin, R. 141 false common belief 145 Fara, M. 9 ﬁctionalism 241–80 agnostic ﬁctionalism 275 intuitionistic strategy 247 mathematical 264–73 modal 242–5, 250 modal fallacies strategy 247–8 restriction strategy 248–50 rigidiﬁer strategy 248 ﬁctionalist epistemology 255–6 Field, Hartry 255, 264, 265–6, 267–8, 269, 270, 277, 279 Fitch, Frederic axioms 14, 20 Cartesian restriction 41–5, 46, 47–8 ‘Deﬁnition of Value’: withdrawal of 45 deﬁnitions 14–15, 16–17, 19, 33–4, 37, 41 empirical necessitation 37 n. generalized paradox 3–4 operators 4, 18, 30–1 theorems 1, 3, 20, 31–3, 36, 37 n., 45–6, 76–8 theory of value, collapse of 47–8 Fitch self-defeat 288, 289, 294 Fitch’s paradox 85, 320–3, 340–1 anonymous referee, identity of 34–7 basic rules 106–7 deﬂationary approach I: weakening the factivity principle 329–31 deﬂationary approach II: weakening the knowability principle 332–8 premises of 321 and proof 320–1 responses to 77–8, 89–90, 205–7 Florio, S. 229 van Fraassen, B. 279 Frege, Gottlob 272 Gabbay, D. 131 game theory 135, 139, 144, 145 Gerbrandy, J. 138

Index global anti-realism 298, 302, 303 G¨odel, Kurt 34–5, 212, 257, 262, 271–3, 277, 279 G¨odel-mapping (G¨odel-McKinsey-TarskiRasiowa-Sikorski mapping) 57, 58, 62, 66, 72–3, 74 G¨odel’s axiom 321 G¨odel sentences 257, 264 Greenough, P. 325 n. 3 Grim, P. 86 Guldmann, F. 62 n. 10 Gupta, Anil 322 Hagen, J. 244 n. 9, 246 n. 15 Hale, B. 243–4, 249 n. 19 Halpern, J. 141 Hand, Michael 8, 9, 197, 207, 214 n. 11 & n. 12, 219 n. 12, 249, 285 n. 3, 291 n. 10, 297 n. 14 Hart, W. D. 5, 9, 39, 45 n. 20, 54, 59, 95, 193 n. 16 Hazlett, Allan 331 n. Herzig, A. 139 Heyting, A. 66 Hilbert spaces 254 Hilpinen, R. 343 n. 5 Hintikka, Jaakko 130, 135, 146, 165, 172, 175 Hoshi, T. 139 Humberstone, L. 5 n. 5 idealism, problem of 300–1 ignorance 1, 29, 35: see also knowledge incompleteness 34–5, 123, 212, 257, 271–2 inconsistency 121–3: see also consistency indeﬁnite extensibility 78–81, 82–4, 85–6, 87 natural numbers 80–1, 82 ordinals 79–80 of propositions 86, 88, 90 real numbers 80 sets 80 indexical sentences 285, 289 indispensability argument 264–5 ineffable truths 150 inference 284 in Fitch’s paradox 95–7, 99–100 global restrictions on rules of 225–6 inscribing 218 intuition: Platonism and 264 intuitionism 6, 56–7, 65, 78, 186–8 intuitionistic logic 186–7 intuitionistic provability 57 irrationality 28 irrealism 321: see also anti-realism; realism

369

Jago, M. 146 James, P. 26 n. 7 Jehle, D. 245 n. 12 Jenkins, C. S. 9, 213 n. 9, 247, 306–9, 317 n. 17 justiﬁcationism 51–2 justiﬁed belief 332–8 reﬂection principles for 333–4, 335, 336 Kamp, Hans 322 Kelly, K. 145 Kelp, C. 9 Kenyon, T. 8, 207 n. 3, 249 knowability 107, 111–12, 117 and anti-realism 209–11 global restriction on 226–30 and inconsistency 123 and knowledge 121–2 and non-omniscience 105–6, 112, 113, 118 and paraconsistent veriﬁcationism 118–21 and theism 207–9, 218, 220–1 and trivial world 111, 112, 113 truth and 184–6 validity of 109, 111, 112, 113, 118–21 validity, non-normal semantics of 114–17 and veriﬁcationism 113–14, 118–21 knowability paradox 1–3, 32–3, 283–5, 302–3 classic puzzle 303–8, 318 and lost logical distinction 209–22 new puzzle 304, 308–16, 318 and omniscience 241 operators and 214–19 plausibility of 324 and syntactic generalization strategy 216–19 knowability principle 1–3, 5–6, 77–8 global restriction 223–38 normalization and 229–30 and omniscience 252–3 and Peirceanism 161–2 performance principle and 295–6 restrictions on 6–7, 187 n. 5, 223–38, 296–300 and temporal/modal analogy 157–9 and temporal/modal combination 159–60 weakening of 332–8 knowability theorems 1, 3, 20, 24–5, 31–3, 36, 37 n., 45–6, 76–8 Knower paradox 101–3, 108 knowing 22, 31 deﬁnition of in terms of believing 26–7 knowledge 39–40, 121–2, 164, 171–3, 178 and abnormal epistemic possibilities 124–5 and desire 40, 41–3 explicit temporal perspectives on 140–1

370

Index

knowledge (cont.) in Fitch’s paradox 95–6 interpretation of ‘know’ 194–6 limits of, and contradiction 102–3 see also ignorance Kooi, B. 138 Krause, D. 261 n. 12, 277 Kripke, S. 114 n. 18, 117 n. 22, 139, 209 n. 4, 335 K¨unne, Wolfgang 35 Kuppfer, M. 285 n. 3 Kvanvig, Jonathan L. 5 n. 5, 8, 9, 197, 204 n. 28, 207 n. 3, 247, 249, 297 n. 14, 304, 309–19, 311, 312, 318–19, 328 n. 12, 337 n. 19 learning (coming to know) theory 145 Lewis, David 242, 243–6, 270 Lewis/Langford: deﬁnitions and theorems 13, 18–19 liar paradox 197 n. 21, 323 de Lima, T. 139 Linsky, B. 3 n. 3, 5, 8, 39, 322 Linstr¨om, S. 9, 189 n. 10, 204 n. 28 Liu, F. 135 logical necessity 23 logical possibility 23, 256–7 McDowell, J. 70 n. 19 McGee, Vann 212 McGinn, C. 5, 45 n. 20, 54, 59, 95, 276 McGlynn, Aidan 315 n. MacIntosh, J. J. 5 n. 4 Mackie, John 5, 45 n. 20, 214, 287, 294, 311 n. 10, 316 n. 15, 334, 335 Makinson, D. C. 167 n. 8 manifestationism 250–1, 298 mapping objection 57–9, 63–4, 71, 73 negative part 58–9, 62, 63, 64, 74, 75 positive part 58, 59, 74 Martin-L¨of, P. 129 mathematical ﬁctionalism 264–73 mathematical propositions: indeﬁnite extensibility of 88 mathematics, philosophy of comprehension principles 261–4, 275–6, 277, 280 and full-blooded Platonism 254–61, 280 mathematical objects, existence of 254, 255–6, 261, 262, 263, 264, 265, 271, 273–4, 275–7, 276–7 and standard Platonism 261–4, 280 meaning theory 73, 300, 325–7 acquisition challenge 327 anti-realist 289–92, 297–8 manifestation challenge 327

realist theories of 325–6 and truth 297–8 Melia, J. 9, 204 n. 28 Meyer, Robert 34–5 modal ﬁctionalism 242–5 modal realism 243–4, 249 & n. 19 molecularism 298 Moses, Y. 141 Moore’s Paradox 36, 130, 132, 165 Moretti, L. 5 n. 4 Muddy Children puzzle 141–3 Murzi, J. 9, 36 Nagel, Ernest 13, 34 n. 7, 35–6, 40–1 natural numbers: indeﬁnite extensibility of 80–1, 82 NBG (von Neumann-Bernays-G¨odel) set theory 263, 267–8, 269, 271–2, 273 necessity 21–3 negation 170, 189 Nolan, D. 242 n. 4, 243 n. 6 & n. 7 nominalism, see mathematical ﬁctionalism non-assertibility 66 non-contradiction 107 non-omniscience: 105–6, 112, 113, 118: see also omniscience Nozick, R. 31 n. 2 object theory 261 n. 13 obligation 27, 213 omnipotence, paradox of 4 omniscience 328–9 full-blooded Platonism and 257–8 and knowability paradox 241 and knowability principle 295–6 see also non-omniscience omniscience principle 1–3, 78, 85 ontological argument 318 Orayen, Ra´ul 323 n. 6 ordinal numbers 79–80, 83–4 ordinary language: value concepts and 21–2 Osborne, M. 145 Pacuit, E. 140 Pagin, P. 189 n. 9 paracompleteness 108–9 and inconsistency 121–2 semantics 110–11 Strong Kleene framework 108–9 and trivial worlds 111, 112, 123 paraconsistency 7, 97–8 paraconsistent veriﬁcationism 118–23 and inconsistency 121–3 and knowability, validity of 118–21 Parry, William 25 n. 6, 36 Parsons, Charles 268, 322

Index partial causation 25–6 Peano axioms 275 Peirceanism 161–2 Percival, P. 6, 9, 99 n. 3, 189 n. 9 & n. 10, 247 n. 17 performance principle: and knowability principle 293, 295–6 Plantinga, Alvin 176–7, 209 n. 5 Platonism 253, 274 comprehension principles in mathematics 261–4 full-blooded 254–61, 280 and intuition 264 standard (traditional) 261–4, 280 plurality of worlds theory 243–4, 246, 247, 249 Poincar´e, H. 322 n. 2 possibility 21–2, 93, 254 causal 23 deontic 23 in Fitch’s paradox 96–7 logical 23, 256–7 lost distinction between actuality and 209–13, 219 possible worlds theory 131, 220, 242–3, 249–50, 270, 306–7, 314 and conjunctive knowability thesis 344–6, 351, 352, 353, 354 possibly true contradictions 109–10, 112, 117 Potter, M. 82 pragmatic self-defeat 286–9 Prawitz, Dag 31 n. 3, 237, 292 n. 11 Preface Paradox 163, 166–8 Priest, G. 8, 100 n. 5, 103 n. 11, 104 n. 2 & n. 3, 110 n. 10, 111 n. 11, 119 n. 23, 125 n. 32 Prior, Arthur 151, 159–61, 167 n. 8 Pritchard, D. 9, 325 n. 3 propositions and conjunction elimination 22–3, 24–5, 27 and conjunction introduction 23, 27 compound 87, 89 indeﬁnite extensibility of 78–9, 86, 88, 90 mathematical 88 sets of 86, 87 simple 87, 88, 90 truth classes of 23–4, 27, 76–7, 164 proving 22 public announcements 132, 134–5, 136 and common knowledge 138, 141–3 Putnam, H. 1, 5, 265 n. 16, 327 n. 10 quantiﬁcation over indeﬁnitely extensible domains 78–9, 81, 82–7

371

and principles of transmissibility 84–5, 86, 87–9 and truth values 78–9, 82–3, 85 Quine, W. V. O. 169, 265 n. 16 Raatikainen, P. 292 n. 11 Rabinowicz, W. 9, 189 n. 10, 204 n. 28 Ramsey, F. 321–2 Rasmussen, S. 6, 7, 54–5, 62 Ravnkilde, Jens 6, 54, 55 Rea, M. 5 n. 4 real numbers 80 realism and decidable sentences 59–61 epistemic 278 n. meaning theories 325–6 modal 243–4, 249 n. 19 semantic 56 recognition-transcendence 289–92, 293–4, 300 relevance logic 349 reliabilism 272–3 Rescher, N. 33 n. 4, 35 n. 9 Resnik, M. 264 n. 15 Restall, G. 9, 114 n. 18, 119 n. 23 restricted realism 61, 62 restriction strategies 74 Dummett 6–7, 73, 305 on rules of inference 225–6 Tennant 8, 41, 55, 71–3 Robinson, A. 275 Rosen, Gideon 242 & n. 4, 243, 244 n. 9, 249, 250 Rosenkranz, S. 47 n. 22, 187 n. 5 Routley, Richard 5, 7, 8, 34, 35, 45 n. 20, 102 n. 9 Rubenstein, A. 145 R¨uckert, J. 9, 158 n. 5, 189 n. 10, 204 n. 28, 306 Russell, Bertrand 5, 79, 163 n. 1, 170 n. 13, 178 n. 17, 321 Russell set 277 Russell’s paradox 315 Russell’s theory of types 17, 321–3 Sack, J. 140 Salerno, J. 5, 6, 6 n. 5, 9, 10, 47 n. 22, 130, 147, 158, 187 n. 5, 207 n. 3, 227–8, 230 n. 5, 231 n. 6, 242 n. 3, 247, 249 n. 20, 250 n. 22, 252, 297 n. 14, 302, 304 n. 2, 322 n. 4, 329 n. 13, 341 n. 3, 343 n. 5 Salerno proof: global restriction 227–8 Segerberg, K. 9, 189 n. 10, 204 n. 28 self-defeat 294, 300 and indexicals 289 pragmatic 286–9

372 self-defeating assertions 286–7, 294 self-fulﬁlling assertions 138–9 self-reference 177 type theory and 167–9, 171–2 semantic anti-realism 53–5, 77, 325–8 and justiﬁed belief 332–8 timid 250–1 semantic realism 56 set theory 274–5, 321–2 NBG (von Neumann-Bernays-G¨odel) 267–8, 269, 271–2, 273 Zermelo-Fraenkel (ZFC) 323 n. 6 Shalkowski, S. 270 Shapiro, S. 264 n. 15, 267 Shope, Robert 178 simpliﬁcation 107 Smith, N. 350 n. 12 Solomon, G. 6 n. 7, 189 n. 9 Sorenson, Roy 36 n. 12, 101 n. 7, 165 n. 5, 168–9 Spicer, Finn 331 n. Stanley, J. 10 Stebel, K. 316 n. 16 Stjernberg, Frederik 166 n. 7 Stone, Jim 44 n.19 striving 22 strong modal ﬁctionalism 242, 247, 250 structuralism 264 n. superassertibility 56 n. Surprise Examination Paradox 138, 163,168–9, 170 n. 12 Szabo, Z. 10 Tarski, Alfred 265, 322, 323 n. 6 Taylor, Richard 221 n. Tennant, Neil 6, 47 n. 22, 54–5, 130, 183–204, 188 n. 7, 207, 223–38, 251 n. 23, 291 n. 10, 300, 327 n. 10 anti-realism 54, 305 anti-realist response to Fitch argument, soft 197–204 anti-realist treatment of Fitch argument, moderately hard 189–97 decidability 190–3, 195, 196, 197 factiveness 225, 230–7 interpretation of ‘know’ 194–5 intuitionistic relevance logic 198–200 restricted knowability principle 197–204, 297–300 restriction: Cartesian 5, 8–9, 41, 47, 55, 71–3, 197–204, 248, 249, 304 n. 2, 305 restriction: global 223–38 restriction: importance of normal form for 229–30, 233–5

Index theism 207–9, 218, 220–1 time-indexing 60, 61, 62, 64–5, 73, 74, 75 timid modal ﬁctionalism 242 n. 4, 250 timid semantic anti-realism 250–1 truth 23 anti-realism and 184–8, 298–92 equivalence thesis for 62, 63 and knowability 184–6 recognition-transcendence of 289–92, 293–4 veriﬁcation and 176–7, 178–9 truth classes of propositions 23–4, 76–7 theorems about 24–5 truthmaker necessitarianism 307 truths ephemeral 150–1 evidence-transcendent 326–8, 332 ineffable 150 Tsohatzidis, S. 285 n. 3 type theory 177–8, 321–3 and belief 169–70 and self-reference 167–9, 171–2 veriﬁcationism and 131 understanding 83, 291, 326–7 unrestricted claims problem 245–6 Usberti, G. 189 n. 9, 204 n. 28 uttering 217 value 15, 27–8 informed-desire theory of 33 and ordinary language 21–2 Vardi, M. 141 veridicality 106, 111, 117 veriﬁcation 193–4 canonical 283 n. and truth 176–7, 178–9 veriﬁcation principle 93–4, 96–7, 101 veriﬁcation procedures: performance of 292–3, 296 veriﬁcationally inconsistent sentences 68–70 veriﬁcationism 45, 77, 108–9, 133, 146, 163–4, 173–6 knowability and 113–14 and knowledge and abnormal epistemic possibilities 124–5 non-normal semantics of validity 114–17 paraconsistent 118–23 and paradoxes 129–30 and Peirceanism 161–2 strong 303, 304, 309, 310, 312, 313 and type theory 131 weak 303–7, 308–13, 318–19 veriﬁcationist thesis 129, 132, 133, 140, 145

Index Walton, D. 5 n. 4 Wansing, H. 7, 341 Warﬁeld, T. 39 n. 16 warranted assertibility 63–4 Wehmeir, K. 158 n. 5 Weiss, Bernhard 51, 55, 64 n. 14, 65, 74, 75 Whitehead, A. N. 78 n. 13, 163 n. 1, 170 n. 13, 178 n. 17 Williamson, Timothy 1, 2, 4, 6, 9, 29 n., 62, 47 n. 22, 53, 57, 62–4, 75, 101 n. 6, 158, 166 n. 7, 169 n. 10, 170 n. 12, 175, 183–204, 206–7, 211, 213 n. 9, 247, 249 n. 20, 273, 297, 302, 309 n. 9, 323 n. 6, 324 n. 1, 329 n. 13 Wittgenstein, Ludwig 323 n. 5, 326–7 Wright, Crispin 5 n. 4, 6, 9, 38 n. 14, 54, 63, 178, 189 n. 9 & n. 10, 193 n. 16,

373

247 n. 16, 288, 294 n. 12, 326 n. 4, n. 5, & n. 6, 327 n. 7, n. 8, & n. 10, 328 n. 11 anti-realism 54, 55 lost-opportunity cases 294 n. provisional biconditionals 173–4, 176–8 on realism 326 n. 4 superassertibility 56 n. veriﬁcationism 173–4 writing 218 Yap, A. 140 Zalta, Ed 261 n. 13 Zermelo, E. 263, 321 Zermelo-Frankel set theory 263, 264 n. 15, 315 n. 14 Zimmerman, T. 285, 286 n. 4

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New Essays on the Knowability Paradox Edited by

JOE SALERNO

1

1

Great Clarendon Street, Oxford ox2 6dp 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 in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With ofﬁces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © the several contributors 2009

The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2009 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, 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 book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Laserwords Private Limited, Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 978–0–19–928549–5 1 3 5 7 9 10 8 6 4 2

—for Rebecca

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Contents List of Contributors Acknowledgements

x xii

Introduction

1

Joe Salerno PA RT I : E A R LY H I S TO RY 1.

Referee Reports on Fitch’s ‘‘A Deﬁnition of Value’’

13

2.

Alonzo Church A Logical Analysis of Some Value Concepts

21

3.

Frederic B. Fitch Knowability Noir: 1945–1963

29

Joe Salerno PA RT I I : D U M M E T T ’ S C O N S T RU C T I V I S M 4.

Fitch’s Paradox of Knowability

51

5.

Michael Dummett The Paradox of Knowability and the Mapping Objection

53

6.

Stig Alstrup Rasmussen Truth, Indeﬁnite Extensibility, and Fitch’s Paradox

76

Jos´e Luis Berm´udez PA RT I I I : PA R AC O N S I S T E N CY A N D PA R AC O M P L E T E N E SS 7.

Beyond the Limits of Knowledge

8.

Graham Priest Knowability and Possible Epistemic Oddities Jc Beall

93 105

viii

Contents PA RT I V: E PI S T E M I C A N D T E M P O R A L O PE R ATO R S : AC T I O N S , T I M E S A N D T Y PE S

9. Actions That Make Us Know

129

Johan van Benthem 10. Can Truth Out?

147

John Burgess 11. Logical Types in Some Arguments about Knowability and Belief

163

Bernard Linsky PA RT V: C A RT E S I A N R E S T R I C T E D T RU T H 12. Tennant’s Troubles

183

Timothy Williamson 13. Restriction Strategies for Knowability: Some Lessons in False Hope

205

Jonathan L. Kvanvig 14. Revamping the Restriction Strategy

223

Neil Tennant PA RT V I : M O D A L A N D M AT H E M AT I C A L F I C T I O N S 15. On Keeping Blue Swans and Unknowable Facts at Bay: A Case Study on Fitch’s Paradox

241

Berit Brogaard 16. Fitch’s Paradox and the Philosophy of Mathematics

252

Ot´avio Bueno PA RT V I I : K N OWA B I L I T Y R E C O N S I D E R E D 17. Performance and Paradox

283

Michael Hand 18. The Mystery of the Disappearing Diamond

302

C. S. Jenkins 19. Invincible Ignorance

320

W. D. Hart

Contents

ix

20. Two Deﬂationary Approaches to Fitch-Style Reasoning

324

Christoph Kelp and Duncan Pritchard 21. Not Every Truth Can Be Known (at least, not all at once)

339

Greg Restall Bibliography Index

355 367

List of Contributors Jc Beall is Professor of Philosophy at the University of Connecticut, and Arché Associate Fellow at the University of St Andrews. Johan van Benthem is University Professor of Logic at the University of Amsterdam and Professor of Philosophy at Stanford University. Jos´e Luis Berm´udez is Professor and Director of Philosophy, Neuroscience and Psychology at Washington University in Saint Louis. Berit Brogaard is Research Fellow at the RSSS Philosophy Program and Centre for Consciousness at the Australian National University, and Associate Professor of Philosophy at the University of Missouri-Saint Louis. Ot´avio Bueno is Professor of Philosophy at the University of Miami. John Burgess is Professor of Philosophy at Princeton University. Alonzo Church was Associate Professor of Mathematics (without tenure) at Princeton University when his contribution to this volume was written in 1945. He was Professor of Mathematics and Philosophy at Princeton, 1961–7, and then Professor of Philosophy and Mathematics at UCLA until he retired in 1990. Church died in 1995. Sir Michael Dummett is Emeritus Professor at the University of Oxford. He was knighted for service to philosophy and racial justice. Frederic B. Fitch was Sterling Professor Emeritus of Philosophy at Yale University when he died in 1987. Michael Hand is Professor of Philosophy at Texas A&M University. W. D. Hart is Professor of Philosophy at the University of Illinois at Chicago. C. S. Jenkins is Arché Associate Fellow at the University of St Andrews, Associate Fellow of the Centre for Metaphysics and Mind at the University of Leeds, and Lecturer at the University of Nottingham. Christoph Kelp has recently completed his doctoral dissertation at the University of Stirling. Jonathan L. Kvanvig is Distinguished Professor of Philosophy at Baylor University. Bernard Linsky is Professor of Philosophy at the University of Alberta. Graham Priest is the Boyce Gibson Professor of Philosophy at the University of Melbourne, and Arché Professorial Fellow at the University of St Andrews. Duncan Pritchard is Professor of Philosophy at the University of Edinburgh.

List of Contributors

xi

Stig Alstrup Rasmussen has formerly held positions at the Universities of Edinburgh and Copenhagen. He obtained his habilitation at the University of Copenhagen in 2004. Greg Restall is Associate Professor of Philosophy at the University of Melbourne. Joe Salerno is Research Fellow at the Australian National University and Associate Professor of Philosophy at Saint Louis University. Neil Tennant is Humanities Distinguished Professor of Philosophy at the Ohio State University. Timothy Williamson is the Wykeham Professor of Logic at the University of Oxford.

Acknowledgements I am indebted to the contributors for their hard work and their patience with the editorial process. Each of them has furthered my understanding of the knowability proofs and other matters modal epistemic. I am extremely grateful to the graduate students that have served as my research assistants: Amy Broadway (Missouri), Heidi Lockwood (Yale), Julien Murzi (Shefﬁeld), Jonathan Nelson (Saint Louis) and Nick Zavidiuk (Saint Louis). They helped me enormously with various aspects of the project—among them, plundering archives, transcribing, compiling the bibliography, typesetting, and proof-reading for typographical and philosophical errors. Thanks also to my graduate students at Saint Louis University for stimulation, and to those grads at the Goethe University of Frankfurt who attended my seminar on the knowability paradox in May 2006. I thank the editor, Peter Momtchiloff, and supporting editors for their assistance and dedication to the project and to two anonymous Oxford University Press readers for their exceedingly helpful comments and suggestions. Many people have helped to improve the volume in one way or another, including Aldo Antonelli, Jc Beall, Scott Berman, Jim Bohman, Susan Brower-Toland, John Burgess, David Chalmers, Roy Cook, Judy Crane, Michael Della Rocca, Herbert Enderton, Saul Feferman, Bas van Fraassen, Anne-Sophie Gintzburger, John Greco, Nick Grifﬁn, Michael Hand, Monte Johnson, John Kearns, Jon Kvanvig, Bernard Linsky, Heidi Lockwood, Ruth Barcan Marcus, Robert Meyer, Gualtiero Piccinini, Graham Priest, Krister Segerberg, Wilfried Sieg, Roy Sorensen, Kent Staley, Jim Stone, Eleonore Stump, Neil Tennant, Achille Varzi and Ted Vitali. Berit Brogaard deserves special mention for being a constant source of feedback. My daughter, Rebecca, has been an endless source of sleepless nights but also inspiration, both of which were needed to complete this project. It is to her that I dedicate the volume. J.S.

Introduction Joe Salerno

T h e K n ow a b i l i t y Pa r a d o x In his seminal paper A Logical Analysis of Some Value Concepts (1963; reprinted, Chapter 2 of this volume), Frederic Fitch articulates an argument that threatens to collapse a number of modal epistemic distinctions. Most directly, it threatens to collapse the existence of fortuitous ignorance into the existence of necessary unknowability. For it shows that there is an unknown truth, only if there is a logically unknowable truth. Fitch called this ‘Theorem 5’, which usually is represented formally as follows: (Theorem 5 ) ∃p(p & ¬Kp) ∃p(p & ¬♦Kp) , where p holds a place for sentence letters; ♦ is normal possibility, read ‘it is possible that’; and K is the epistemic operator, ‘it is known (by someone [like us] by some means or other at some time) that’. The theorem rests on tremendously modest modal epistemic principles, which we will turn to shortly. The converse of Theorem 5 is modest as well. So Theorem 5 does the interesting work in erasing the logical difference between there being truths forever unknown and there being truths logically unknowable. The contrapositive of Theorem 5 is better known as the knowability paradox: (Knowability Paradox)

∀p(p → ♦Kp) ∀p(p → Kp).

If each truth is knowable in principle, then it follows logically that each truth is at some time known. That’s the result. It is thought to be paradoxical for a number of related reasons. First, it refutes all too easily interesting brands of anti-realism which are committed to the knowability principle, ∀p(p → ♦Kp). It refutes them since the knowability principle entails the obviously false omniscience principle, ∀p(p → Kp). The knowability principle has been claimed for a number of historic non-realisms, among them Michael Dummett’s semantic anti-realism, Hilary Putnam’s internal realism, the logical positivisms of the Berlin and Vienna Circles, Peirce’s pragmatism, Kant’s transcendental idealism, and Berkeley’s metaphysical

2

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idealism. How strange that the knowability principle, and every brand of nonrealism that avows it, are only as plausible as the exceedingly implausible, and obviously false, omniscience principle. An extension of Fitch’s result, found in Williamson (1992: 68), shows that a traditional strengthening of the knowability principle forecloses on the very distinction between what is possible and what is actual. Roughly, if truth is possible knowledge then possibility is actuality.¹ The paradoxicality is that sophisticated forms of anti-realism could be so easily refuted. A second reason to regard the proof as paradoxical is that it threatens to erase the logical distinction between the knowability principle and the omniscience principle. More speciﬁcally, the proof logically collapses the relatively moderate and plausible claim that each truth can be known into the apparently stronger and unbelievable claim that each truth is in fact known. The claims seem to carry distinct logical commitments, but they do not if Fitch’s result is valid. Fitch’s result presupposes the following principles. Knowing a conjunction requires knowing each of the conjuncts: (A) K (p & q) Kp & Kq Knowing entails truth: (B) Kp p Theorems are necessarily true: (C) If p, then p And, a necessarily false proposition is impossible: (D) ¬p ¬♦p The proof may be characterized this way:

At the top of the tree we suppose for reductio that the Fitch-conjunction, p & ¬Kp, is known. By (A), it follows that each conjunct is known. The third line demonstrates an application of factivity, (B), to the right conjunct of the second line. In the face of the ensuing contradiction, we discharge and deny our only assumption. By necessitation, (C), and then by (D), we conclude with the impossibility of our initial assumption—giving, ¬♦K (p & ¬Kp). Now suppose the knowability principle, ∀p(p → ♦Kp), and take the following instance: (p & ¬Kp) → ♦K (p & ¬Kp). This together with the above ¹ More carefully, Williamson shows this: if necessarily something is true if and only if it is knowable, then necessarily p is possible if and only if p.

Introduction

3

theorem, ¬♦K (p & ¬Kp), entails ¬(p & ¬Kp), which may be generalized to ∀p¬(p & ¬Kp). The classical equivalent is the omniscience principle, ∀p(p → Kp). At a glance:

In sum, if all truths are knowable, then all truths are known: ∀p(p → ♦Kp) ∀p(p → Kp). T h e Ge n e r a l i z e d Pa r a d o x Fitch generalized the knowability result, showing that any operator O that is both factive and closed under conjunction-elimination, generates the following aporia: ∀p(p → ♦Op)

.. .

∃p(p & ¬Op)

⊥ To prove this Fitch begins with Theorem 1, which holds of any factive operator O that is closed under conjunction-elimination: (Theorem 1) ¬♦O(p & ¬Op). Theorem 1 generates the above aporia.² Others have noted that it is not just factive, conjunction-distributive operators that validate Theorem 1 and generate the aporia. Belief, for instance, is closed under conjunction-elimination but is not factive. Yet arguably a belief-instance of Theorem 1 is provable, giving ¬♦B(p & ¬Bp).³

In this way the corresponding aporia is generated for the belief operator: ∀p(p → ♦Bp) ∃p(p & ¬Bp) .. . ⊥ ² To see how, substitute p & ¬Op for p in ∀p(p → ♦Op). By Theorem 1, it follows that ¬(p & ¬Op). This in turn may be may be generalized, giving ∀p¬(p & ¬Op), or equivalently ¬∃p(p & ¬Op). ³ See, for instance, Linsky (1986; and Ch. 11 of this volume).

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That is, the plausible notion that any truth could be believed is inconsistent with the truism that some truths are not ever believed. Such proofs about belief avoid unrestricted factivity principles in favor of restricted principles about the transparency of beliefs about one’s own beliefs. To take another example, a knowledge-version of the result may be derived without the conjunctiondistributivity principle (Williamson 1993). Most generally, then, a Fitch-aporia, or Fitch paradox, is generated for any operator O just when 1. The conjunction p & ¬Op is un-O-able: ∀p¬♦O(p & ¬Op); 2. The O-ability principle, ∀p(p → ♦Op), is plausible; and 3. Clearly, some truths are un-O-ed: ∃p(p & ¬Op). Operators that seem to generate Fitch-aporias include It is written truthfully on the board that Somebody brought it about that God brought it about that The laws of nature made it the case that It is believed that It is thought that So, for instance, the paradox of omnipotence may be seen, logically, as a special case of Fitch’s paradox. It says, roughly, that God can do anything that is in fact done, but only if God does in fact do everything. Another example: any truth can in principle be thought, but only if every truth is (at some time) thought. This latter result, like the result about belief, requires (in lieu of factivity) a principle that avows some minimal transparency of one’s thoughts about one’s thoughts. T h e Vo l u m e , C o n t r i b u t i o n s a n d L i t e r a t u re We here turn to some traditional and developing treatments of the paradox. The earliest version of the knowability proof appears in a 1945 referee report for the Journal of Symbolic Logic (printed here as Chapter 1). Its author, Alonzo Church, anonymously conveyed the proof to Fitch. The proof had the effect of undermining a certain deﬁnition of ‘value’ that Fitch was articulating—a deﬁnition that is trivialized if there are unknowable truths. So the proof originates in a context that is very different from the one in which we discuss the proof today. We think of the knowability paradox today either as an all-too-quick refutation of anti-realism or as a logical collapse of apparently distinct philosophical commitments. More on the more recent debate in a moment. Church offers a number of potentially promising ways to block the proof. He is most sympathetic to a rejection of closure principles for knowledge and belief, and a fortiori

Introduction

5

the principle that knowledge is closed under conjunction-elimination. This is principle (A) in our earlier presentation of the proof. So Church ultimately takes the knowability proof to be invalid—dare I say, paradoxical. However, Church’s proposal does not help Fitch, since Fitch is deeply committed to necessary logical connections between the relevant propositional attitudes. Church considers that one may alternatively appeal to Russell’s theory of logical types, which would have the effect of blocking special instances of the conjunctive distributivity principle—principle (A). The appeal to types foreshadows Linsky (Chapter 11 of this volume) and Hart (Chapter 19 of this volume). However, Church notes that the type-theoretic approach, like the rejection of closure principles, is antithetical to the goals of Fitch’s manuscript. Fitch had a very different kind of reply in mind. He responds to the referee report with a letter to the editor, in which he restricts the relevant class of true propositions to ones that it is ‘empirically possible’ to know. Fitch’s deﬁnition of value is thus resuscitated. His restriction strategy foreshadows Neil Tennant (1997), where we ﬁnd an analogous restriction to the class of true propositions that it is logically possible for somebody to know. We will discuss Tennant’s restriction in a moment. It should be noted here that Church was unimpressed with Fitch’s restriction strategy, and in a subsequent referee report (also in Chapter 1) attempted a version of the knowability proof that respects Fitch’s restriction. The debate between Fitch and Church is tracked in Salerno (Chapter 3). Church’s argument against Fitch’s restriction strategy, I argue, is critically ﬂawed. Part I of the volume is dedicated to this, the early history of the Church–Fitch paradox of knowability. Chapter 1 is the pair of referee reports from 1945. They record one side of a dialog between Fitch and the referee regarding the paper submitted by Fitch to JSL. Chapter 2 is Fitch’s seminal 1963 paper, shaped in no small part by those reports. Fitch’s paper has been the logical fuel or foil for the literature on the knowability paradox. Essay 3 is my understanding of the ﬁrst two essays. It offers an account of why Fitch included the knowability result in the 1963 paper. Part II is about Michael Dummett’s semantic anti-realism. The ﬁrst wave of reactions to Fitch’s 1963 paper, including Hart and McGinn (1976), Hart (1979), Mackie (1980), and Routley (1981), had a common theme. They all aimed to use Fitch’s proof to discredit various forms of veriﬁcationism, the view that all meaningful statements (and so all truths) are knowable.⁴ The knowability principle is commonly taken throughout the literature as a particularly clear expression of Hilary Putnam’s internal realism (1981) and Michael Dummett’s ⁴ An exception is Walton (1976), whose aim was to draw lessons in the philosophy of religion. For related discussion, see Plantinga (1982); Humberstone (1985); MacIntosh (1991); Kvanvig (1995, and 2006); Rea (2000); Wright (2000); Cogburn (2004); Bigelow (2005); Brogaard and Salerno (2005); and Moretti (2008).

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anti-realism (1959b; 1973, and elsewhere). The Fitch paper then threatens these forms of anti-realism. Since Williamson (1982) and Rasmussen and Ravinkilde (1982), however, we ﬁnd various proposals to vindicate at least Dummettian anti-realism. Fitch’s reasoning is classically, but not intuitionistically, valid. Speciﬁcally, the move from ¬(p & ¬Kp) to p → Kp (i.e., the ﬁnal step in our version of the proof ) is intuitionistically unacceptable, since it harbors an application of double-negation elimination—i.e., ¬¬p p. Leading developments in Dummettian anti-realism favor intuitionistic revisions to classical logic.⁵ As such, Dummettian anti-realism is said to evade the unwelcome classical consequences of Fitch. The proposal is further developed in Williamson (1988b; 1990; and 1992). An objection to the intuitionistic strategy is found in the view that the intuitionistic consequences of Fitch’s reasoning are as bad, or almost as bad, as the classical consequences. The objection is developed in Percival (1990).⁶ The main intuitionistic consequence is p → ¬¬Kp, which says that no truths are forever unknown. Some equivalent formulas include ¬(p & ¬Kp), which denies that there are unknown truths, and ¬Kp → ¬p, which says that anything forever unknown is false, and ¬(¬Kp & ¬K ¬p), which denies that there are any forever undecided statements. The potentially irksome consequence, which is a focus of Wright (1993a: 426–7) and Williamson (1994a), can be put this way. The intuitionistic anti-realist lacks the resources to express the apparent truism that there may be truths that never in fact will be known, formally ∃p(p & ¬Kp). That is because the inconsistency derivable from the joint acceptance of the knowability principle and ∃p(p & ¬Kp) is intuitionistically acceptable. In Chapter 4 Dummett embraces the intuitionistic consequences without regret. The paper defends p → ¬¬Kp as the best expression of semantic antirealism.⁷ In a letter to the editor of this volume, Dummett explains that the intuitionistic anti-realist as I conceive of him or her, does not think it irksome that the notion ‘never in fact’ cannot be expressed by the use of the intuitionistic logical constants. Rather he or she thinks that the only meaning that can be given to ‘never’ is that expressible by the intuitionistic logical constants. So there is no worry and no frustration. (Letter: September 27, 2005)

For insightful discussion of the intuitionistic use of ‘never’, see Williamson (1994a). Incidently, Dummett does not endorse the position articulated in his (2001), which proposes a restriction of the knowability principle to ‘basic’ or atomic ⁵ For alternative formulations of the anti-realist argument against classical logic, see Tennant (2000); Salerno (2000); and Wright (2001). ⁶ Important further discussion and a reply appears in DeVidi and Solomon (2001). ⁷ Cf., DeVidi and Solomon (2001), which offers a defense of this very position on behalf of the Dummettian anti-realist. Dummett embraces the truth of p → ¬¬Kp in much earlier work, including (1977: 339 [2000: 236]).

Introduction

7

sentences. Dummett tells me that he wrote that paper to dispel the myth that Fitch’s paradox is an objection to any form of anti-realism. In Chapter 5 Stig Rasmussen further investigates and defends Dummett’s newly favored knowability principle, p → ¬¬Kp. The centerpiece of the discussion is the ‘mapping objection,’ which points out that Gödel’s 1933 mapping of intuitionistic logic into S4 fails to preserve the original formulation of the knowability principle, and that this fact counts against the original formulation as an expression of intuitionistic anti-realism. In Chapter 6 José Bermúdez argues that the Dummett (2001) position is well-motivated. The position restricts the knowability principle to atomic statements, and deﬁnes intuitionistic truth inductively from there. Bermúdez offers an instructive account of Dummett’s development in (1990) and (1996). There Dummett attempts to clarify the notion of indeﬁnite extensibility of such concepts as set, natural number, and real number, and argues that only intuitionistic logic can illuminate a proper understanding of the notion. It is argued that if this is correct, then the Dummett (2001) theory of truth is well-motivated, and so, we have a principled solution to the knowability paradox. Part III is dedicated to paraconsistency and paracompleteness. The paraconsistent approach to the paradox is ﬁrst suggested in Richard Routley (1981). While considering the liar (‘This very statement is not true’), the knower (‘This very statement is not known’) and Fitch’s proposition, ♦K (p & ¬Kp), Routley entertains, but does not endorse, a uniform treatment: What the hardened paraconsistentist says is that [the liar] and ♦K (p & ¬Kp), though inconsistent, are nonetheless coherent, that this is how things are: some (but not too many) inconsistencies hold true. (1981: 112, n. 26)

Routley does not endorse the approach. His actual position is that Fitch’s result is valid and that it indicates a necessary limitation of human knowledge. Fitch’s result shows us that if there is in fact an unknown truth then there is a logically unknowable truth. On the assumption that our actual ignorance is a contingent matter, it is unclear whether the resulting unknowability is contingent or necessary. However, if necessarily we actually fail to know some truths, as Routley argues, then it follows by Fitch’s main argument and the closure of necessity (over necessary implication) that, necessarily, some truths are unknowable. The passing insight about paraconsistency emerges in the context of Routley’s more central discussion of the necessary limits of knowledge. The paraconsistent approach is ﬁrst defended in Beall (2000), where it is argued that the knower sentence provides independent evidence that knowledge is inconsistent. For the concept entails that Kp & ¬Kp, for some p. Further, it is argued that without a solution to the knower we should accept contradictions of this form and go paraconsistent. To this end Wansing (2002) deﬁnes a paraconsistent, constructive, relevant, modal, epistemic logic that evades Fitch.

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The section of this volume on paraconsistency constitutes the most recent developments of the paraconsistent treatment of the problem. In Chapter 7, Graham Priest develops the Routley/Beall proposal by countenancing the mere possibility of truth-value gluts and appealing to a paraconsistent logic with excluded middle. Beall, in Chapter 8, compliments the development by exploring alternatives, some of which avoid the epistemic oddities of Priest’s framework. Beall’s centerpiece is a semantic framework that is paracomplete, but not paraconsistent, and avoids a commitment even to the mere possibility of truth-value gluts. Part IV is an exploration of temporal and epistemic analogs of Fitch’s reasoning. The strategy is to translate the modalities in the knowability principle into a favored temporal or epistemic logic, and to draw lessons from there about the plausibility of the knowability principle and the result in which it ﬁgures. Johan van Benthem (Chapter 9) does this by placing the result in a dynamic epistemic setting—a setting in which the truth values of our epistemic attributions vary over time with the performance of various actions, such as announcements. The essay is a more thorough development of van Benthem (2004). John Burgess (Chapter 10) translates the Fitch modalities into various Priorian temporal modalities. Each of these two approaches offers, not a rejection of Fitch’s proof, but an investigation of the problematic nature of the corresponding knowability principle. Bernard Linsky (Chapter 11) proposes that we block Fitch’s result by appealing to a theory of types in our account of epistemic and doxastic reasoning. Interestingly, this is one of the proposals that Alonzo Church (in Chapter 1) considers when entertaining objections to the knowability proof. Linsky shows that the theory of types systematically treats a wide variety of contemporary paradoxes of knowledge and belief. Part V is dedicated to Neil Tennant’s Cartesian restriction strategy. Tennant’s position is that intuitionistic logic alone will not free anti-realism from the grips of Fitch. His well-discussed proposal is to restrict the knowability principle to Cartesian propositions, that is, propositions that it is not provably inconsistent to know. Objections to the proposal include Hand and Kvanvig (1999), Williamson (2000b), and DeVidi and Kenyon (2003). For replies see Tennant (2000a; and 2000b). Further motivation for Tennant’s proposal can be found in Jon Cogburn (2004) and Igor Douven (2005). In Chapter 12, Williamson continues the debate, speciﬁcally against Tennant (2001a), and renews his pessimism about the prospects for a successful defense of semantic anti-realism (Cf. Williamson 2000b). Debate with Williamson continues in Tennant (forthcoming). Chapter 13 is Kvanvig’s renewed discontent with Tennant’s (and any other) restriction to the knowability principle. As he sees it, the real paradoxicality is not that Fitch’s result threatens anti-realism, but that it threatens to collapse the very distinction between the existence of unknown truth and the existence of unknowable truth. The section is completed

Introduction

9

by Chapter 14, which is Tennant’s current position—a modiﬁcation of the Cartesian restriction strategy. Part VI is about modal and mathematical ﬁctionalism. We learn in Brogaard (Chapter 15) that modal ﬁctionalism is threatened by Fitch’s paradox. Otávio Bueno (Chapter 16) evaluates the relevance of Fitch’s paradox in the epistemology of mathematics. He argues that the mathematical ﬁctionalist must contend with the unwelcome consequences of Fitch. Part VII, Knowability Reconsidered, includes papers that reconsider the antirealist thesis about the knowability of truth. There is a history of attempts to either refute or reformulate anti-realism in reaction of Fitch. I mentioned some refuters earlier. The reformulater is one who rejects, or offers an alternative to, the knowability principle as a characterization of anti-realism. They include Edgington (1985), Melia (1991), Wright (2000), Hand (2003), and Jenkins (2005), among many others. Michael Hand (Chapter 17) further develops his 2003 proposal that Dummett’s anti-realist is not committed to the knowability principle, owing to the fact that it carelessly blurs semantic conditions about veriﬁcation procedures with pragmatic conditions about the performance of such procedures. C. S. Jenkins (Chapter 18) agrees that the knowability principle fails as an expression of anti-realism. Her own statement of anti-realism (2005) is echoed here, but her primary concern is to take issue with Kvanvig (2006), in which it is argued that the real paradoxicality of Fitch’s proof is the modal collapse that occurs in the reasoning from the knowability principle to the omniscience principle. W. D. Hart (Chapter 19) takes Fitch’s proof to be evidence for realism. He argues that the prospects are not good for a solution coming from the theory of types. Christoph Kelp and Duncan Pritchard (Chapter 20) offer some hope for an anti-realism that endorses a justiﬁed believability principle in place of the knowability principle. They evaluate the thesis that, for all true propositions, it must be possible to justiﬁably believe them. An alternative weakening of the knowability principle is proposed by Greg Restall (Chapter 21). His principle states that, for every truth p, there is a collection of truths, such that (i) each of them is knowable and (ii) their conjunction is equivalent to p. Restall proves that this formulation evades the paradox, and draws lessons about the operant notion of possibility. I regret that the volume is incomplete. It includes no extensive discussion of Dorothy Edgington’s important 1985 proposal, in which the knowability principle is reformatted as a thesis about the knowability of actual truth. Important criticisms are found in Wright (1987a (2nd edn., 1993: 428–32)), Williamson (1987a; 1987b; 2000a, ch. 12), and Percival (1991). Developments of Edgington’s proposal are found in Rabinowicz and Segerberg (1994), Linström (1997), Rückert (2004), Fara (forthcoming), and Murzi (manuscript). Related proposals, that focus on the modal semantics of Fitch’s paradox, include Kvanvig (1995; 2006), Brogaard and Salerno (2006) and Costa-Leite (2006). These latter three approaches diagnose various modal fallacies. Kvanvig appeals to issues about

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when we are licensed to substitute into modal contexts. Brogaard and Salerno appeal to Stanley and Szabo’s (2000) theory of quantiﬁer domain restriction, according to which there is hidden structure in quantiﬁed noun phrases. CostaLeite appeals to the fusion of Kripke frames—the insight being that knowability is not to be understood compositionally out of one-dimensional possibility and knowledge operators.

Pa r t I E a r l y Hi s t o r y

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1 Referee Reports on Fitch’s ‘‘A Deﬁnition of Value’’ Alonzo Church

The referee reports transcribed below were handwritten by Alonzo Church to his co-editor, Ernest Nagel, of the Journal of Symbolic Logic. They were issued in 1945 in response to a paper by Frederic Fitch, ‘‘A Deﬁnition of Value,’’ which was not published. They contain the earliest formulations of the modal epistemic result today known as ‘‘Fitch’s knowability paradox.’’ The bracketed numerals, [n.], indicate the original page numbers. Our appendix is a list of the cited principles. They either originate in Lewis and Langford (1932) or are hypothesized by us to be the principles from Fitch’s original manuscript, which was not found. The original reports are located in the Ernest Nagel Papers, Box 1, Arranged Correspondence, Church, Alonzo. Rare Book and Manuscript Library, Columbia University. They are printed here in full by their permission and by kind permission of Alonzo Church, Jr. We are grateful to Nick Zavediuk for his assistance in the transcription process.

Fi r s t Re f e re e Re p o r t [1.] It seems to me that the role and meaning of Professor Fitch’s ‘SC’ is seriously in need of clariﬁcation. It is not sufﬁcient merely to take ‘SC’ as primitive and undeﬁned. It must be contemplated that ultimately there is either a deﬁnition of ‘SC’ or an elaborate set of empirical postulates about it; otherwise particular empirical necessitations such as ‘‘brakeless trains are dangerous’’ could not be decided. Perhaps Fitch means to say that there is one absolutely determined set of ‘‘all the valid laws of empirical sciences,’’ such that the currently accepted laws of the currently known empirical sciences are an approximation to a certain subset of this set, and that SC is something like the conjunction of all the laws of this set. But this belief in an ultimate set of absolutely valid empirical laws is held by hardly any contemporary empirical scientist. And I think that the recent history of the empirical sciences, especially physics, renders such a belief indefensible. The only alternative I see is to take a particular formalized system of empirical science (say a system which uniﬁes the empirical sciences as they are known today),

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and to take SC to be the conjunction of the primitive propositions of this system. This is acceptable, but it must be realized that it makes empirical necessitation relative to a particular system. Accepting this emendation, I go on to Fitch’s Def. 3. [2.] Following the line of Fitch’s thought, let me call a proposition empirically impossible if SC strictly implies its negation. (This makes empirical impossibility equivalent to the negation of empirical possibility and is therefore consistent with Fitch’s Def. 6.) Then it may plausibly be maintained that if a is not omniscient there is always a true proposition which it is empirically impossible for a to know at time t. For let k be a true proposition which is unknown to a at time t, and let k be the proposition that k is true but unknown to a at time t. Then k is true. But it would seem that if a knows k at time t, then a must know k at time t, and must also know that he does not know k at time t. By Def. 2, this is a contradiction. Now an empirically impossible proposition empirically necessitates every proposition. Therefore, the argument runs, by taking q in Def. 3 to be k , it may be inferred that everything is of value to a at time t. Thus Def. 3 is reduced to a triviality. In spite of the plausibility of the preceding argument I think Fitch has a good defense (but only one). This defense is that there is no law of psychology according to which one who believes a proposition must believe all its logical consequences; on the contrary, historical counter-examples are well known. To be sure, one who believes a proposition without believing its more obvious consequences is a fool; but it is an empirical fact that there are fools. It is even possible [3.] that there might be so great a fool as to believe the conjunction of two propositions without believing either of the two propositions; at least, an empirical law to the contrary would seem to be open to doubt. On this ground it is empirically possible that a might believe k at time t without believing k at time t (although k is a conjunction one of whose terms is k). Unfortunately this defense compels Fitch to abandon his Ax. 1. And, what is more serious, it lights the way to a second and opposite objection to Def. 3. If there is no empirical law according to which one who believes a proposition must believe its logical consequences, it would seem that by the same token there is no empirical law according to which a person’s desires must be in reasonable accord with that person’s beliefs. If someone desiring to recover from a certain disease, and knowing the one and only course of action which will lead to recovery, nevertheless does not desire that course of action, we may call that someone a fool; but again the fact is that fools there be. It is a historical fact that there have been persons who desired to avoid smallpox, who knew the medical efﬁcacy of vaccination as a preventive, and who nevertheless violently resisted vaccination (therefore presumably did not desire it). I conclude [4.] that there is no valid law of psychology according to which anything whatsoever about my desires may be inferred from the fact that I know so-and-so. It follows by Def. 3 that nothing

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15

is of value to a at time t, and again Def. 3 is reduced to a triviality. This is my second objection to Def. 3, and the one to which I attach the greater weight. Since Def. 3 is the core of the paper, I hold that the entire paper is therefore untenable. By this it is not to be understood that I disapprove of the idea of applying symbolic logic to value theory. However severe my criticism of Fitch for what I hold to be logical ﬂaws, my criticism is still more severe of those philosophers who offer similar deﬁnitions of value in vague verbal form without the attempt at even so much accuracy as may be attained by the use of certain elementary notations of symbolic logic. For these latter escape the kind of criticism I level against Fitch only by making their statement so vague as to render all criticism uncertain. The very fact that an attempt by Fitch to state formally what I take to be a rather ordinary sort of deﬁnition of value leads to these logical difﬁculties is an indication of the need for at least some elementary symbolic logic here. Let me say also that, in order to meet Fitch on his own ground, I have accepted uncritically what seems to be his notion of [5.] proposition, although it is well known that the notion of proposition is uncertain and in need of clariﬁcation. I am willing to concede, at least as a possibility, that one way to obtain clariﬁcation of the notion is to plunge directly into the use of propositions and to clear up individual difﬁculties as they arise. Finally, I note that Fitch makes a medical error on page 4 of the manuscript, in implying that quinine is the only cure for malaria. An entirely different drug, atabrine, is as a matter of fact also an effective cure. It seems to me that according to ordinary usage it would be said that quinine is valuable to the malaria sufferer, even if there does exist an alternative method of cure by means of atabrine. This observation may reveal another deﬁciency in Def. 3. But in view of more serious objections it seems unnecessary to go into this.

Se c o n d Re f e re e Re p o r t 1. It is not clear to me why Fitch thinks that, to quote his letter, ‘‘In order to show that a’s ignorance of k is empirically necessary, he would ﬁrst have to show that a’s ignorance of k is empirically necessary.’’ The fact is that the quoted statement is false. To enforce my point, let me put the matter quite formally: Assume: k. Assume also: ∼(a KNt k). Def.: k = (k & ∼ (a KNt k) ). By the foregoing Def., and Fitch’s Th. 3: (a KNt k ) EN (a KNt k). By Fitch’s Def. 2, and Lewis-Langford 11.2: (a KNt k ) SI k .

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Hence by the Def. of k , and Lewis-Langford 11.2, 11.6: (a KNt k ) SI ∼(a KNt k). Hence by Lewis-Langford 12.42: (a KNt k) SI ∼(a KNt k ). Hence by Fitch’s Th. 1: (a KNt k) EN ∼(a KNt k ). The transitive law for EN follows from Lewis-Langford 15.1, 16.2, 11.6. Hence from the foregoing step and step 4 above we get: (a KNt k ) EN ∼(a KNt k ). Hence by the law of double negation and Fitch’s Def. 5: ∼((a KNt k ) EC (a KNt k )). Hence by Fitch’s Def. 6: ∼EP(a KNt k ). However, it follows from our assumptions: k . From this now it follows (for the reason explained in my previous report, and as I [2.] understand Fitch to admit) that Fitch’s Def. 3 is untenable as it now stands, and must be altered if the paper as a whole is to be maintained. Now let us consider the revised form of Def. 3 which Fitch proposed in his letter of February 26 and afterwards abandoned. As I understand it this is: Def. 3R. (a VLt p) = (Eq) [q & EP(a KNt q) & [(a KNt q) EN (a DSt p)]]. I think that a reductio ad absurdum of Def. 3R is possible along the same lines as that I have given for Def. 3. At least, let me attempt it, and leave it to Fitch to say where if anywhere I have assumed something he would not admit. I shall show as a consequence of Def. 3R that instant death is of value to a at time t. In other words, if p is ‘‘a dies at time t,’’ I shall show that a VLt p . Assume that a does not desire instant death at time t (because in the contrary case, if we assume that it is empirically possible for one to know one’s own desires, the conclusion a VLt p is obvious). Nevertheless it is empirically possible for a to desire instant death at time t, both (1) because it is empirically possible that a should be insane, and (2) it is empirically possible that a’s external circumstances [3.] at time t might have been so dreadful as to compel even a sane man to desire instant death. Assume also that a is not omniscient, and let k be something which is true but unknown to a at time t. Deﬁne k as before. Then as before k is true but a’s ignorance of k is empirically necessary. Let q be the disjunction (a DSt p ) ∨ k . Then q is true. I suppose Fitch would admit (a KNt p) EN (a KNt (p ∨ q)). At least this seems to be entirely in the spirit of his Th. 3, and it is hard to see how he could maintain one and deny the other. Also I suppose that logical consequences of the empirically possible are empirically possible, and that it is empirically possible for one to know one’s own desires.

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17

Thus because a DSt p is empirically possible, therefore a KNt (a DSt p ) is empirically possible, and therefore a KNt q is empirically possible. Because (a DSt p ) EN (a DSt p ) and k EN (a DSt p ), therefore q EN (a DSt p ). Hence because (a KNt q ) EN q , it follows that (a KNt q ) EN (a DSt p ). Finally, taking, in Def. 3R, p to be p , and q to be q , we get the conclusion that a VLt p . The case of instant death is of course [4.] chosen only for illustration. In general, under Def. 3R, everything is of value to a at time t which it would be empirically possible for a to desire at time t under any empirically possible circumstances (however remote from the actual circumstances). This is little if any less disastrous than the situation under Def. 3, that everything whatever is of value to a at time t. Of course the foregoing refutation of Fitch’s deﬁnition of value is strongly suggestive of the paradox of the liar and other epistemological paradoxes. It may therefore be that Fitch can meet this particular objection by incorporating into the system of his paper one of the standard devices for avoiding the epistemological paradoxes. If this is possible it will involve a drastic rewriting of the paper, not just a footnote here and there. To my further objection—that there is no law of psychology according to which it can be inferred from the fact that a knows something that therefore a desires something—Fitch replies by pointing out that a might know that a desires p. If, however, Fitch consents to adopt one of the standard devices for avoiding the epistemological paradoxes, this reply will no longer be open to him. For example, on the basis of Russell’s original [5.] theory of types, ‘‘a desires p’’ is of higher order than p, whereas the two ‘‘something’’ ’s in my assertion must of course be understood as of the same order. On the basis of Tarski’s resolution of the epistemological paradoxes, the distinction between language and meta-language has roughly the same effect. I insist therefore that there is no known law of psychology according to which ‘‘a desires p’’ is ever a necessary consequence of ‘‘a knows q.’’ Moreover, in the light of every-day experience (summed up in the commonly heard conclusion, ‘‘Some people are utterly unreasonable’’), it seems unlikely that there is a valid law of psychology of that sort remaining to be discovered. If some of us think that there is a notion of value in spite of the fact that some people are utterly unreasonable, it is because we think we know how to distinguish between what is reasonable and what is unreasonable. The problem is whether this distinction between reasonable and unreasonable can be deﬁned in non-valuational terms, or whether this or some like value-theoretic concept must be accepted as primitive (undeﬁned). I do not think that Fitch has solved the problem. The assumption that there is an absolute set of valid empirical laws, SC, to which the accepted laws of the empirical sciences are [6.] approximations in some sense, is of course a piece of metaphysics. I have no objection to metaphysics per se.

18

Alonzo Church

But this particular piece of metaphysics has more opponents than adherents, and it seems that a deﬁnition of value which presupposed it would be of interest only to a very restricted circle. I do not understand why Fitch objects to avoiding the metaphysical issue by making his deﬁnition of value relative to a particular (comprehensive) system of empirical science. But this is a side-issue, in view of the existence of more serious objections. As to the matter of quinine and atabrine: it seems that, according to Fitch, if a is a malaria sufferer who has equal access to quinine and atabrine, then both ‘‘quinine is of value to a’’ and ‘‘atabrine is of value to a’’ are false. Moreover it may be that there is some drug which is a quicker and more certain cure for malaria than either quinine or atabrine, which is easily accessible to a (if he only knew), but whose properties in this respect are still undiscovered. If so, then not even the disjunction, quinine or atabrine, is valuable to a. It seems to me that such a notion of value departs so far from the everyday notion that it is hardly justiﬁed to use the same word for it. Finally, Fitch’s plan of adding [6.] short postscripts to his paper commenting on particular objections by the referee does not seem to me a good one. So far as the objections either are valid or represent misunderstandings likely to be duplicated by others, they should be met (if that is possible) by alterations in the body of the paper.

Appendix Joe Salerno and Julien Murzi Fi t c h’s Op e r a t o r s

p SI q = p ≺ q = p strictly implies q p EN q = p empirically necessitates q EPp = p is empirically possible p EC q = p is empirically consistent with q aKNtp = a knows at time t that p aBtp = a believes at t that p aVLtp = a values at t that p aDStp = a desires at t that p L e w i s a n d L a n g f o rd ( 1 9 3 2 ) Deﬁ n i t i o n a n d T h e o re m s

(p ≺ q) =df ∼ ♦(p & ∼q) 11.2 (p & q) ≺ p

Referee Reports on Fitch’s ‘‘A Deﬁnition of Value’’

19

11.6 ((p ≺ q) ≺ (q ≺ r)) ≺ (p ≺ r) 12.42 (p ≺ ∼q) ≺ (q ≺ ∼p) 15.1 ((p ⊃ q) & (q ⊃ r)) ≺ (p ⊃ r) 16.2 ((p ≺ q) & (p ≺ r) & T) ≺ (p ≺ (q & r)): T = ((q & r) ≺ (r & q)) Fi t c h’s De ﬁ n i t i o n s¹

Def. 2∗ Def. 2 is Fitch’s deﬁnition of knowledge. All we know from Church’s use is that it justiﬁes the principle that a’s knowing at time t that p strictly implies p: aKNtp ≺ p.² Def. 3∗ aVLt p =df ∃q(q & (aKNtq EN aDStp) ). Value is what one would desire given sufﬁcient knowledge: it is valuable to a at t that p if and only if there is a true proposition q, such that a’s knowing at t that q empirically necessitates a’s desiring at t that p.³ Def. 3R aVLt p =df ∃q(q & EP(aKNtq) & (aKNtq EN aDStp) ). Value is what one would desire given sufﬁcient knowledge: it is valuable to a at t that p if and only if there is a truth q that it is empirically possible to know and a’s knowing at t that q empirically necessitates a’s desiring at t that p.⁴ Def. 5∗ (p EN ∼ p) ≺ ∼ (p EC p). Necessarily, if p empirically necessitates ∼p, then p is not (empirically) consistent with itself.⁵ Def. 6∗ ∼ (p EC p) =df ∼ EPp. p is not (empirically) consistent with itself just in case p is not empirically possible.⁶ ¹ An asterisk, ‘∗ ’, indicates that the principle does not appear explicitly in the reports, and therefore, that we have hypothesized its content. ² Church’s applications appear in Report 1: 2 and Report 2: 1. ³ Our formulation of Def. 3 is based on Church’s trivialization argument against it. Compare Report 1: 2 and Report 2: 1–2. ⁴ Report 2: 2. ⁵ Report 2: 1. ⁶ Report 1: 2, and Report 2: 1.

20

Alonzo Church Fi tc h’s Ax i o m s a n d T heo rem s

Ax. 1∗ (aBtp & (p EN q) ) ≺ aBtq Belief is closed under ‘‘empirically necessary’’ implication: necessarily, if a believes at t that p and p empirically necessitates q, then a believes at t that q.⁷ Th. 1∗ (p ≺ q) ≺ (p EN q) Strict implication strictly implies empirical necessitation: necessarily, if p strictly implies q then p empirically necessitates q.⁸ Th. 3∗ aKNt (p & q) ≺ (aKNtp & aKNtq) Knowing a conjunction strictly implies knowing the conjuncts: necessarily, if a knows at t that both p and q, then a knows at t that p and a knows at t that q.⁹ ⁷ The discussion at Report 1: 2–3 suggests that Ax. 1 is this closure principle for belief. Alternatively, it is an unrestricted closure principle for knowledge (viz., knowledge is closed under necessary empirical implication). ⁸ See for instance, Church’s use in Report 2: 1. ⁹ Report 2: 1.

2 A Logical Analysis of Some Value Concepts¹ Frederic B. Fitch

The purpose of this paper is to provide a partial logical analysis of a few concepts that may be classiﬁed as value concepts or as concepts that are closely related to value concepts. Among the concepts that will be considered are striving for, doing, believing, knowing, desiring, ability to do, obligation to do, and value for. Familiarity will be assumed with the concepts of logical necessity, logical possibility, and strict implication as formalized in standard systems of modal logic (such as S4), and with the concepts of obligation and permission as formalized in systems of deontic logic.² It will also be assumed that quantiﬁers over propositions have been included in extensions of these systems.³ There is no intention to provide exhaustive logical analyses, or to provide logical analyses that reﬂect in detail the usage of so-called ordinary language. This latter task seems impossible anyhow because of the ambiguities of ordinary language and the obvious inconsistencies and irregularities of usage in ordinary language. Furthermore, the term ‘ordinary language’ is itself rather vague. Whose ordinary language? Should English be preferred to Chinese? Various arguments that invoke English or Latin grammatical usage are seen to be without foundation from the standpoint of Chinese. Just as the concepts of necessity and possibility used in so-called ordinary language correspond in some degree to the concepts of necessity and possibility Chapter 2 was ﬁrst published in the Journal of Symbolic Logic 28/2, 135–42 (1963) and is reproduced by permission of the Association for Symbolic Logic. ¹ An earlier draft of this paper was presented as a retiring presidential address to the Association for Symbolic Logic; read before the Association at Atlantic City, New Jersey, December 27, 1961. ² For example see A. R. Anderson, The formal analysis of normative systems, Technical Report No. 2, Contract No. SAR/Nonr-609(16), Ofﬁce of Naval Research, Group Psychology Branch, 1956; also, by the same author, A reduction of deontic logic to alethic modal logic, Mind , n.s. vol. 67 (1958), pp. 100–3. ³ Such quantiﬁers can be introduced by methods analogous to those used in R. C. Barcan (Marcus), A functional calculus of ﬁrst order based on strict implication, this Journal, vol. 11 (1946), pp. 1–16; The deduction theorem in a functional calculus of ﬁrst order based on strict implication, ibid., pp. 115–18; and F. B. Fitch, Intuitionistic modal logic with quantiﬁers, Portugaliae mathematica, vol. 7 (1948), pp. 113–18. See also, R. Carnap, Modalities and quantiﬁcation, this Journal, vol. 11 (1946), pp. 33–64.

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Frederic B. Fitch

used in modal logic, so too it is to be hoped that the ordinary language concepts of striving, doing, believing, desiring and knowing will correspond in some degree to the concepts that we will partially formalize here. Also, just as there are various slightly differing concepts of possibility and necessity corresponding to differing systems of modal logic, so too there are presumably various slightly differing concepts of striving, doing, believing, and knowing, having differing formalizations. We begin by assuming that striving, doing, believing, and knowing all have at least some fairly simple properties which will be described in what follows, and we leave open the question as to what further properties they have. First of all, we assume that striving, doing, believing, and knowing are twotermed relations between an agent and a possible state of affairs. It is convenient to treat these possible states of affairs as propositions, so if I say that a strives for p, where p is a proposition, I mean that a strives to bring about or realize the (possible) state of affairs expressed by the proposition p. Similarly, if I say that a does p, where p is a proposition, I mean that a brings about the (possible) state of affairs expressed by the proposition p. We do not even have to restrict ourselves to possible states of affairs, because impossible states of affairs can be expressed by propositions just as well as can possible states of affairs. In the case of believing and knowing, there is surely no serious difﬁculty in regarding propositions as the things believed and known. So we treat all these concepts as two-termed relations between an agent and a proposition. In a similar way, the concept of proving could also be regarded as a two-termed relation between an agent and a proposition. For purposes of simpliﬁcation, the element of time will be ignored in dealing with these various concepts. A more detailed treatment would require that time be taken seriously. One method would be to treat these concepts as a three-termed relation between an agent, a proposition, and a time. Another method would be to avoid specifying times explicitly, but rather to use a temporal ordering relation between states of affairs. This latter method might be more in keeping with the theory of relativity, in either its special or general form. As a further step of simpliﬁcation we will often ignore the agent and thus treat each of the concepts under consideration as a class of propositions rather than as a two-termed relation. For example, by ‘striving’ we will mean the class of propositions striven for (that is, striven to be realized), and by ‘believing’ we will mean the class of propositions believed, relativizing the whole treatment to some unspeciﬁed agent. But the agent can always be speciﬁed if we wish to do so, and we can replace classes by two-termed relations. A class of propositions (in particular such classes of propositions as striving, knowing, etc.) will be said to be closed with respect to conjunction elimination if (necessarily) whenever the conjunction of two propositions is in the class so are the two propositions themselves. For example, the class of true propositions is closed with respect to conjunction elimination because (necessarily) if the

A Logical Analysis of Some Value Concepts

23

conjunction of two propositions is true, so are the propositions themselves. If α is a class closed with respect to conjunction elimination, this fact about α can be expressed in logical symbolism by the formula, (p) (q) [ (α[p & q] ) [ (αp) & (αq) ] ] , where ‘’ stands for strict implication. We assume that the following concepts, viewed as classes of propositions, are closed with respect to conjunction elimination: striving (for), doing, believing, knowing, proving.

For example, in the case of believing we assume: (p) (q) [ (believes[p & q] ) [ (believes p) & (believes q) ] ]. Here are some further concepts which are evidently closed with respect to conjunction elimination: truth, causal necessity (in the sense of Burks),⁴ causal possibility (in the sense of Burks), (logical) necessity, (logical) possibility, obligation (deontic necessity), permission (deontic possibility), desire for.

A class of propositions will be said to be closed with respect to conjunction introduction if (necessarily) whenever two propositions are in the class, so is the conjunction of the two propositions. If α is a class closed with respect to conjunction introduction, this fact about α can be expressed in logical symbolism by the formula, (p) (q) [ [ (αp) & (αq) ] (α[p & q] ) ]. Except for causal, logical, and deontic possibility, all the concepts so far regarded as closed with respect to conjunction elimination could perhaps also be regarded as closed with respect to conjunction introduction, or some varieties of them could. For present purposes, however, we do not need to commit ourselves on this matter except to say that truth and causal, logical, and deontic necessity are all indeed closed with respect to conjunction introduction. A class of propositions will be said to be a truth class if (necessarily) every member of it is true. If α is a truth class, this fact about α can be expressed ⁴ A. W. Burks, The logic of causal propositions, Mind , n.s. vol. 60 (1951), pp. 363–82.

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in logical symbolism by the formula, (p)[(αp) p]. The concepts truth, causal necessity, and logical necessity are clearly truth classes. It also seems reasonable to assume that doing, knowing, and proving are truth classes, and so we make this assumption. Thus, whatever is true or causally or logically necessary is true; and (as we assume) whatever is done, known, or proved is also true. The following two theorems about truth classes will be applied to some of the above-mentioned truth classes in subsequent theorems. Theorem 1. If α is a truth class which is closed with respect to conjunction elimination, then the proposition, [p & ∼(αp)], which asserts that p is true but not a member of α (where p is any proposition), is itself necessarily not a member of α. Proof. Suppose, on the contrary, that [p & ∼(αp)] is a member of α; that is, suppose (α[p & ∼(αp)] ). Since α is closed with respect to conjunction elimination, the propositions p and ∼(αp) must accordingly both be members of α, so that the propositions (αp) and (α(∼ (αp) ) ) must both be true. But from the fact that α is a truth class and has ∼(αp) as a member, we conclude that ∼(αp) is true, and this contradicts the result that (αp) is true. Thus from the assumption that [p & ∼(αp)] is a member of α we have derived contradictory results. Hence that assumption is necessarily false. Theorem 2. If α is a truth class which is closed with respect to conjunction elimination, and if p is any true proposition which is not a member of α, then the proposition, [p & ∼(αp)], is a true proposition which is necessarily not a member of α. Proof. The proposition [p & ∼(αp)] is clearly true, and by Theorem 1 it is necessarily not a member of α. Theorem 3. If an agent is all-powerful in the sense that for each situation that is the case, it is logically possible that that situation was brought about by that agent, then whatever is the case was brought about (done) by that agent. Proof. Suppose that p is the case but was not brought about by the agent in question. Then, since doing is a truth class closed with respect to conjunction elimination, we conclude from Theorem 2 that there is some actual situation which could not have been brought about by that agent, and hence that the agent is not all-powerful in the sense described. Theorem 4. For each agent who is not omniscient, there is a true proposition which that agent cannot know.⁵ Proof. Suppose that p is true but not known by the agent. Then, since knowing is a truth class closed with respect to conjunction elimination, we conclude from Theorem 2 that there is some true proposition which cannot be known by the agent. ⁵ This theorem is essentially due to an anonymous referee of an earlier paper, in 1945, that I did not publish. This earlier paper contained some of the ideas of the present paper.

A Logical Analysis of Some Value Concepts

25

Theorem 5. If there is some true proposition which nobody knows (or has known or will know) to be true, then there is a true proposition which nobody can know to be true. Proof. Similar to proof of Theorem 4. Theorem 6. If there is some true proposition about proving that nobody has ever proved or ever will prove, then there is some true proposition about proving that nobody can prove. Proof. Similar to the proof of Theorem 4, using the fact that if p is a proposition about proving, so is [p & ∼(αp)]. This same sort of argument also applies to the class of logically necessary propositions, since this is a truth class closed with respect to conjunction elimination. Thus by Theorem 1 we have the result that every proposition of the form [p& ∼ p] is necessarily not logically necessary, and hence necessarily possibly false, where ‘’ denotes logical necessity. In other words, the proposition ∼ [p & ∼ p] is true for every proposition p.⁶ In particular, if p is a true proposition which is not necessarily true, then [p & ∼ p] is a true proposition which is necessarily possibly false. I now wish to describe a relation of causation, or more accurately, partial causation, which will be used in giving a deﬁnition of doing in terms of striving and a deﬁnition of knowing in terms of believing, as well as some other deﬁnitions. I will assume that partial causation, expressed by ‘C’, satisﬁes the following axiom schemata C1–C4: C1. C2. C3. C4.

[[p C q] & [q C r]] [p C r]. [p & [p C q]] q. [p & [[p & q] C r]] [q C r]. [[p C q] & [p C r]] ≡ [p C [q & r]].

(transitivity) (detachment) (strengthening) (distribution)

Here ‘p ≡ q’ is deﬁned as ‘[p q] & [q p]’. I will also employ an identity relation among propositions and will employ the following axiom schemata I1–I9 for this identity relation:⁷ I1. [[p = q] & (. . . p . . .)] (. . . q . . .). I2. p = p. ⁶ This result in slightly different form is to be found in the two papers by Anderson cited above. He uses it in constructing a model of deontic logic in alethic modal logic and attributes it to W. T. Parry, Modalities in the survey system of strict implication, this Journal, vol. 4 (1939), pp. 137–54, Theorem 22.8. ⁷ It is interesting to observe that I2–I9 may be used to serve as postulates for an algebra like Boolean algebra but somewhat weaker, provided that the identity symbol is regarded as a symbol for equality in such an algebra and that (in place of I1) there are added postulates to the effect that equality is symmetrical and transitive, and that the negates, conjuncts, and disjuncts of equal elements of the algebra are equal. Also, there should be a postulate to the effect that there are at

26 I3. I4. I5. I6. I7. I8. I9.

Frederic B. Fitch p = ∼∼ p. p = [p & p]. [p & q] = [q & p]. [p & [q & r ] ] = [ [p & q] & r ]. [p & [q ∨ r ] ] = [ [p & q] ∨ [p & r ] ]. p = [ [p & q] ∨ p]. [∼ p & ∼ q] = ∼ [p ∨ q].

Notice that we do not have such theorems as p = [p & [q ∨ ∼ q]] and p = [p ∨ [q & ∼ q]]. Only a few of the axiom schemata listed above will be directly relevant in what follows. The ones most relevant are C2, C4, I1, and I6. The property expressed by C3 reﬂects the fact that C is only partial causation. If C were total causation, then C3 would clearly be unacceptable. It should also be remarked that C need not be regarded as restricted to relating states of affairs that have space-time location, but may relate any state of affairs (e.g., a mathematical truth) to other suitable states of affairs. Otherwise, the sort of knowledge deﬁned below would be knowledge only of states of affairs that have space-time location. Using the relation C, a deﬁnition of doing in terms of striving will now be given. It is perhaps best to regard this deﬁnition merely as an axiom schema that provides a necessary and sufﬁcient condition for doing, and similarly in subsequent deﬁnitions. As before, reference to the agent and to time are omitted for simplicity. D1. (does p) ≡ ∃q[(strives for [p & q]) & [(strives for [p & q]) C p]]. This means that an agent does p if and only if there is some (possible or impossible) situation q such that the agent strives for p and q, and a result of this striving is that p takes place. Using I1, I6, C4, and properties of existence quantiﬁcation, it is easy to show that this deﬁnition gives the result that doing is closed with respect to conjunction elimination. A deﬁnition of knowing in terms of believing is now given: least two unequal elements of the algebra. Such an algebra provides an algebraic formulation for the Anderson–Belnap system of ﬁrst degree entailments with quantiﬁers omitted (A. R. Anderson and N. D. Belnap, Jr., First degree entailments, Technical Report No. 10, ibid., 1961, since the assertion that p entails q can be deﬁned as the assertion that p equals the conjunction of p with q, or equivalently as the assertion that q equals the disjunction of q with p. This algebra was suggested to me by a list of theorems on page 21 of my paper, A system of combinatory logic, Technical Report No. 9, ibid., 1960, and in part also by some discussions with Anderson. It also bears a close relation to the system of my paper, The system C of combinatory logic, Technical Report No. 13, ibid., 1962 (also forthcoming in this Journal). The system of ﬁrst degree entailment including quantiﬁers was also arrived at independently by Miss Patricia A. James and myself as a modiﬁed form of the system of my book Symbolic logic (New York, 1952) prior to the Anderson–Belnap formulation of that system. This alternative approach to the system of ﬁrst degree entailment is sketched on p. vii of Miss James’s doctoral dissertation, Decidability in the logic of subordinate proofs (Yale University, 1962).

A Logical Analysis of Some Value Concepts

27

D2. (knows p) ≡ ∃q[p & q & [[p & q] C (believes [p & q])]]. This means that an agent will be said to know p provided that p and some (possibly other) situation q are both true, and provided that the fact that they are both true causes the agent to believe the fact that they are both true. Thus the known fact p must be causally efﬁcacious (as part of the conjunction [p & q]) in bringing about the agent’s belief that [p & q] is the case, and hence that p itself is the case, since belief is assumed closed with respect to conjunction elimination. It is easy to show that knowing, as thus deﬁned, is a truth class closed with respect to conjunction elimination. Ability to do can be deﬁned in the following way: D3. (can do p) ≡ ∃q[ (strives for[p & q] )Cp]. This deﬁnition can be shown to give the result that ability to do is closed with respect to conjunction elimination. Obligation to do can be deﬁned in terms of doing and the concept of obligation as expressed by the operator ‘0’ of deontic logic, as follows: D4. (should do p) ≡ 0 (does p). Obligation to do, as thus deﬁned, can be shown to be closed with respect to conjunction elimination and also with respect to conjunction introduction. I now wish to propose a deﬁnition of desire, as follows: D5. (desires p) ≡ ∃q[ (believes(can do[p & q] ) )C(strives for[p & q] ) ]. This means that an agent desires a situation p if his belief that he can achieve the conjunction of p with some (possibly other) situation causes him to strive for that conjunction of situations. Desire as thus deﬁned can be shown to be closed with respect to conjunction elimination. A concept of value, which I now wish to consider, can be deﬁned in the following way: D6. (value p) ≡ ∃q∃r [q & [ (knows q)C (strives for [p &r ] ) ] ]. This means that a situation p is a value for an agent if (and only if) there is an actual situation q and situation r such that if the agent knows q then he will strive for the conjunction of p and r. In knowing q the agent may be supposed to have all the knowable relevant information concerned with the effect of his striving for the conjunction of p and r, and if this knowledge causes him to strive for this conjunction, it must be because this conjunction, and in particular p itself, is of value to him. To see why q may be supposed to contain all the knowable relevant information for the purpose at hand, let us suppose, on the contrary, that q does not contain all such relevant information. Then there might be some additional information s such that knowledge of the conjunction of q and s would cause the agent not to strive for any conjunction of the form

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Frederic B. Fitch

[p & t]. But in the hypothetical case that the agent knew [q & s], he would also know q because of the fact that knowing is closed with respect to conjunction elimination, and this knowledge of q, by assumption, would cause him to strive for [p & r]. Thus he would be caused to strive for [p & r] and also caused not to strive for [p & r], and the assumption that he could know such a proposition as [q & s] leads to an absurdity. Hence q may be regarded as containing all the knowable relevant information. It can be shown easily that value as thus deﬁned is closed with respect to conjunction elimination. The objection might be raised against the above deﬁnition of value that the agent must be assumed to be rational, since otherwise he might have all the relevant knowledge to enable him to make a choice in his own interest, and yet, being irrational, he would be caused by this knowledge to make some other choice and to strive for some outcome that would not be of value to him. One way, and perhaps the only way, to attempt to meet this objection is to maintain that all irrationality is due to lack of sufﬁcient knowledge, so that the having of sufﬁcient relevant knowledge already rules out any relevant amount of irrationality. According to this view, any sort of insanity would be curable simply by giving the patient sufﬁcient knowledge of himself and of the world around him. This view would not deny that in practice there might be insuperable obstacles that prevent the communication of this knowledge to the patient, but the existence of such obstacles would not prove that irrationality was not essentially a lack of knowledge. This deﬁnition of value of course does not guarantee that there are any values in this sense, though it seems to me not unreasonable to assume that there may be values in this sense. A more difﬁcult problem is the problem of the comparison of values, that is, the problem of greater and less among values. This problem will not be dealt with here. YALE UNIVERSITY

3 Knowability Noir: 1945–1963 Joe Salerno

The literature on the knowability paradox emerges in response to a modal epistemic proof ﬁrst published by Frederic Fitch in his famous 1963 paper, ‘‘A Logical Analysis of Some Value Concepts.’’ Theorem 5, as it was there called, threatens to collapse a number of modal and epistemic differences. Let ignorance be the failure to know some truth. Then Theorem 5 collapses a commitment to fortuitous ignorance into a commitment to necessary ignorance. For it shows that the existence of truths in fact unknown entails the existence of truths necessarily unknown. The converse of Theorem 5 is trivial (if truth entails possibility), so Fitch goes most of the way toward erasing any logical difference between the existence of fortuitous ignorance and the existence of necessary unknowability. More exactly, it is the contrapositive of Theorem 5 that is today referred to as the knowability paradox. The contrapositive tells us that any truth can be known but only if every truth is in fact known. As such it collapses sophisticated anti-realism into naive idealism—a philosophical difference we may wish to preserve even if we are not sympathetic to anti-realism. Further, and with slightly strengthened resources, Fitch’s proof threatens to dissolve the very distinction between what is possible and what is actual.¹ A special thanks to those who have assisted my archival research, including Aldo Antonelli, John Burgess, Michael Della Rocca, Herbert Enderton, Bernard Linsky, Heidi Lockwood, Ruth Barcan Marcus, Julien Murzi and Bas van Fraassen. An extra special thanks to Julien Murzi, who as my research assistant in the Fall of 2005 helped me to identify and think more clearly about the famous anonymous referee reports, which are central to the present paper. For discussion and/or assistance I am also grateful to many others, including Scott Berman, Berit Brogaard, Judy Crane, Susan Brower-Toland, David Chalmers, Solomon Feferman, Nick Grifﬁn, Michael Hand, Monte Johnson, Jon Kvanvig, Matthias Lutz-Bachmann, Robert Meyer, Andreas Niederberger, Gualtiero Piccinini, Graham Priest, Krister Segerberg, Wilfried Sieg, Roy Sorensen, Kent Staley, Jim Stone, Neil Tennant, Achille Varzi, Nick Zavediuk, anonymous readers for Oxford University Press, and audience members at the Paciﬁc APA in Portland (March 24, 2006), the Goethe University of Frankfurt (May 15, 2006), the Institute for Logic, Language and Computation at the University of Amsterdam (May 23, 2006), and the Namicona Epistemology Workshop at the University of Copenhagen (August 22, 2006). Thanks also to my department at Saint Louis University for granting time and resources to research and write the paper. ¹ See Williamson (1992: 68) for the proof. What Williamson shows, precisely, is this: (p ↔ ♦Kp) T +K ♦p ↔ p, which says that if, necessarily, a proposition is true just in case it is knowable,

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Fitch’s 1963 paper is an enigma in itself. Although much has been written about its knowability proofs, virtually nothing has been said about Fitch’s understanding of their signiﬁcance. That is because Fitch provides, but never comments on, the ﬁnding. Indeed, the paper appears to change subjects midway. In the ﬁrst half we ﬁnd some knowability proofs and general lessons about concepts that share certain logical properties with the concept of knowledge. In the second half we ﬁnd a logical analysis of a particular concept of value, which happens not to share the relevant logical properties with the concept of knowledge. Why does Fitch develop and include the knowability results in a paper whose primary goal is to articulate a logical analysis of value? It initially appears that the knowability considerations have nothing to do with Fitch’s ﬁnal analysis. The thesis of the present paper is that Fitch’s intent was to pinpoint a disruptive set of logical properties that lend themselves to the trivialization of conditional analyses. Or, at the very least, Fitch included the central theorems to demonstrate a sort of conditional fallacy that threatens, although not irredeemably, against his own analysis of value. If this is right, then Fitch does not take the knowability proofs to be paradoxical, but instead takes them to be a lesson about how intensional operators interact, surprisingly, to thwart the efforts of conditional analyses. Fitch’s demonstration of the knowability proofs may be understood as a logical lesson in how to avoid the so-called ‘‘conditional fallacy’’ in philosophical analysis. My reading of Fitch is based on unpublished papers archived at Yale, Columbia and Princeton. The important documents include a pair of reports from 1945 (Chapter 1 of this volume), in which an anonymous referee conveyed to Fitch the knowability proof. The handwriting of the draft to the editor gives away its author, which is unmistakably Alonzo Church. The subsequent debate between Fitch and Church paints a clearer picture of what Fitch, by 1963, perceived to be the philosophical signiﬁcance of the so-called paradox of knowability. The archival documentation puts us in a position, for the ﬁrst time, to articulate and evaluate a lost chapter in the history and philosophy of logic—the early history of the knowability paradox. T h e 1 9 6 3 Pa p e r : W h a t’s He Bu i l d i n g i n T h e re ? The published literature begins with Fitch’s 1963 paper. Here Fitch investigates intensional operators that are factive and closed under conjunction-elimination. An operator O is factive just when its application implies truth: (Factivity)

(Op → p)

then it follows in modal system T (augmented with minimal epistemic resources) that a proposition is possible just in case it is true.

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The formula says, necessarily, if Op then p. Factive operators include ‘it is true that,’ ‘it is known that,’ ‘it is perceived that,’ and ‘it is necessary that.’ By contrast, ‘it is believed that’ is not factive, since believing p does not require the truth of p. An operator is closed under conjunction-elimination (or is conjunctiondistributive) just when it applies to a conjunction only if it applies to the corresponding conjuncts: (&-E Closure)

(O(p & q) → (Op & Oq) )

Both knowledge and belief are conjunction-distributive in this sense, since knowing/believing a conjunction requires knowing/believing each of the conjuncts. Fitch’s concern primarily is with ‘knows,’ which is both factive and conjunction-distributive.² Fitch’s concern in the ﬁrst half of the paper is only with operators that satisfy these two principles and the theorems in which they ﬁgure. He proves six theorems. Their content is discussed below. The philosophical signiﬁcance of each theorem, if any, I for now leave open, since Fitch did not comment on their signiﬁcance. The ﬁrst two theorems are perfectly general. I paraphrase the ﬁrst: Theorem 1: for any factive propositional operator O that is closed with respect to &-E, ¬♦O(p & ¬Op). Fitch proves here that there is always an un-O-able proposition, when O has the aforementioned logical properties. The proof is well rehearsed in the literature for the case of knowledge. Substituting the knowledge operator K for O gives us a theorem about the unknowability of any Fitch-conjunction, p&¬Kp. The unknowability may be stated this way: ¬♦K (p & ¬Kp). The demonstration follows:³ (1) K ( p & Kp) (&-E Closure) Kp & K Kp (Factivity & trivial logic) K p & Kp (1) K (p & Kp) (Normal Modal Logic) K (p & Kp) At the top of the tree we suppose for reductio that the Fitch-conjunction, p & ¬Kp, is known. By the closure of knowledge under &-E, it follows that each conjunct is known. The third line demonstrates an application of factivity to ² There are few exceptions to the received view that ‘knows’ is both factive and conjunctiondistributive. Robert Nozick (1981), for instance, articulates a concept of knowledge that is not conjunction-distributive. ³ The Genzen–Prawitz notation is preferred throughout for the perspicuity of logical dependencies.

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the right conjunct of the second line. In the face of the ensuing contradiction, we discharge and deny our only assumption. By necessitation and the duality of the modal operators, we conclude with the impossibility of that assumption. Fitch-conjunctions are unknowable! And more generally, conjunctions of the form p & ¬Op are un-O-able, when O is factive and conjunction-distributive. Fitch’s second perfectly general theorem says this: for the aforementioned operators, O, if p is a true proposition that is not O-ed, then p & ¬Op is a true proposition that is un-O-able. Theorem 2: for any factive operator O that is closed under &-E, if p is true but un-O-ed, then that it is an un-O-ed truth is itself un-O-able. (p & ¬Op) → ¬♦O(p & ¬Op). The result follows trivially from Theorem 1. The remainder of the theorems, Theorems 3 through 6, are special cases or consequences of the above perfectly general results. Theorem 3: If an agent a is all-powerful in the sense that anything that is true could have been brought about by a, then everything that is true was brought about by a: ∀p(p → ♦aBp) → ∀p(p → aBp). B is the factive, conjunction-distributive operator ‘brought it about that.’ Theorem 3 follows from Theorem 1, substituting B for the operator variable, O. The next two theorems are the knowability proofs. Theorem 4 is credited by Fitch to an anonymous referee. Theorem 4: for each agent that is not omniscient, there is a true proposition that that agent cannot know: ∃p(p & ¬aKp) → ∃p(p & ¬♦aKp). Theorem 4 is the contrapositive of Theorem 3, replacing the knowledge operator, K , for B. The next theorem is Theorem 5. It or its contrapositive is most often equated with the knowability paradox. It is a modiﬁcation of Theorem 4. Theorem 5: If there is a true proposition which nobody knows (or has known or will know) to be true, then there is a true proposition which nobody can know to be true: ∃p(p & ∀a¬aKp) → ∃p(p & ∀a¬♦aKp). Theorem 5 strengthens both the antecedent and the consequent of Theorem 4. It does this by generalizing over subjects in both places. Theorem 5 is then slightly more interesting when we detach the consequent, since it commits us to the existence of a truth that cannot be known by anyone. When we suppress the quantiﬁers ranging over subjects, as is standardly done for ease of exposition, Theorems 4 and 5 say the same thing—viz., if there is an unknown truth then

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there is an unknowable truth. Although the referee conveyed to Fitch Theorem 4 and thereby informed this entire section of Fitch’s paper, the slightly more interesting Theorem 5 (i.e., the so-called knowability paradox) and the perfectly general theorems are owed, at least in part, to Fitch. Fitch’s ﬁnal result, Theorem 6, just is Theorem 5, replacing ‘knows that’ with ‘proves that’ and stipulating that our propositions p are themselves about proving. Theorem 6: If there is some true proposition about proving that nobody has proved or ever will prove, then there is some true proposition about proving that nobody can prove: ∃p(p & ∀a¬aPp) → ∃p(p & ∀a¬♦aPp), where our propositional variables range over propositions about proving. The set of six theorems in their own right constitute an interesting development in the logic of intensional operators and action, and they play a role in current developments of modal epistemic logic.⁴ Fitch, as we mentioned, does not comment on their signiﬁcance. If the ﬁrst half of Fitch’s 1963 paper is about the logic of un-O-ability, then what is its connection to the apparently unrelated subject that occupies Fitch in the second half of the paper? The second half of the paper is concerned to articulate a logical analysis of an informed-desire theory of value. Informed-desire says, roughly, that something is valuable to a subject just when she would desire it if she had all the relevant information.⁵ Fitch’s ﬁnal analysis in 1963 roughly is this: s values p just when there is a truth q, such that, necessarily, if s knows that q then s strives for p. The logical analysis appears as Deﬁnition 6 : (D6 ) Vp

iff ∃q(q & (Kq → Sp) ).⁶

D6 is the centerpiece of the 1963 paper, but Fitch develops analyses of other propositional operators as well. He offers, for instance, deﬁnitions of ‘knows,’ ‘does,’ ‘can do,’ and ‘desires.’ In each of these cases he employs the strict (causal) conditional in the analysis and ends with considerations about whether or not the main operator (or deﬁniendum) is factive and conjunction-distributive. It is only in the case of his causal deﬁnition of knowledge, D2, that we ﬁnd an ⁴ See, for instance, Rescher (2005) and van Benthem (Chapter 9 of this volume). ⁵ The counterfactual gloss appears in an earlier draft of the paper (1961) and is meant to capture a causal reading. Fitch borrows from the logical analysis of causation found in William Burks (1951). In so doing, Fitch (1961: 6a) explains that the relevant sense of ‘A causes B’ is strict implication in a modal system such as S2 or M. ⁶ Some liberty is taken here with the formalism. Fitch uses ‘C’ for ‘(partially) causes’ rather than the necessary conditional, although it is clear from the text and from the 1961 address that a strict conditional reading is adopted. (See n. 5.) Moreover, there are epicycles in Fitch’s analysis that involve other propositional variables. Not being relevant to the present discussion, they are suppressed.

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operator that is both factive and conjunction-distributive. The main attraction, i.e., the analysis of value, however, is conjunction-distributive but not factive. Importantly, ‘knows’, which, again, is the only factive, conjunction-distributive operator deﬁned in the second half of the paper, ﬁgures in Fitch’s deﬁnition of value. So, a desideratum for understanding Fitch is this: the signiﬁcance of the knowability theorems must carry a lesson about the role played by ‘knows’ in Fitch’s analysis of value. But which lesson? Why in a paper about how to articulate an informed-desire theory is Fitch concerned to prove the knowability results? The question can be answered more carefully once we have uncovered the lost history of the proofs. So we leave this section with the central question, to which we will return. What’s he building in there?

W h o Di s c ove re d Fi t c h’s Pa r a d o x ? Another curiosity of Fitch’s 1963 paper is the identity of the famous anonymous referee, to whom Fitch credits the ﬁrst of the two knowability results. Following Theorem 4, Fitch tells us, This theorem is essentially due to an anonymous referee of an earlier paper, in 1945, that I did not publish. This earlier paper contained some of the ideas of the present paper. (1963: 138, n. 5)

That is all that Fitch says on the matter. The present section reveals more. We ﬁnd that Fitch’s 1945 paper was titled ‘‘A Deﬁnition of Value’’ and submitted to the Journal of Symbolic Logic in January or February of 1945.⁷ And, although so many recent papers mention the anonymous referee (under that description), few have published speculation about his identity. According to Richard Routley (Sylvan), [Robert] Meyer conjectures, what seems to me unlikely, that Anon[ymous] = G¨odel. (1981: 110, n. 12)

Routley’s skepticism is not explained. Why think that G¨odel is an unlikely suspect for authorship of the result? G¨odel was not ofﬁcially an editorial consultant for JSL in 1945. More interestingly, as John Burgess noted to me, it was unlikely that the editors would have asked G¨odel to referee a paper, since his perfectionism would have prevented him from returning a report in a timely manner. As for Robert Meyer, he recently admitted that he does not recall having ever discussed Fitch’s paradox with Routley, but notes that he ‘‘would have been struck by the strong whiff of the G¨odel formula,’’⁸ which says of itself that it is true but unprovable. Meyer is recalling the ﬁrst incompleteness result, which ⁷ The title is mentioned in Nagel (1945b).

⁸ Personal correspondence.

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demonstrates that, for any consistent, sufﬁciently strong theory T in the language of arithmetic, there are truths unprovable in T . For any such theory T , we ﬁnd that there is a sentence p such that p is true but unprovable in T : p & ¬PTp . The resemblance to the anomalous Fitch-conjunction, p & ¬Kp, catches one’s attention here. The G¨odel-conjunction and the Fitch-conjunction are analogous epistemic claims. Both advocate that the truth of some proposition p cannot be established by certain means. An important difference of course lies, ﬁrst, in the self-reference that is indicative of the G¨odel sentence and, second, in the epistemic terminology. For G¨odel the terminology is ‘‘unprovable in T .’’ For Fitch the notion is, less formally but more generally, ‘‘unknowable.’’ G¨odel promises a truth that could never be proven in T . Fitch promises a truth that could never be known by any means. Wolfgang K¨unne (2003: 425, n. 159) brieﬂy considers the hypothesis that G¨odel was the originator of the knowability result but notes that Fitch’s result is ‘‘in one respect more ambitious’’ than G¨odel’s theorem. K¨unne’s suggestion, I believe, is a claim about the relative logical strength of the respective claims to unknowability. G¨odel shows us that there is a truth that cannot be proven in T , but of course this does not entail that the truth could not be proven by some other means. Whereas, via Fitch we may conclude that there is a truth, viz., the Fitch-conjunction, p & ¬Kp, that is unknowable, full stop. On the contingent assumption that p & ¬Kp is true, it does follow by Fitch’s result that p & ¬Kp is an unknowable truth. And so, it follows that it could not be proven in any consistent theory strong enough for arithmetic. The problem with taking the Fitch conclusion to be logically stronger is that the existence of unknowable truths depends on the existence of some ignorance, which arguably is a contingent matter. However, some have contended that the existence of some ignorance is logically necessary.⁹ If it is necessary, then the conclusion of the knowability result is in fact stronger than G¨odel’s ﬁrst incompleteness theorem. The G¨odel-hypothesis is the only candidate in the literature. In the Summer of 2005, however, an altogether different hypothesis emerges. The hypothesis was prompted by found correspondence between the 1945 coeditors of JSL. In a letter dated March 6, 1945 Ernest Nagel updates Alonzo Church:¹⁰ I made a copy of your report on Fitch’s ms. (on the assumption that his receiving your handwritten version would destroy your anonymity) and sent it to him with the statement that his ms. in its present form was not acceptable for publication by the JSL. He replied two days later—I enclose his letter; and yesterday he returned the ms. with another letter ⁹ See, for instance, Routley (1981) and Rescher (2005: Appendix 2). ¹⁰ Thanks to John Burgess for his assistance in searching the Alonzo Church Papers and for identifying this letter. Thanks to Herbert Enderton for bringing the Church Papers to my attention.

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appended. I do not think he has met either of your two fundamental objections—indeed, his reply to the second difﬁculty seems to me to evade the issue rather completely. I am sending you the material for any further comments you may wish to make. (Nagel 1945a)

The letter indicates a number of things. In 1945 Church refereed a paper written by Fitch; the author of the report was anonymous to Fitch; and Fitch’s paper was (at least, at this stage) not being accepted for publication. The evidence is circumstantial, but if this was the paper in question and there were no other referees on the job, then it would seem that Church was the anonymous referee who conveyed the knowability proof to Fitch in 1945. The Nagel letter led me to the Ernest Nagel Papers (Columbia University), where my research assistant, Julien Murzi, very quickly identiﬁed the referee report in October 2005. The document had not previously been identiﬁed. It was composed in Church’s trademark vertical handwriting, and thereby conﬁrmed that Church indeed was the referee. In the excerpt below we ﬁnd the earliest known formulation of the knowability proof. Church writes: it may plausibly be maintained that if a is not omniscient there is always a true proposition which it is empirically impossible for a to know at time t. For let k be a true proposition which is unknown to a at time t, and let k be the proposition that k is true but unknown to a at time t. Then k is true. But it would seem that if a knows k at time t, then a must know k at time t, and must also know that he does not know k at time t. By Def.2, this is a contradiction.¹¹ (1945; Report 1, p. 2)

In sum, if a person a is not omniscient (that is, if there is a truth unknown to a), then there is a truth unknowable to a. It is evident that this result becomes the ﬁrst knowability result, Theorem 4—the very result that Fitch credits to the anonymous referee. It is not surprising that Church was the author of the report and its main proof, which is often taken to be about the logical limits of knowledge. It would be understated to say that Church thought deeply about such matters. He formalized the concept of effective calculability (1936a) and proved the undecidability of ﬁrst-order logic (1936b). Possible inﬂuences on Church’s thought in 1945 include G¨odel’s work from the prior decade and the interactions the two philosophers had in Princeton in the years leading up to 1945. In JSL William Parry (1939: 140) had proved Theorem 22.8: ¬♦¬♦(p → ¬♦¬p), which is equivalent to ¬♦(p & ¬p) —the core of Theorem 4, replacing all occurrences of with K . There was also Moore’s Paradox (1942: 543), which reveals the peculiarities of propositions of the form, ‘p but I don’t believe p.’¹² The critical documents found in the Nagel Papers actually include two referee reports, which we will label chronologically, Reports 1 and 2 (or R1 and R2). ¹¹ Church refers here to Def. 2, which appears to be Fitch’s deﬁnition of knowledge. As can be seen from the context, Church employs it to exploit the factivity of knowledge. ¹² Thanks to Roy Sorensen for information about this related problem and its earliest source.

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They are included in their entirety as the ﬁrst chapter of this volume. Report 2 was written by the same hand as Report 1. The second, but not the ﬁrst, report was signed by the author. The originals were apparently seen only by Nagel in his capacity as editor, who typeset them to preserve Church’s anonymity. In large part they consist of a series of trivialization arguments against Fitch’s analysis of value. Some of these arguments utilize the knowability result quoted above. The next section evaluates the ideas central to Report 1. T h e Fi r s t Re f e re e Re p o r t

A trivialization of Fitch’s analysis The knowability result was developed by Church to trivialize Fitch’s analysis of ‘a values p at time t,’ which is referred to in Report 1 as ‘Def.3.’ No statement of Def.3 appears in the report, but the context allows us to reconstruct the deﬁnition as follows: (Def. 3)Vp iff ∃q(q & (Kq → Dp) ) The formula tells us that it is valued (or is valuable to a subject) that p just when there is a truth q, such that knowing q necessarily implies desiring p.¹³ The analysis tells us, for instance, that it is valuable to me that I take my migraine medication if it is true that the medication will stop the pain and knowing that it stops the pain leads me to desire that I take the medication. Church’s criticism of the analysis begins with the acknowledgment that we are non-omniscient—that there are some truths p that an agent a does not know (at time t). Formally, for some p, it is true that: (1) p & ¬Kp. By the familiar result it is impossible for a to know both that p is true and that p is not known by a (at t). (2) ¬♦K (p & ¬Kp). Conditionals with impossible antecedents are necessarily true. So, from (2), it follows that: (3) (K (p & ¬Kp) → r ) , where r is any proposition you like. ¹³ From Church’s report we learn that Fitch employs a notion of ‘empirical necessitation’ rather than ‘strict implication’ in the right-hand side of the deﬁnition and distinguishes between the two notions throughout his paper. Fitch’s strict implication is Lewis and Langford’s. The modality is governed by S2. Fitch’s empirical necessitation, by contrast, appears to be a weaker notion. At the very least, Fitch’s Th. 1 appears to be a principle stating that strict implication entails empirical necessitation. See, for instance, the application of Th. 1 in Report 2, page 1. I suppress Fitch’s distinction between strict implication and empirical necessitation throughout. The issues here do not hang on the decision.

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Let r be ‘It is desired by a that s’ or just ‘Ds’. Then: (4) (K (p & ¬Kp) → Ds). Hence, from (1) and (4) it follows that there is a truth q such that knowing q strictly implies desiring s: (5) ∃q(q & (Kq → Ds) ). Therefore, by Def.3, s is valued: (6) Vs. And since s was arbitrarily chosen, it therefore follows that everything is valued. In sum, if there is a truth unknown to a then a values everything. At a glance the result is this, where q is the true conjunction p & ¬Kp. q ∃q(q &

¬ Kq (Kq →Ds) (Kq →Ds)) (Def.3) Vs

Church’s argument illustrates the mistake in Fitch’s analysis. The mistake tends to occur when we deﬁne concepts in conditional terms. This, the so-called ‘‘conditional fallacy,’’ is not unrelated to the paradoxes of implication. Classical conditionals behave strangely when their antecedents are false or impossible. More speciﬁcally, but without attempting to characterize all and only cases of the fallacy, the conditional fallacy is a mistake that occurs just when the antecedent of the conditional deﬁniens is not always logically independent of the deﬁniendum. That is, instances of the analysis include cases where the deﬁniendum contradicts, entails or is entailed by the antecedent of the conditional deﬁniens. Such conditions will sometimes effect a surprising disparity in truth value between the deﬁniens and the deﬁniendum.¹⁴ This is what gets Fitch’s analysis into trouble. The conditional embedded in his deﬁnition, Vp iff ∃q(q & (Kq → Dp) ), has instances where the antecedent, Kq, is not logically independent of the deﬁniendum, Vp. And that is because there are instances of the antecedent that are logically impossible and so entail any proposition whatsoever. A fortiori, such instances necessarily imply the deﬁniendum. The mistake in Fitch’s analysis results from his failure to detect the logical anomaly of unknowable truth. For the existence of unknowable truth is the logical phenomenon responsible for the surprising trivialization of Fitch’s analysis. Fitch later takes to heart this lesson of philosophical analysis. The lesson will play a critical role in Fitch’s 1963 paper. ¹⁴ This understanding of the fallacy is informed by Shope (1978) and Wright (2000).

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In the second referee report Church considers blocking the above trivialization by appealing to Russell’s theory of types. In so doing Church foreshadows Linsky (Ch. 11, this volume) and Hart (Ch. 19, this volume). However, Church dismisses the option as contrary to Fitch’s purposes, since an employment of the theory of types would invalidate closure principles central to Fitch’s paper. As we will see in the next section, Church has independent reason for rejecting these closure principles.

Closure principles for knowledge and belief In the ﬁrst report Church foreshadows what he takes to be Fitch’s only good defense against the trivialization argument, and that is to question the validity of closure principles for propositional attitude operators. Speciﬁcally, he denies that there is a ‘‘law according to which one who believes a proposition must believe all its logical consequences’’ (Report 1: 2). Church questions here the validity of the principle that belief is closed under logical consequence. His intention, though, is to question the justiﬁcation for the principle that belief is closed under conjunction-elimination. Church writes: To be sure, one who believes a proposition without believing its more obvious logical consequences is a fool; but it is an empirical fact that there are fools. It is even possible that there might be so great a fool as to believe the conjunction of two propositions without believing either of the two propositions; at least an empirical law to the contrary would seem to be open to doubt. On this ground it is empirically possible that a might believe k at time t without believing k at time t (although k is a conjunction one of whose terms is k).¹⁵

Church denies that belief is necessarily closed under conjunction elimination. It is unclear, however, how this is supposed to help Fitch. The trivialization argument never utilizes a closure principle for belief. It utilizes, instead, a closure principle for knowledge. And, of course, it would be a fallacy of division to suppose that the concept of belief has a certain logical property P (e.g., closure under logical consequence) just because (1) belief is a component of knowledge and (2) knowledge has P.¹⁶ An alternative, non-fallacious reading of Church’s passage is that he simply means ‘‘knowledge’’ when he speaks of belief. In that case Church simply questions whether knowledge is closed under conjunction-elimination. However, this limited closure principle is harder to reject than the more general principle that knowledge is closed under logical consequence. That is because ‘‘knowing p’’ ¹⁵ R1: 2–3. ¹⁶ More recent instances of this very closure-fallacy in epistemology are detected by Ted Warﬁeld (2004).

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and ‘‘knowing q’’ are implicit in ‘‘knowing p & q.’’ So it is doubtful that Church has offered Fitch a ‘‘good defense’’ of the trivialization argument. But let us suppose that he has and consider one further point about the denial of closure in this context. Directly after his articulation of the problems with taking belief to be closed under conjunction-elimination and offering the defense on Fitch’s behalf, Church makes the following claim: Unfortunately this defense compels Fitch to abandon his Ax. 1. And, what is more serious, it lights the way to a second and opposite objection to Def. 3. If there is no empirical law according to which one who believes a proposition must believe its logical consequences, it would seem that by the same token there is no empirical law according to which a person’s desires must be in reasonable accord with that person’s beliefs. (R1: p. 3)

The consequence of rejecting closure principles for belief, according to Church, is that it invites a skepticism about other principles that express necessary connections between propositional attitudes, in particular between knowledge and desire. And without such necessary connections, the right-hand side of Fitch’s analysis is never satisﬁed, and so, the theory is trivialized in the opposite direction. Nothing is of value to anyone! Church’s point is overstated. Surely there may be laws about our propositional attitudes, even if belief/knowledge is not closed under logical consequence more generally. That is, for all we know, some principles other than the unrestricted closure principles justify necessary connections between our propositional attitudes. Fitch, in fact, gives the following example in reply to Church: necessarily, if it is known that I desire that p, then I desire that p. So sometimes knowledge does necessitate desire. The example is an instance of the principle that knowledge necessarily implies truth. With it Fitch proves trivially that there are some necessary connections between knowledge and desire. Fitch’s example is cited in Church’s second referee report (R2: 4). We learn from Nagel’s letter of March 6 that Fitch replied to the ﬁrst referee report with two letters and a revised manuscript. These documents, like Fitch’s initial submission, are yet to be found. In any case Nagel was unimpressed by them. Recall Nagel’s remark to Church: ‘‘I do not think [Fitch] has met either of your two fundamental objections—indeed, his reply to the second difﬁculty seems to me to evade the issue rather completely. I am sending you the material for any further comments you may wish to make.’’ The second difﬁculty, recall, was Church’s animadversions to closure principles and other ‘‘laws’’ relating propositional attitudes. I do not see that Fitch’s point—about factively knowing that one desires something—evades Church’s difﬁculty ‘‘completely,’’ but I will not pursue the issue further. Fitch’s very revealing reply to the other difﬁculty, i.e., the trivialization argument from unknowable truth, is summarized in the second referee report. We turn to that document next.

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T h e Se c o n d Re p o r t

Fitch’s Cartesian restriction strategy In reply to Nagel’s March 6 letter, Church issued a second referee report. From it we learn of Fitch’s reactions to the trivialization argument in Report 1. Fitch’s analysis said that it is valuable that p just in case there are truths that would, if known, necessitate the desire that p. Church showed us that, vacuously, there will be such truths, since there are truths that it is impossible to know. The natural reply is to restrict the class of truths to those that it is possible to know. Presumably it is only the knowable truths that should ﬁgure in causal relations between knowledge and desire. In reply to the ﬁrst report, Fitch endorses this insight by offering an alternative restricted theory of value, Def. 3R: (Def. 3R)

Vp iff

∃q(q & ♦Kq & (Kq → Sp) )¹⁷

The restricted analysis says that something p is of value to a subject a just when there is some knowable truth q that would, if known, necessitate a’s desiring that p. Let us call this ‘Fitch’s Cartesian restriction strategy,’ because it foreshadows Neil Tennant (1997), where the restriction is proposed under that name to block the knowability paradox. Tennant deﬁnes a Cartesian proposition p as one for which Kp is not provably inconsistent. Tennant (2001) considers versions of the restriction in terms of what it is metaphysically possible to know. The Cartesian restriction on the relevant class of truths blocks the problematic unknowable truth, ‘p & ¬Kp,’ from consideration. The fact that knowledge of it vacuously implies an arbitrary proposition becomes inconsequential.

Church’s objection to the Cartesian restriction strategy In the second report Church announces that ‘‘a reductio ad absurdum of Def 3R is possible along the same lines as that I have given for Def 3.’’ Church claims there is a Cartesian truth that trivializes Fitch’s restricted analysis. He begins by noting that, for some p, p is an unknown truth. So (i) Dp ∨ (p & ¬Kp) is true, for an arbitrary proposition p . That is, (i) follows from our nonomniscience. After all, if p & ¬Kp is true, then so is the weaker claim, Dp ∨ (p & ¬Kp). Church goes on to argue that: (ii) Proposition (i) is Cartesian. ¹⁷ The amended analysis appears in Report 2: 2. The above formulation of Fitch’s Def. 3R differs from Church’s in that I substitute ♦ for ‘EP’, which reads, ‘it is empirically possible that.’ Also, I continue to replace ‘EN’ or ‘empirically necessitates’ with the necessary material conditional and drop the variables ranging over subjects.

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We shall evaluate this and the next premise in a moment. The ﬁnal premise is that knowing proposition (i) strictly implies the desire that p : (iii) (K (Dp ∨ (p & ¬Kp) ) → Dp ) If premises (i), (ii) and (iii) are all correct, then it follows that there is a Cartesian truth q such that, necessarily, if q is known then p is desired. By Fitch’s Cartesian restricted theory of value, Def. 3R, it would follow that p is valuable, for arbitrary p . We may generalize. If an agent is non-omniscient, then everything whatsoever is valuable to her! Here we consider the premises of Church’s argument. Premise (i) is trivial. If p & ¬Kp is true for some p, then so is Dp ∨ (p & ¬Kp), by disjunctionintroduction. What about premise (ii)? It says that the awkward disjunction, given by (i), is Cartesian, i.e., can be known. Church begins his defense of this premise with the reasoning that any desire is possible (Report 2: pp. 2–3): (a) for any p , ♦Dp . But then knowledge of that desire is possible: (b) for any p , ♦K (Dp ). And so, by the closure of knowledge under disjunction-introduction: (c) for any p , ♦K (Dp ∨ (p & ¬Kp)). Church defends premise (a) by noting that anything, even one’s instant death, can be desired, since it is possible to be insane or in a position less fortunate than one brought about by instant death. By similar reasoning, we might argue further that even contradictions can be desired. So an arbitrary proposition can be desired. But how does Church get from premise (a) to premise (b)? He seems to be assuming that, necessarily, any possible desire is a knowable desire. That is, if it is possible for one to desire that p then it is possible for one to know that one desires that p . Note that there are some implicit principles being invoked. We uncover them by asking what it takes to justify the principle that any possible desire is a knowable desire? Perhaps Church believes that, necessarily, any desire can be known. So, necessarily, if p is desired, then it is possible to know that p is desired: (Dp → ♦K (Dp ) ) On this reading, Church assumes an unrestricted knowability principle about desire. Notice that this is not sufﬁcient to license the move from line (a) to line (b). For the assumption that it is possible to desire p , together with the above principle, by minimal normal modal reasoning, entails only that it is possible that it is possible that Dp is known: ♦♦K (Dp ).

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The S4 axiom is then needed to reduce this to ♦K (Dp ). So if the operant notion of possibility satisﬁes the S4 axiom and in fact, necessarily, any desire is knowable, then it follows that, necessarily, any possible desire is a knowable desire. With this latter principle in hand, premise (b) does in fact follow from premise (a). Premise (c) then follows from premise (b) on the assumption that knowledge is closed under disjunction-introduction. Kp K (p ∨ q) The principle, Church tells us, ‘‘seems to be entirely in the spirit of [Fitch’s] Th. 3.’’ (Report 2: 3) Th. 3 we may hypothesize to be the principle stating that knowledge is closed under conjunction-elimination. Church puts these principles on a logical par. Presumably he is onto the fact that both are instances of the principle that knowledge is closed under obvious logical consequence. There is, however, the objection that these two closure principles are not on a par. ‘a knows both that p and q’ and ‘a knows p and a knows q’ are implicit in one another. Arguably, they say the same thing. Such gives us reason to think that knowledge is closed under conjunction-elimination, despite the problems with thinking that knowledge is closed more generally under logical consequence. By contrast, ‘a knows p’ and ‘a knows p ∨ q’ are not implicit in one another. The latter is not implicit in the former, since q may embed concepts that are not grasped by one who understands ‘a knows p’. So it is not decisive that knowledge should be closed under disjunction-introduction, even if it is closed under conjunction-elimination. Now Fitch never commits himself to the S4 axiom, and need not be committed to the closure of knowledge under disjunction-introduction, even if he does accept its closure under conjunction-elimination. So Church’s argument for premise (ii) is not decisive. His logical assumptions are not trivial.¹⁸ We turn to Church’s justiﬁcation for premise (iii). How does Church prove that (K (Dp ∨ (p & ¬Kp) ) → Dp )? The reasoning here is troubling. It can be found on page 3 of Report 2. Church notes that, by the factivity of knowledge, knowing Dp ∨ (p & ¬Kp) entails Dp ∨ (p & ¬Kp). And further that each of these disjuncts implies Dp . So, by proof-by-cases, Dp . Therefore, if Dp ∨ (p & ¬Kp) is known, then Dp . Resting on no contingent assumptions, Dp ∨ (p & ¬Kp) necessarily implies Dp . Here is the reasoning at a glance: ¹⁸ Incidentally, there is a more modest defense of premise (ii). That is, it is metaphysically possible to know premise (i) for the following reason. It is possible to know Dp , recognize that Dp entails Dp ∨ (p & ¬Kp), and thereby come to know Dp ∨ (p & ¬Kp). If this is right, then it is after all possible to know Dp ∨ (p & ¬Kp). That is, premise (i) is Cartesian.

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Joe Salerno K (Dp ∨ (p &¬Kp)) Dp ∨ (p & ¬Kp)

(2) (1)

Dp Dp (2) K (Dp ∨ (p &¬Kp)) →Dp ( K (Dp ∨ (p &¬K p))→Dp )

(1) p &¬K p Dp (1)

If this is correct then there is a Cartesian truth, such that knowing it necessitates desiring p , for any proposition p . By Fitch’s Cartesian restricted theory of value, therefore, p is valued. The theory trivializes. If this is Church’s argument, then we have to reject it. Something went wrong in the proof-by-cases. The right disjunct p & ¬Kp does not imply Dp . Church may be confusing the proposition p & ¬Kp with K (p & ¬Kp). The latter strictly implies everything, since it is impossible. So obviously K (Dp ∨ (p & ¬Kp) ) strictly implies Dp . Perhaps that is what Church intended, and the mistake can be chalked up to a misprint. However, with the supposition of a misprint in the formulation of Church’s proof-by-cases one must make corresponding adjustments to the ﬁrst part of Church’s argument. The truth that must be shown to be Cartesian is now Dp ∨ K (p & ¬Kp). It is Cartesian. It is knowable because its left disjunct is. But it is not true. Or at least it is not true for an arbitrary desire, Dp . Therefore, the trivialization argument comes apart; it fails against Fitch’s Cartesian restricted theory of value.¹⁹ Other items that appear in the second referee report include (1) a more formal (Lewis and Langford style) proof of the central knowability result that appeared in the ﬁrst report; (2) some mention of the similarity of the trivialization arguments to the liar and set-theoretic paradoxes and the standard devices for resolving them; (3) a brief mention of a problem of accepting the factivity of knowledge while embracing a theory of types; (4) further discussion of Fitch’s concept of empirical necessity; and (5) some counterexamples to Fitch’s theory of value that do not hinge on the knowability theorems. I will not comment on these items. Fitch does not seem to have directly addressed Church’s ﬁnal trivialization argument against the Cartesian restricted theory. In Nagel’s last letter to Church ¹⁹ Jim Stone (in personal correspondence) constructs an argument for premise (iii) in Church’s spirit. It presupposes the transparency of desire (i.e., that all desires are known, Dp → K (Dp), and all failures to desire are known, ¬Dp → K (¬Dp) ). It also presupposes that knowledge is closed under disjunctive syllogism. It goes like this. Suppose K (Dp ∨ (p & ¬Kp)), and suppose for reductio that ¬Dp . By the transparency of desire, K (¬Dp ). Since a disjunction and the negation of the left disjunct are both known, it follows, by the closure of knowlege under disjunctive syllogism, that the right disjunct is known—giving K (p & ¬Kp). But that is impossible. So by classical reductio, Dp . Hence, by conditional proof, K (Dp ∨ (p & ¬Kp)) → Dp . So a defense of premises (iii) may be given and Church’s master argument may be rehabilitated, although the text does not warrant crediting this argument to Church. An even more modest master argument against Fitch’s restricted analysis is formulated in the ﬁnal section of this paper.

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on the matter (April 13, 1945), we learn that Fitch has withdrawn his paper owing to ‘‘a defect in my deﬁnition of value’’ and because ‘‘the paper should be rewritten anyhow.’’

Fi t c h i n t h e Si x t i e s The 1945 Church–Fitch debate helps to explain some things about the 1963 paper. One question is about the intended signiﬁcance of the knowability results. Why does Fitch include them? Consider again the knowability theorems, which say, roughly, that there is an unknowable truth if there is an unknown truth. Fitch presents them in passing but does not comment on their signiﬁcance. Of course, that he demonstates the proofs without comment indicates that he takes them to be valid and not paradoxical. Moreover, it is obvious that there are unknown truths, and so, by the relevant theorems, it would seem that we are meant to recognize the existence of unknowable truths. The insight is intrinsically interesting, but the question regards its role in the paper. One interpretation is that Fitch is offering a refutation of veriﬁcationism, the thesis that all meaningful statements (and so, all truths) are veriﬁable. Indeed, this is how the early literature interprets Fitch.²⁰ To be consistent, this reading requires us to argue, analogously, that there are implicit conclusions that Fitch wishes us to draw from the other theorems. Recall that Theorem 3 shows that if there is an omnipotent being, then he has in fact done everything. Presumably, Fitch would expect us to conclude from this that there is no omnipotent being (or that he is not supremely good, or that there is no free will, or something of the sort). However this is an unlikely reading of Fitch’s intent, as it marks the theorems, including the knowability proofs, as a curious tangent from the paper’s primary goal. The 1963 paper is not a defense of any metaphysical position, not even a defense of the informed-desire theory of value. Rather, it aims to articulate the logical content of that theory. It would be odd, in a paper with that purpose, for Fitch to prove the absurdity of veriﬁcationism or disprove the existence of God. More to the point, this interpretation of the theorems tells us nothing about the factive, conjunction-distributive role played by ‘knows’ in Fitch’s analysis of ‘value.’ So it would seem that the key to understanding Fitch lies elsewhere. ‘Knows’, unlike the other concepts that Fitch deﬁnes in the second half of the 1963 paper (including the concept of value), is both factive and conjunctiondistributive. For this reason it gives rise to the existence of unknowable truth. Fitch wishes us to recognize the existence of unknowable truths for logical, not metaphysical reasons. Unknowable truth is the hallmark of the kind of trivialization that Fitch wishes to avoid in 1963—the very kind of trivialization ²⁰ See, for instance, Hart and McGinn (1976); Hart (1979); Mackie (1980); and Routley (1981).

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that Church wielded against him in 1945. Through the 1945 exchange Fitch recognizes that conditional analyses harbor grave pitfalls. When the dominant propositional operator of the antecedent of a conditional deﬁnition is factive and conjunction-distributive, then there will be an instance of the conditional analysis whose antecedent is impossible. But then the antecedent will not be logically independent of the deﬁniendum (whatever it is), and consequently, trivialization threatens. For the special case, the moral of the knowability theorems, is then to beware of this fallacy in the conditional understanding of the informed-desire theory. My explanation of why Fitch included the knowability theorems in the paper is supported by the fact that Fitch does in fact heed the warning by protecting against the fallacy. Directly following the formal articulation of his 1963 analysis of value, Vp iff ∃q(q & (Kq → Sp) ), Fitch explains that to avoid absurdity, q may be regarded as containing all the knowable relevant information. (Ch. 2, this volume: 28)²¹

We see here the very Cartesian restriction that Fitch attempted in 1945, although Fitch includes it here without much remark. It appears then that the reason that the knowability theorems are included in the ﬁrst half of the paper is to explain the need for the Cartesian restriction that emends the ﬁnal analysis in the second half. Recall that in 1945 Fitch attempted in this way to restrict his analysis in reply to Church’s ﬁrst referee report. But the attempt was met with an overwhelmingly negative second report. At the time Fitch decided to withdraw his paper, even though the second report, as we have seen, was critically ﬂawed. However, Fitch must have recognized the errors of Church’s second report. For by 1963 he was perfectly happy with a Cartesian-restriction in his ﬁnal analysis of value. Why did Fitch wait so long to publish the analysis? I believe that skepticism about non-trivial necessary connections between knowledge and desire kept Fitch sufﬁciently worried, and that it was not until Burks (1951) offered a logical analysis of causal conditionals that Fitch believed himself to have the logical resources to explain the relevant modal relation. This is supported by the fact that Fitch makes extensive use of Burk’s analysis in both his presidential address to the Association for Symbolic Logic (1961) and his 1963 publication. We have seen that the role of the knowability theorems in Fitch’s paper do carry lessons about the role played by ‘knows’ in Fitch’s ﬁnal analysis of value. These are the aforementioned lessons about whether and how to protect against the conditional fallacy. In 1963 what Fitch is building in there is an analysis of value that is sheltered from this fallacy. With this interpretation of the knowability proofs, we ﬁnd an account of the early history and initial, perceived signiﬁcance of the so-called knowability paradox. ²¹ See other mentions of the restriction on page 27.

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A g a i n s t Fi t c h’s C a r t e s i a n Re s t r i c t i o n Church’s result shows us that there are unknowable truths, and that these truths serve to trivialize Fitch’s 1945 theory of value. Fitch responds by restricting his theory to truths that it is possible to know. The problem with the restriction, as Church attempted to show in his second report, is that there are knowable truths that trivialize the restricted theory. The trick is to come up with a knowable truth q, such that q is weaker than the unknowable truth, p & ¬Kp, and such that q trivializes the theory. Church’s choice of such a proposition was Dp ∨ (p & ¬Kp). It is knowable, but as I argued it fails to do the job that Church set for it. And that is because knowing that proposition does not necessitate an arbitrary desire. Church fails to trivialize the restricted theory, but he was right in thinking it can be done. There are in fact knowable truths weaker than p & ¬Kp that serve to trivialize Fitch’s restricted theory of value. They are truths of the following form: (1) p & (Kp → q) which says both that p and that knowing p implies q. That will be true whenever there is an unknown truth—i.e., whenever: (2) p & ¬Kp is true. So whenever p &¬Kp is true for some sentence p, p & (Kp → q) will be true for an arbitrary sentence q. And that is because a false proposition (in this case Kp) materially implies any proposition.²² Therefore, for some proposition p, the following is true: (3) p & (Kp → Sq). Moreover, knowledge of its truth necessarily implies that q is strived for: (4) (K (p & (Kp → Sq) ) ) → Sq.²³ And ﬁnally, p & (Kp → Sq) is Cartesian. That is, it is possible to know p & (Kp → Sq), even though it is not logically possible to know the logically stronger proposition, p &¬Kp. In sum, if there is an unknown truth p, then p & (Kp → Sq) is a knowable truth, and, necessarily, knowing it necessitates striving for q. But then there is a knowable truth, p, that satisﬁes (Kp → Sq). Consequently, the right-hand ²² Such consequences of the Fitch-conjunction are used by Williamson (2000b: 110–12) and Brogaard and Salerno (2006: 266–7) against Tennant’s (1997) Cartesian restriction strategy. Interesting discussion also appears in Rosenkranz (2004), although his prescription is for the Cartesian restriction strategist to reject normal modal logic. ²³ The proof of (4) is straightforward. It requires that K be factive and closed under conjunctionelimination.

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side of Fitch’s theory of value is vacuously true. It follows by the restricted theory of value that q is valued, for arbritrary q. The theory collapses! The formalism below more perspicuously demonstrates the collapse of Fitch’s 1963 theory of value. After all, p & (Kp → Sq) is a knowable proposition; knowing it necessarily implies Sq; and it is true if p &¬Kp is true, for some p. p & ¬K p p & ( K p → Sq) (p & ( K p → Sq) & ∃q(q &

( K (p & (K p → Sq)) → Sq) (K (p & (K p → Sq)) → Sq) (K q → Sq)) (D6) Vq ∀qV q

Although the above argument trivializes Fitch’s theory of value, it does not uncover a conditional fallacy. The conditional’s antecedent K (p & (Kp → Sq) is logically independent of the deﬁniendum Vq. There is, however, a conditional fallacy that the 1963 analysis perpetrates. This is demonstrated by a different version of the above argument. Just replace all occurrences of the formula p & (Kp → Sq) in the above proof with ( ∗ )p & (Kp → (Vq &Sq) ). (∗ ) appears to be Cartesian; knowing it is not logically independent of the deﬁniendum, Vq; knowing (*) strictly implies Sq; and it succeeds in trivializing the theory, since Sq and its embedded proposition, q, were chosen arbitrarily. There are instances of Fitch’s deﬁnition of value where the antecedent of the relevant conditional is not logically independent of the deﬁniendum. If a lesson of Church’s 1945 result is not to commit the conditional fallacy in philosophical analysis, then by 1963 Fitch had appreciated the danger but his analysis had not satisfactorily protected against it.

Pa r t I I Dum m e t t’s C o n s t r u c t i v i s m

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4 Fitch’s Paradox of Knowability Michael Dummett

Fitch’s paradox of knowability runs as follows. The constructivist or (as I have been calling him) justiﬁcationist believes that every true statement is capable of being known to be true. This may be symbolized by: (A) p → ♦ Kp, where ‘K’ means ‘is, has been or will be known by somebody’. We should normally think that there are many true statements that will never be known to be true; favourite examples concern the parity of the number of a large set of objects. Replacing ‘p’ by ‘p & ¬Kp’ we obtain: (B)

(p & ¬Kp) → ♦K(p & ¬Kp).

But K(p & ¬Kp) is contradictory and hence impossible; hence ¬♦K(p & ¬Kp), and accordingly: (C) ¬(p & ¬Kp), whence no true statement will never be known: (D) p → ¬¬Kp, which by classical logic implies that every true statement will eventually be known: (E) p → Kp, contrary to our strong intuition. It follows that principle (A) cannot be maintained, and hence that the constructivist/justiﬁcationist is wrong. That is essentially Fitch’s reasoning. What is wrong with it? The fundamental mistake is that the justiﬁcationist does not accept classical logic. He is happy to accept principle (D), provided that the logical constants are understood in accordance with intuitionistic rather than classical logic. In fact, in line with the inspired suggestion of Bernhard Weiss, he will prefer (D) to (A) as a formalization of his view concerning the relation of truth to knowledge. When A is a mathematical statement, ‘¬A’ is usually explained as meaning ‘Given a proof

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of A, we could derive a contradiction’, where ‘we could derive’ is to be glossed by ‘using anything we have already proved’. It follows that ‘¬A’ may be read as ‘It is in principle impossible to prove A’. In a more general context, ‘¬A’ may be read as ‘It is in principle impossible for us to be in a position to assert that A’ or ‘There is an obstacle in principle to our being able to assert that A’, where ‘in principle’ is to be glossed by ‘in the light of all that we already know’. Hence ‘¬¬A’ means ‘There is an obstacle in principle to our being able to deny that A’, where denying that A is asserting that ¬A. It follows that ‘¬¬KA’ means ‘There is an obstacle in principle to our being able to deny that A will ever be known’, in other words ‘The possibility that A will come to be known always remains open’. That this holds good for every true proposition A is precisely what the justiﬁcationist believes. This is the principle expressed by (D); and (D) captures the relation which the justiﬁcationist believes to obtain between truth and knowledge. He is not concerned to deny that there may be true propositions which will in fact never be known. This is something that cannot be expressed by means of intuitionistic logical constants. We cannot capture this proposition by introducing a binary operator Kn (p, t ) to mean ‘it is known at time t that p’. Intuitionistically interpreted, ‘∀t ¬Kn (A, t )’ holds good only if there is a general reason why it cannot be known at each time t that A, that is, precisely if ¬KA. What the justiﬁcationist wants to deny is not that there are true propositions that will always happen to remain unknown, but that there are true propositions that are intrinsically unknowable: for instance one stating the exact mass in grams, given by a real number, of the spanner I am holding in my hand. With (D) interpreted intuitionistically and adopted in place of (A) as the principle connecting truth and knowledge, there is now no paradox. (C) indeed will hold good; but when understood intuitionistically, it is no longer contrary to intuition.

5 The Paradox of Knowability and the Mapping Objection Stig Alstrup Rasmussen

I According to Timothy Williamson, the Paradox of Knowability—or Fitch’s Paradox (Fitch 1963)—is not really a paradox from any point of view. Despite what is suggested by a cursory glance at the reasoning involved, the argument presents no genuine paradox to either semantical realist or semantical anti-realist (Williamson 1982, 1992, and 2000a, Ch. 12). I agree. However, Williamson’s various treatments of the alleged paradox leave open several issues. Furthermore, at least some alternative proposals deserve serious consideration. Originally, the paradox-generating piece of reasoning was launched—and later revived—as a refutation of any form of idealism adhering to the thesis that all truths are in principle knowable. The supposedly suitable formal rendering of the thesis is (1)

(∀p) (p → ♦Kp).

The (second-order) quantiﬁer ranges over propositions, the modal operator indicates possibility, and ‘K’ is rendered as (although the reading of ‘K’ turns out to be one of the trouble-spots): (K) Kp, if and only if it is currently known that p. Furthermore, we are not concerned with the bearer of any purported bit of knowledge. This issue is relevant to some questions, but these will not be ours. Thesis (1) is supposedly characteristically held by a Dummettian anti-realist. The realist, on the other hand, supposedly balks at (1), if (1) is thought of as holding unrestrictedly. Now Dummett nowhere seems to saddle his anti-realist with precisely (1). However, his various characterizations of the semantical anti-realist have usually been taken to imply a commitment, on the part of the anti-realist,

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to some such thesis.¹ In any case, Dummett’s Victor (Dummett 2001) or Neil Tennant’s ‘naive’ anti-realist (Tennant 2002: 135) are certainly thus committed; and, e.g., Crispin Wright will unavoidably be taken to underwrite a similar view, on behalf of anti-realism (Wright 1992: Chs 1–2). Indeed, (1) can seem to hold trivially, once truth is equated with warranted assertibility, or the like. This was among our (mistaken) contentions in Rasmussen and Ravnkilde (1982: 436–7 n. 77). It was even earlier that most of the above (rightly) thought it worth while to revive the thesis and the attendant putative paradox (Hart and McGinn 1976 and Hart 1979). The set of issues involved in the paradox has generated a good deal of discussion. Overall, the argument has always been thought of as an attempted reductio ad absurdum of, speciﬁcally, anti-realism-cum-idealism, albeit often as an unsuccessful one. Let us kick off by agreeing on what the argument is. Plausibly, there are facts not currently known. (2)

(∃p) (p & ¬Kp).

Also, as Plato is often credited with having pointed out, knowledge is factive. The usual formal rendering of this is: (F)

(∀p) (Kp → p).

And knowledge distributes over conjunctions: (D)

(∀p) (∀q) (K(p & q) → (Kp & Kq) ).

Now to the Basic Version (BA) of the troublesome derivation, which purports to show that (1), (2), (F) and (D) form an inconsistent set (We use Lemmon-style natural deduction, as presented in Read and Wright 1994. I ﬁrst published this version in Rasmussen 1997): 1 2 3 1 1,3 6 6 6 6 6 6

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

(∀p) (p → ♦Kp) (∃p) (p & ¬Kp) q & ¬Kq (q & ¬Kq) → ♦K(q & ¬Kq) ♦K(q & ¬Kq) K(q & ¬Kq) Kq & K¬Kq Kq K¬Kq ¬Kq Kq & ¬Kq

A (anti-realism?) A (trivial?) A 1 ∀E 3,4 →E A 6 SI (D) 7 &E 7 &E 9 SI (F) 8,10 &I

¹ Dummett’s latest statement of his ofﬁcial position concerning anti-realist truth known to me is in Dummett 2005, especially p. 673.

The Paradox of Knowability and the Mapping Objection 1,3 1,3 1,2

(12) (13) (14)

Kq & ¬Kq ⊥ ⊥

5,6,11 12 2,3,13

55 ♦E ¬E ∃E

Assumptions (1) and (2) are framed in a manner consonant with Fitch’s original formulation. The argument (BA) works classically, as well as intuitionistically. Often, the adherent to classical logic is presented as blaming the contradiction on (2): 1 1

(15) (16)

¬(∃p) (p & Kp) (∀p) (p → Kp)

2, 14 15

¬I SI

And therefore the anti-realist is supposedly in the position of having to conclude from his cherished thesis (1) that all truths are currently known; which is patently false. So, the thought is, (1) is false, and semantical anti-realism untenable. The move from (15) to (16) is, of course, intuitionistically invalid. This matters, as pointed out in Rasmussen and Ravnkilde (1982: 436–7 n. 77) and Williamson (1982), since the anti-realist is presumably committed to a logic no stronger than the intuitionistic one. Interestingly, the most compelling kind of proof of this appeals to the anti-realist’s commitment to (1) (Wright 2001: 66 n. 24).² So, we may note in passing that a principle very much like (1) seems to be ﬁrmly entrenched within the anti-realist way of thinking. However, the logical revisionism seemingly inherent in anti-realism merely exempts its adherent from endorsing the ﬁnal step in the (extended) argument (BA); and the derivation (1) through (14) may seem bad enough. At least one of (1) and (2) must go. If the derivation is regarded as a reductio of (1) speciﬁcally, then it is tempting to adopt the restriction strategy: the anti-realist can have (1), but only as applied to a suitably restricted set of propositions. Neil Tennant restricts the range of the quantiﬁer to so-called Cartesian propositions, i.e., propositions that admit of coherent anti-realist assertion (Tennant 1997/2002: ch. 8, and 2002). Dummett has proposed a more subtle restriction, according to which (1) can be taken to hold primitively for basic propositions only; whereas its fate in cases involving more complex substitution-instances is up for discussion (Dummett 2001). On the other hand, Bernhard Weiss has put forth a proposal according to which there should at least be no objection to taking the paradox as a reductio of (2), although he tinkers with (1) as well (cf. note 14 below). Williamson, however, takes the paradox as having been not properly stated. One crucial move in his restatements over the years, starting with Williamson 1982, is the introduction ² The argument is to the effect that, given (1), the law of excluded middle entails intuitionistically that all sentences are in principle decidable. A previous, less formal, version of the argument (Wright 1992: 41–3) misﬁres, since it shows only that the decidability in principle of all sentences follows from (1), combined with the universal applicability of bivalence. Cf. my discussion in Rasmussen 2002.

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of an explicit time-index on the epistemic K-operator. This measure has met with a favourable reception by others, as indeed it will in the present paper.

II Before turning to these various approaches, two points deserve mentioning. First, if the above basic version of the paradox is supposed to be effective against anti-realism, then it seems unavoidable that the semantical realist is equally vulnerable. True, the realist will not endorse (1), in general. However, if the quantiﬁer is restricted to, say, effectively decidable sentences of number theory, the case for his having to endorse (1), as thus restricted, is at least as strong as the case for holding that anti-realism resides in the general adherence to (1). At the same time, the realist has no quarrel with (2). On the contrary, to him (2) captures the truism that reality outstrips human knowledge. But if so, the basic version of the paradox is not suitable as a refutation of anti-realism, speciﬁcally. Furthermore, the point tends to reinforce the feeling that perhaps something has gone seriously wrong, probably with both (1) and (2), since the anti-realist is in any case not happy with (2). Second, and importantly for nearly everything in the sequel, once we realize that intuitionistic and classical logic are both in play, the question arises whether the paradox is supposed to be couched in one or the other. Since the derivation itself is valid in both, this might seem not to matter. But this is far from the truth, since the choice of logic has a bearing on the interpretation of the assumptions responsible for the paradox, viz. (1) and (2). Until 1982 it seems to have been presupposed that the logic is classical. But, as pointed out in the above, the anti-realist is likely to insist on an intuitionist construal of the logical constants. Once this is realized, the imputation to the anti-realist of (1) seems puzzling. Suppose attention is restricted to the ﬁeld of mathematics, where truth-values may be assumed to be monotonous.³ Assumption (1) appears to be an attempt to capture that, to the intuitionist, all mathematical truths are provable, in the ³ Famously, the defeasibility of empirical propositions gives rise to their anti-realistically conceived truth-values failing to be monotonous. Wright’s attempt to deal with this problem in terms of superassertibility (Wright 1993b: 414–16 and 1992: 48–61) seems to me to misﬁre. Either superassertibility is monotonous, in which case the notion appears to be uanavailable to the anti-realist. Or superassertibility does not generate monotonous appraisals. If so, superassertibility is hardly an improvement on garden-variety assertibility. The difﬁculty very nearly surfaces in Wright (1992: 54 n. 17). Wright seems to plump for the ﬁrst horn of the dilemma, since ‘p’ is supposed to be superassertible, only if we currently have a warrant for supposing that, as a matter of fact, a defeater will never turn up for the now assertible ‘p’; and Wright clearly assumes that, occasionally, such a warrant may be had. This notion is unlikely to curry favour even with the semantical realist. But he, of course, has no sleepless nights over the non-monotonicity of assertitibility, however construed, since he has available to him a nice monotonous notion of (possibly veriﬁcation-transcendent) empirical truth. Wright still seems to hold on to the notion of superassertibility being useful (Wright 2006: 56–9).

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absolute sense of ‘provable’ (i.e., not provable in some given formal theory, but quite generally). We may further assume that intuitionistic provability is in question. But Gödel has shown how to represent intuitionistic provability in classical modal logic S4. And now (1) looks suspiciously as a misguided attempt, using the Gödel–McKinsey–Tarski–Rasiowa–Sikorski mapping (Gödel 1933 and Troelstra 1986), to translate into classical modal logic S4 some sentence, ‘q’, which the anti-realist might wish to assert intuitionistically.⁴ But (1) maps no such ‘q’. Consequently, (1), or some descendant of (1) like (1∗ ) below, cannot be thought of as a rendering in classical logic of any intuitionistic claim—put forth, perhaps, for the beneﬁt of the realist, who might not otherwise know what to make of anti-realist sayings. But, equally, the point about the Gödel-mapping serves to alert us to the further fact that intuitionistic logical constants are already (weakly) intensional.⁵ This raises the question as to whether the anti-realist would wish to introduce intensional operators, as in (1), into an explicit, intuitionistically construed statement of his fundamental principle. If not, (1) fails, as a rendering of anti-realism, even when read intuitionistically. Furthermore, the fault seems to be structural in such a way as to tell against the aforesaid restriction strategy, as well as against Williamson’s overall response to the paradox, on behalf of the anti-realist. Call this dual worry the mapping objection. It should be stressed that the objection is not decisive against proposals such as Williamson’s (1992). However, the objection offers a diagnosis in terms of the genealogy of principle (1), which strongly suggests that the principle is an amphibious hybrid between the points of view of realism and of anti-realism.⁶ Let us rehearse what the mapping objection is. The anti-realist is supposed to face a paradox, preliminarily shaped as in (BA). Quite independently of what the anti-realist might make of assumption (2), we consider the fate of (1). The anti-realist has two general ways of taking assumption (1), according as the logical constants are read classically or intuitionistically. In consequence, his opponent ⁴ There are several such mappings, bearing out that ϕ is an intuitionistic theorem, if and only if the map, ϕ ∗ , is a theorem of classical S4. I use Gödel’s original preferred mapping: For atomic ϕ, ϕ ∗ = ϕ; (ϕ & ψ) ∗ = ϕ ∗ & ψ∗ ; (ϕ v ψ) ∗ = ϕ ∗ v £ψ∗ ; (ϕ → ψ) ∗ = ϕ → ψ∗ ; and (¬ϕ) ∗ = ¬ ϕ ∗ . The constants are intuitionistic, when ﬂanking ‘=’ to the left, classical when ﬂanking identity to the right. ⁵ Only weakly intensional, because although intuitionistic sentential logic is of course not truthfunctional, sequents such as the ‘paradoxes of material implication’ hold good intuitionistically. The S4-mapping (or the set of such mappings) brings out exactly how, and to what extent, intuitionistic constants are intensional, modulo our comprehension of the intensionality built into standard modal logic. ⁶ There is a question as to whether these considerations generalize to the non-mathematical case. There is no telling, as long as we do not have available to us an anti-realistically convincing non-monotonic logic for criterially based assertions, and for empirical generalizations. However, it is not unreasonable to expect that, once such a logic becomes available, it makes sense to search for a modal mapping into standard (not necessarily classical) logic analogous to the Gödel-mapping. There is even a research programme here: ﬁnd the non-monotonic logic that maps well into some reasonable modal logic.

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has a choice as to which reading to impute as the one to which the anti-realist is committed. From the realist point of view, the anti-realist therefore faces a dilemma. The anti-realist now confronts this dilemma by denying that either horn presents him with an insurmountable difﬁculty. Common to the anti-realist response to both parts of the challenge is that he, as an anti-realist, works in intuitionistic logic. The mapping objection just is the anti-realist’s response to the putative dilemma, in those terms. III The positive part of the anti-realist mapping objection takes assumption (1) as having been stated in classical logic. The idea then has to be that (1), or something very similar to (1), captures in the terminology of classical logic some principle central to anti-realist concerns, when stated intuitionistically. So, in short, (1) is a putative translation into classical logic, as reinforced by intensional operators, of an intuitionistically stated principle with a claim to the adherence of the anti-realist. We assume, here, that the resources of intuitionistic logic include nothing more than the usual intuitionistic logical constants. But if this is the picture, we already know, at least in the mathematical case, how to translate intuitionistic claims into intensionalistic classical ones: the Gödel-mapping gives us precisely what is known and—with a reservation—needed. The reservation is that this gives us no purchase on the case of non-monotonous anti-realist truth. But it is safe to ignore this case. If the realist wishes to challenge the anti-realist, the challenge must work, even where the anti-realist has the strongest and best understood case. This is the one of mathematics. The crucial point is that (1) cannot be taken as a plausible case of a classical translation of anything the anti-realist is committed to claiming intuitionistically. IV The negative part of the mapping objection is this. The anti-realist is now asked to accept that he is committed to endorsing a principle very close to (1), as construed intuitionistically. The anti-realist could reasonably claim that, when read intuitionistically, and no matter how we construe the non-standard operators, the assumption now seems to self-destruct by intensional overkill. After all, the standard intuitionistic logical constants are already (weakly) intensional, the intensionality being of an epistemically constrained sort to boot. The attempt to impute to him a commitment to (1) looks, from this point of view, as really a possibly misguided call for providing a better semantical account of the constants of intuitionistic logic. The matter is not just one of applying Occam’s razor to ensure the utmost paucity of basic logical operators. The whole enterprise looks

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like an attempt to patch up a job badly done, when the anti-realist endorsed intuitionistic logic as his standard of reasoning in the ﬁrst place.⁷ Nothing so far said, or to be claimed in the sequel, constitutes a decisive case against adding overtly intensionalistic operators to standard intuitionistic logic, either in general, or in order to hit upon a version of (1) acceptable to the anti-realist. For instance, the claim will not be that Williamson’s admirable efforts to do so are incoherent. At least if it is, this claim would have to be made good by a different kind of argument, in fact a proof to that effect.⁸ The state of play is, therefore, that the negative part of the mapping objection is highly suggestive of the thought that thus improving the arsenal of intuitionistic idiom lacks motivation, if the latter is supposed to derive from a need, on the part of the anti-realist, to face the putative paradox of knowability. On the other hand, the positive part of the mapping objection is decisive against the attempt to refute anti-realism by appeal to a classically construed version of (BA). This, quite apart from the time-indexing consideration, undercuts the presentations of the paradox in Fitch (1963), Hart (1979), and Hart and McGinn (1976). It is worthy of note in passing that the intuitionistic readings of the epistemiclogical principles (F) and (D) under the S4-mapping become the classically construed (FI) and (DI), respectively: (FI) (DI)

Kp → p K(p & q) → (Kp & Kq)

Both of these are valid, if (D) and (F) are valid in epistemic logic based on classical logic. For instance, on the assumption of ‘Kp’, it follows by (F), MPP and ‘p → p’ that p. By necessitation (justiﬁed by the assumptions both being S4—modal—trivially so in the case of ‘Kp’, and plausibly so for (F), which is a conceptual truth, if it is a truth at all), we arrive at ‘p’; whence (FI) by Conditional Proof. The derivation of (DI) is similar. V Now let us brieﬂy consider the claim that the paradox ought to bother the realist as much as it should the anti-realist. The problem was that the realist seems to have to insist on (2), while he must have (1), once the quantiﬁer-domain is restricted to effectively decidable sentences. The only reasonable reply seems to ⁷ Cf. Note 20 for an alternative view of the negative half of the mapping objection. ⁸ One might speculate that if there really is a need for Williamson’s epistemic modal intuitionistic logic, (1∗ ) as taken to be a formula in his logic, must be a translation as to provability of a formula in a hitherto unknown anti-realistically acceptable logic, ARL, which does not itself contain overt modal operators. The anti-realist might then accept (1∗ ) as a rendering, in Williamson’s logic, of a claim he wishes to make in ARL. But now we are speculating.

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be that, strictly speaking, the realist has neither. This is shown by extracting and making explicit the time-indexing in the K-operator, which is plausible in any case. The force of (1) ought to be that if p is the case, then there comes a time, t, such that p is possibly known at t (initial second-order quantiﬁers are usually left out, in the sequel): (1∗ )

p → (∃t)♦Kt p.

The reading of ‘K’ in (2) is however different, viz. as in (K) above. The point here is that there are true propositions not known now (at to ): (2∗ )

p & ¬Kto p

Since t may differ from to , {( 1 ∗ ) , ( 2 ∗ )} is consistent (quite apart from the modal world shift involved), and this result survives Fitch’s diagonalization move (substituting (2∗ ) for ‘p’ in (1 ∗ )). Note that a uniform reading of ‘K’ cannot be obtained by taking the operator to be really ‘Kto ’, as it occurs in (1), or by thinking of the time-index as somehow absorbed into the possibility-operator. The point of (1) cannot be that of afﬁrming that, if p, then it is possible to come to realize that we now know that p, the ‘now’ being rigid across alethically possible worlds. True, the unfortunate Jones may come to realize, truly, that he has really known since last week that his wife is unfaithful. But such cases of ‘having known all along’ are not pertinent to present concerns. The ‘K’ occurring in (1) is therefore parametrically time-indexed. And since this automatically calls for a binding by means of a quantiﬁer, the time-index becoming explicit as a variable in the process, the plausible construal of (1) is as in (1∗ ). The possibility of reading ‘K’ univocally in (1) and (2) as ‘(∃t)Kt ’ is considered below but ignored initially (cf. Section VIII below). Substantially the same point about the realist’s best stance to the putative paradox, as it concerns a domain of effectively decidable sentences, may be put in terms of an answer to the following query. The point of this is to highlight what the time-indexing mechanism is supposed to achieve in contexts such as the one under consideration. The objection to saddling the realist with even the restricted version of the paradox is that, to make this stick, the restriction to effectively decidable sentences, ‘p’, does not sufﬁce. The paradox-generating reasoning will get no grip, unless relevant instances of (2), with decidable ‘p’, are themselves guaranteed to be effectively decidable by the decidability of ‘p’ itself. Since we are now in the business of making good the claim that the realist seems to have a prima facie problem over some domain of sentences, whatever they be, if the anti-realist does generally, we are free to stipulate restrictions on the domain of the quantiﬁers occurring in (1) and (2). So, assumption (1) concerns, for immediate purposes, effectively decidable sentences only. In fact, (1) is supposed to hold truistically, in the sense that it simply states that

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we are now considering only such sentences as pass the positive test of being effectively decidable. Whether the truism can really be put in this way may be a moot point. But we are committing no injustice to the realist by assuming this, unless he himself has sinned grossly by saddling the anti-realist with (1). Assuming, then, that ‘p’ is effectively decidable, does it follow that ‘p & ¬Kp’ is, too? This obviously depends on the reading of the K-operator. Since we are employing the technical notion of effective decidability, let us lay it down that we are concerned with sentences representable in Robinson Arithmetic. Suppose, further, that ‘Kp’ is interpreted as, roughly, that we have a proof of ‘p’. This either means that somebody has a proof right in front of him. Or that there is a proof occurring on a ﬁnite list of proofs stored somewhere suitable. Or it means that a proof occurs on such a list or can be recursively generated from whatever we already have accessible to us, and so occurs as a member of a recursively speciﬁable list of (potential) proofs. On this last and weakest reading, ‘Kp’ is tantamount to our having access to a recursively enumerable, but possibly inﬁnite, list of proofs. On any of these readings of ‘K’, ‘Kp’ is effectively decidable. This does not depend on the decidability of ‘p’ itself. But the decidability of ‘p & ¬Kp’ does: the latter is effectively decidable, if ‘p’ is, on any of the three suggested readings of ‘K’. There are no other plausible readings of ‘K’, in the present context. On any of them, the required decidability of instances of (2) emerges. To assert (2) is to assert that there are effectively decidable truths the proofs of which occur on no currently available recursively enumerable list of proofs. Any set of the kind considered ‘forces’ at time t whatever has a proof in the set, St . And Sto therefore ‘forces’ whatever has a currently available proof. The time-index serves the sole purpose of indicating states of information permitting the assertion of the decidables in question. Presumably these states will admit of an ordering in terms of monotonous extensions over time. It deserves note that, even if ‘p’ is effectively decidable, it does not follow that ‘(∃t)♦Kt p’ is. Whether there is a proof in some admissible extension of our current stock of proofs depends heavily on how we construe admissibility. The question of a given ‘p’ having a proof occurring as a member in the union of all admissible monotonous extensions of the set of currently acceptable proofs of any sentence of a similar kind seems destined to be undecidable, even assuming that the set of proofs at any given stage is recursively enumerable. But if ‘p’ is supposed to be guaranteed to admit of effective proof, then such a proof there must be; and this is all the restricted version of (1) is supposed to claim. The upshot is that there is no paradox, from the point of view of the restricted realist.⁹ There are as yet unproven truths of which we can guarantee that a proof may be produced. But it certainly looked as if there had to ⁹ The ‘restricted realist’ is of course not restricted as to his realism. The restriction concerns the set of sentences for which the realist might seem committed to (1).

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be one, since (BA) was supposed to create trouble for the anti-realist, and since the challenge is structurally the same to realist and anti-realist alike. We also pick up from the above that time has no role to play, except—but crucially—as an index along an ordering of states of information. In the mathematical case, these may be assumed to increase monotonously over the index.

VI In the above section, we assumed pro tempore that the anti-realist faces a paradox. Furthermore, if so, the restricted realist would be up against a structurally similar problem. But the restricted realist can meet the challenge head on. Now, if time-indexing solves the problem for the realist in the restricted case, might not the same strategy work for the anti-realist, quite generally?¹⁰ This is what happens in Timothy Williamson’s epistemic, modal intuitionistic logic: (1∗ ) is retained, while the paradox is blocked by jettisoning (2∗ ) (Williamson 1992). Williamson provides a semantics for his sophisticated and complicated logic. The logic has a few somewhat weird features. However, I shall not discuss the details, merely put on record that I ﬁnd no incoherence in Williamson’s resulting position. On the contrary, Williamson’s 1992 proposal survives as a live option, for all I have to say. The worry is, simply, that he may be barking up the wrong tree, since the negative part of the mapping objection applies to (1∗ ), if it applies to the original (1). To spell this out more fully, one has to wonder about the motivation for introducing epistemic and alethic modal operators into an anti-realistically acceptable version of the claim that truth is epistemically constrained, once it is realized that the anti-realist will put his statement of the constraint in terms of intuitionistic logic. To repeat, the intuitionistic ‘standard’ logical constants are already intensional. Furthermore, the intensionality is inherently of an epistemic kind. To attempt anything along the lines of (1) therefore seems to be intensional overkill. Furthermore, the Gödel-mapping shows this up as not innocuous. (1) and its time-indexed descendants appear to be attempts to reﬂect in the object-language what is better left to meta-linguistic sayings. This is related to the superstition that the anti-realist, in contradistinction to the realist, has some special problem with the equivalence thesis for truth (ET):¹¹ (ET)

It is true that p, if and only if p.

¹⁰ Finn Guldmann took this line, as an unofﬁcial opponent at the oral defence of my Danish Doctoral Thesis (Habilitation) at the University of Copenhagen, October 2004. My response was mainly in terms of what I now call the mapping objection, which remains the basis of my quarrel with, inter alia, this approach. ¹¹ Cf. Rasmussen 2002.

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He is supposed to do considerably better, if he adopts, instead, the more anti-realist seeming equivalence thesis for warranted assertibility (EA):¹² (EA)

It is warrantedly assertible that p, if and only if p.

But ‘warranted assertibility’ is not, in general, a monotonous property. It is not, when we are dealing with, for instance, defeasible assertions. And so, the right-to-left reading of (EA) fails. And the converse implication may look ﬁshy as well, if we ignore the fact that the anti-realist will of course insist on an intuitionistic reading of the conditional. Suppose, then, that we lay it down that we are dealing with monotonous (assertions of) ‘p’, and that the bi-implication in (EA) is to be construed intuitionistically. It is notable that the right-to-left reading of (EA) now looks like something one might express more precisely, or at least symbolically, in the shape of (1∗ ), as read intuitionistically. If so, the anti-realist is in trouble over (EA), unless Williamson’s 1992 proposal is acceptable. However, the negative part of the mapping-objection suggests that the proposal is not in order. Furthermore, the worry seems to arise primarily from the attempt to represent in the intuitionistic object-language the meta-linguistic property of assertibility. In fact, it is difﬁcult not to read (EA) as not already an attempt to do just that. In the light of this it seems misguided to insist that (EA), in particular, captures the essence of anti-realism. It plainly does not constitute an improvement on (ET), even from the point of view of the anti-realist. On the contrary, anti-realist and realist alike can have (ET) as a regulative principle, with the usual (mostly Tarskian¹³) caveats. Their differences will come out in their (meta-linguistic) remarks concerning ‘truth’. But why cannot these differences be reﬂected at the same linguistic level, at some level? The mapping objection provides, or at least attempts to summarize, the answer. There is no doubt that the anti-realist wishes to say something amounting to ‘truth’ being epistemically constrained. So, following Wright (1992: 41), it seems safe to impute to the anti-realist adherence to (EC): (EC)

If p, then it is warrantedly assertible that p.

However, this is exactly what (1) and its descendants were supposed to capture all along. The positive part of the mapping objection points out that if the implication is taken classically, (EC) translates no claim the anti-realist would wish to put forth. On the other hand, if the implication is read intuitionistically, the sense of (EC) is, at best, that if ‘p’ is warrantedly assertible, then it is warrantedly assertible that ‘p’ is warrantedly assertible. That looks rather like ¹² My discussion to some extent draws on, without echoing, the one in Wright 1992: chs 1–2. Wright arrives at results quite different from mine. ¹³ Later developments (cf. the excellent discussion in McGee 1991) seem to me to be irrelevant to present concerns. In view of Dummett 1991: 72, I think it likely that Dummett would concur.

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an instance (for ‘It is assertible that . . .’) of the characteristic axiom of modal logic S4—p → p—and is hardly to be taken seriously as a shot at any speciﬁcally anti-realist insight. The negative half of the mapping objection is a way of saying that this kind of worry is unavoidable, however we tinker with (1). In addition to the worry over (1) in Williamson and in general, a problem arises over the anti-realist standing of (2) and its cognates. The latter worry is best discussed in connection with a proposal put forth by Bernard Weiss, which has the immediate advantage over its competitors that, on the proposal, (1) is rewritten in a way such as to pre-empt the mapping objection.

VII Weiss proposes, on the anti-realist’s behalf, to substutute (1W) for (1): (1W)

p → ¬¬Kp

The logical constants are of course to be construed intuitionistically. From (1W) the negation of (2) follows easily in intuitionistic logic: (N2)

¬(p & ¬Kp)

So, anti-realism, sc. (1W), and (2) trivially lead to contradiction, and no appeal to the Fitch substitution is called for in the proof. The proposal has a good deal to recommend it. First, it is remarkably simple. Second, the mapping objection fails to get a grip, since (1W) exploits the intensionality of intuitionistic constants. No need for modal operators, as intuitionistic negation is already intensional. Third, the anti-realist certainly does wish to claim that if p, then it is absurd to rule out that ‘p’ might one day admit of proof (become ‘forced’); and this appears to be the content of (1W). Yet all is not well with the proposal.¹⁴ The need for disambiguating ‘K’ by time-indexing is still in force. Once these indices are introduced, we arrive at: (1W∗ )

p → ¬¬(∃t)Kt p

and (N2∗ )

¬(p & ¬Kto p)

(N2∗ ) does not, however, follow from (1W∗ ). It is true that (N2∗∗ ) does follow: (N2∗∗ )

¬(p & ¬(∃t)Kt p)

¹⁴ I should stress that I know of Weiss’s proposal only from a brief account in a letter from Michael Dummett, dated 5 June 2005. It has been suggested to me that Williamson broached a similar idea somewhere. That may be. For the purposes of the present paper, I take Williamson’s position to be the one emerging in those of his contributions adverted to in the text.

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The latter is however not the intended reading of (N2).¹⁵ So, we seem to be back to relying on the time-indexing for doing most of the work for the anti-realist struggling to establish that his position is at least coherent. To forestall a possible objection to the above, suppose that Weiss’s proposal ought to read that (1W∗∗ ) leads to (N2∗ ). (1W∗∗ )

p → ¬¬Kto p

This has the merit that (1W∗∗ ) really does entail (N2∗ ) intuitionistically. However, it seems unlikely that the precisiﬁcation (1W∗∗ ) is the intended reading of (1W). Its import is that if ‘p’ is true, then it is refutable that ‘Kto p’ will never turn out to be ‘forced’. So, it is incoherent to rule out now that we didn’t know that p all along. To revert to the case of Jones and his unfaithful wife, it is plausible to claim incoherently that he will never come to realize that he really knew now (rigidly construed). For instance, Jones may come to realize on Thursday the truth that his wife has been unfaithful since Tuesday. And, supposing that his wife really is unfaithful, it is probably refutable on Tuesday to rule out that Jones will not realize the fact on Wednesday. For there must be evidence, concerning both the wife and Jones himself, which cannot be ruled out to come within the ambit of Jones’s cognizance. As before, such cases are however of no relevance to present concerns. A case closer to those is the following. Suppose that the intuitionist mathematician accepts that Andrew Wiles ﬁrst proved Fermat’s Last Theorem. So he accepts that the theorem is true, but that this was not known in, say, 1960. In fact, it is decidably false that the theorem was known in 1960. But if so (1W∗∗ ) and the facts generate a contradiction: it is both the case that ‘¬K 1960 p’ and (by modus ponens) that ‘¬¬K 1960 p’, with the relevant substitution for ‘p’. It will perhaps be objected that this application, with modus ponens, of (1W∗∗ ) was not available in 1960, on the ground that the relevant instance of ‘p’ was not then assertible. True, but in those days the consequent of (1W∗∗ ) was decidably false, as indeed it remains to this day. The fate of the Fermat conditional would then seem to depend on the status of ‘p’, i.e., of the truth status of Fermat’s theorem itself. In 2008 the theorem is taken to be true, and the conditional consequently false. In 1960, by contrast, the conditional had no truth-value. In conclusion, (1W∗∗ ) is not what the anti-realist wants; and (1W∗ ) ought to be adopted as our ofﬁcial reading of (1W). Now, Weiss plainly is quite happy with anti-realism entailing (N2), by which he presumably means (N2∗ ). But it is far from clear that the intuitionist wishes to endorse this position. Let us review what the anti-realist has to say about (2∗ ), (N2∗ ), and (16∗ ): (16∗ )

p → Kto p.

¹⁵ The proposal of adopting this reading is nevertheless discussed below, in Section VIII.

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And bear in mind the following: it is impossible to say within the ordinary language of intuitionistic mathematics that a [true, SAR] proposition has not yet been proved. (Dummett 1991: 78)

Dummett’s word ‘say’ turns out to be a weasel-word, in the present context. It might seem that (2∗ ), existentially quantiﬁed, expresses exactly that there are as yet unknown truths. I think we should allow that it does. For the quantiﬁed (2∗ ), construed constructively, translates into (G2∗ ): (G2∗ )

p & ¬Kto p

on the Gödel-mapping introduced above (Section II). (G2∗ ) is neither contradictory nor nonsensical. So, the proposition expressed by (2∗ ) is intuitionistically coherent. But (2∗ ) can never be asserted by anti-realist lights, for any interesting ‘p’. On Heyting’s informal semantical account of the intuitionistic constant, we can assert a conjunction, only if we are in a position to assert both conjuncts simultaneously (Heyting 1971: Ch. VII; and Dummett 2000: Section 1.2); and we patently cannot afﬁrm that ‘p’ is not now known, while also maintaining that ‘p’ is now assertible. It is no use tinkering with not presently known assertibles, as is clear from the mathematical case. There is no way of coherently maintaining that we have an assurance of the provability of a mathematical proposition, of which we can rule out that we have current knowledge, sc. proof. And so, the intuitionist cannot, and does not, assert that there are stable counter-examples to (16∗ ), although he obviously has no wish to assert that principle. By the same token, he will not wish to assert Weiss’s (N2∗ ), i.e. the negation of (2∗ ). Notice that the intuitionistic non-assertibility of (2∗ ) does not ensure that (N2∗ ) is also non-assertible. (2∗ ) is not intuitionistically contradictory. The point about (N2∗ ) may therefore perhaps be put in terms of Dummett’s notion of the ingredient sense of a sentence (Dummett 1991: 47–50). The ingredient sense of (2∗ ), as it occurs as the antecedent in ‘If (2∗ ), then q’ (for any sentence ‘q’) is perfectly coherent. The conditional might well be intuitionistically assertible, for a suitable choice of ‘p’ and ‘q’. The same then goes for the negation of (2∗ ), since the negation of ‘p’ is deﬁnable as ‘p →⊥’. So, even if (2∗ ) is never intuitionistically assertible, its negation might well be. It should be borne in mind that the interesting cases of sentences, ‘p’, are those that are anti-realistically problematic, i.e., sentences such that we can issue no guarantee that we shall ever be in a position to decide them either way. Bivalence, conceived of as the schema (BIV) (BIV)

Either ϕ is determinately true or ϕ is determinately false

is not applicable to such sentences, according to the anti-realist. On the other hand, the adherent to anti-realism does not envisage that the sentences in question constitute permanent, stable counter-examples to (BIV). Semantical anti-realism resides in agnosticism as regards the applicability to such sentences, non-KED

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sentences, using an abbreviation for ‘not-Known-to-be-Effectively-Decidable’ suggested by Jens Ravnkilde (Rasmussen and Ravnkilde 1982: section 3 and ‘Appendix’; and Rasmussen 1990: section 2). What is involved here is of course not the Church/Turing notion of effective decidability, or not just that notion. If a sentence is effectively decidable in that sense, it is KED in our terms. The converse implication however fails. We follow what we take to be Dummett’s view: a sentence is KED, just in case we are in a position to issue a present guarantee that nothing debars us, in principle, from deciding its truth-value; and the sentence is non-KED if and only if we have no present guarantee that we shall ever be able to decide it, yet are not in principle debarred from doing this.¹⁶ In the case of the anti-realistically problematic sentences, we are dealing with non-KEDs. Suppose, then, that ‘p’ is an anti-realistically problematic sentence, a non-KED sentence. Since we are able to decide, at each given point in time, whether or not we know that p, at that time, ‘Kto p’ is decidable. In fact, the latter is ex hypothesi false, in the case before us. But then the fate of (2∗ ) as to decidability is determined exclusively by that of ‘p’. In consequence both (2∗ ) and its negation (N2∗ ) are non-KED sentences. Among the entailments of this is that Weiss’s principle (N2∗ ) does not hold in the relevant cases. This, by the way, does not bring in its train that (1W∗ ) must be abandoned as well. For, although (1W∗∗ ) entails (N2∗ ), (1W∗ ) does not.

VIII It was claimed above that (N2∗∗ ) is not the intended reading of (N2). This is historically correct. However, suppose we stipulate (N2∗∗ ) as the intended reading. This is tantamount to taking (2), in the basic version of the paradox, as really amounting to (2∗∗ ): (2∗∗ )

p & ¬(∃t)Kt p

¹⁶ Occasionally, Dummett has treated of sentences of which we can presently rule out that they will ever be decided, as if they were of the sort to make the anti-realist frown upon attributing to them a determinate truth-value. But such sentences are not non-KED. Rather they monotonously lack both truth and falsity, hence occupy the territory of either possessing a third stable truth-value (in the manner of past-tense sentences with Lukasiewics) or lacking a determinate meaning (the Continuum Hypothesis of set theory would be a candidate). The sentences that engage our interest are however fully determinate as to meaning and lack a determinate truth-value. I believe that I am in genuine disagreement with Dummett over how best to capture what characterizes the kind of sentence of special interest to the Dummettian anti-realist. This may appear a bit steep. Surely, Dummett knows best what is a Dummettian anti-realist? The trouble arises because there are two ways in which to characterize the Dummettian anti-realist. The positive way, which is the one I adopt, is that of regarding the anti-realist as a generalized intuitionist, the latter thought of in Dummettian terms. The negative way is the one often taken by Dummett, especially since around 1980. According to this account, the anti-realist is anybody who deviates from Fregean semantic realism. However, when it comes to the question of how to deal with sentences of speciﬁc kinds, there is probably no substantial disagreement between Dummett and myself. Pursuing these matters will in any case take us too far aﬁeld.

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This interpretation would be in line with the perception in some quarters that the upshot of the paradox is to the effect that if all truths are knowable, then all truths are actually known at some stage by somebody. Now the introduction of the bearers of knowledge is, as before, just inconsequential clutter. Also, even if the paradox is seen as a reductio of (2∗∗ ), there is no intuitionistic transition to (16∗∗ ): (16∗∗ )

p → (∃t)Kt p

Consequently, if there is a threatening version of the paradox in the ofﬁng here, it has to be because of the fate of (2∗∗ ). Two questions present themselves. (i): What is the intuitionistic status of (2∗∗ ) and (N2∗∗ )? And (ii): Is (2∗∗ ) actually inconsistent with the anti-realist’s preferred version of (1)? Now if (1) is interpreted along the lines of Weiss’s amended (1W∗ ), then (N2∗∗ ) unquestionably follows. This is all right, it seems, since the intuitionist can hardly claim that we can know, of a speciﬁc ‘p’, that nobody will ever know that p; and afﬁrm, in the same breath, that p. So we may rule out the intuitionistical assertibility of (2∗∗ ); and, as noted above, the intuitionist certainly would wish to assert (1W∗ ) in any case. In all of this, no semblance of a paradox surfaces. If, on the other hand, we proceed from assumption (1∗ ) and (2∗∗ ), no formal contradiction emerges, give or take the Fitch-substitution. The worst we get is the conjunction of ‘(∃t)♦Kt p’ and ‘¬(∃t)Kt p’, and these evidently form a consistent set, unless we interpret ‘possibility’ as ‘realized at some accessible actual time’. However, if we were to interpret the diamond according to this suggestion, (1∗ ) would amount to (16∗∗ ). But the whole point of the paradox was to point out trouble for the anti-realist he did not realize he faced. If he is supposed to adopt (16∗∗ ) as an assumption, the paradox becomes pointless. Parallel remarks apply to the realist who combines adherence to (1∗ ), as restricted to the case of decidable sentences, with an endorsement of (2∗∗ ). Occasionally, it is suggested that the intuitionist, perhaps the anti-realist generally, is in any case in trouble over (2∗∗ ). Perhaps all the above shows impeccably from intuitionistic principles that (2∗∗ ) can be ruled out. Still, the suggestion goes, there are cases where this must be wrong. The matchbox right in from of me contains a number n of matches. This is perfectly decidable. Hence, classical logic applies: either the number of matches equals n, or it does not. But nobody ever bothers to count the matches. Instead, a demented child dispenses with the matchbox forever by throwing it into the open ﬁre. A body-count of matches is henceforth out of the question. Yet it is pre-incident true that there is n matches in the box, yet nobody will ever know the size of n. So, we have a clear-cut case of an instance of (2∗∗ ) on our hands; and even the anti-realist ought to be able to assert as much.¹⁷ ¹⁷ I owe the example to an anonymous referee. Dummett’s well-worn example of the late Jones who never in his lifetime had an opportunity to acquit himself with bravery, or otherwise, would

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It deserves note in passing that, if this were all so, anti-realism would be in trouble independently of the alleged paradox under consideration. But it is not so. Realist intuitions have been allowed to interfere with the description of the alleged trouble-case. There is, here, an anti-realistically illicit appeal to the distributivity of counterfactual antecedents through disjunction; or an appeal to truth-value links of dubious anti-realist merit. Possibly both. Of course the anti-realist is not going to allow the derivation from¹⁸ If anybody had counted, then the number of matches would have turned out to be either n or not, to If anybody had counted, the number of matches would have turned out to be n or If anybody had counted, the number of matches would have turned out different from n. serve equally well (Dummett 1963: 147–50). Can we now assert that Jones was brave? That he was not? Dummett unfortunately comes down on the side of committing the anti-realist to the latter, on the ground that we will never know. The anti-realist should sternly remain agnostic about the truth and falsity of either claim, because we cannot rule out that we will come to know of Jones’s bravery (or the opposite) indirectly. If we can rule this out, the proposition that Jones was brave is not anti-realistically problematic, just decidably false (or true). Incidentally, there are even worse cases. Consider the case of a cat in a box, the latter being wired in such a way that the (potential) cat drops out through the bottom whenever the lid is raised. Even supposing that there now is a cat in the box, we shall never know by any obvious means (cf. Percival 1990 and Melia 1991 for similar examples). The proper anti-realist response to such purported straight counter-examples to (1) as ‘There is a cat in the box’ is, as before, that if the sentence is to be thought of as non-KED, it just lacks a truth-value. If we decide that the predicament would appear to be permanent, we must introduce a third stable truthvalue. I have elsewhere (Rasmussen 1997: 149) called such sentences veriﬁcationally inconsistent: the very attempt to verify them turns them into falsities, or at least inﬂuences whatever truth-value they might have possessed unexamined. Such sentences are generally of little interest anti-realistically, although nice questions of a related sort turn up in connection with the interpretation of quantum mechanical measurements. However, in the quantum-mechanical kind of case the mechanism responsible for affecting the truth-value seems to be not just unknown, but perhaps absent. ¹⁸ This sort of case has been well researched, in mathematical as well as in empirical contexts. Vide Dummett 1963, 1973, 2000: 267–9; Wright 1980: ch. XII; Rasmussen and Ravnkilde 1982: Section 3; and Williamson 1988a. The counterfactual fallacy involved receives attention quite independently of issues to do with anti-realism in Lewis 1973: Section 3.4. However, Conditional Excluded Middle surely fails anti-realistically. The paramount concern in contexts such as the matchbox case is to avoid turning non-KED sentences into presently decidable falsities. The temptation is obvious: since the child burned the matchbox, we shall never know the number of matches. But if this it what we think, the sentence is anti-realistically false, hence not problematic. It has been relegated to the domain of the decidable. The truth of it is, however, that we cannot now rule out future availability of evidence about the already discarded box, which will support that the number of matches contained in it was likely to have amounted to some speciﬁc number. For instance, we cannot rule out that some EU bureaucrat has kept a record that will some day turn up. Or some such thing. The story does not itself have to be likely for the sentence to stay anti-realistically problematic.

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And so, the requisite ‘p’ (‘The number of matches equals n’) admits of neither anti-realist assertion nor denial in the circumstances. Nothing like an assertible instance of (2∗∗ ) is in the ofﬁng. The presumably only alternative way of making obligatory the anti-realist assertibility of the requisite instance of (2∗∗ ), would be via truth-value links between the past and present tense statements. Since it is now true that the matchbox contains either n matches or not, it must be true tomorrow that the now discarded box yesterday contained either n matches or not. This too has been looked into.¹⁹ The tense-theoretic anti-realist plainly has a problem with defending the truth-value links customarily accepted. The usual tack adopted is either to hold this against the anti-realist, or to struggle to defend the anti-realist’s entitlement to the habitual links. Although I cannot here enter into the matter, my line would be that the anti-realist cannot in general expect to maintain these links, just as he cannot adopt the principle of bivalence. For present purposes, this means that there is no reason why the anti-realist should be committed to the assertion of anything to do with the number of matches in yesterday’s discarded box of matches. It is decidably true now that the number of matches in the box equals n, say. So, bearing in mind that our concern is now with tense, rather than with mathematical decidability, had we counted, we would now have come up with that result. But we did not count, and the box has been discarded. The number is therefore henceforth no longer a decidable issue. Nobody will ever know, at least not by the direct expedient of counting. But if so, we can no longer claim that there must be a fact as to what the number was. The requisite ‘p’ has become devoid of truth-value on temporal grounds, despite the mathematical decidability of the original question. Yet again, no anti-realistically assertible instance of (2∗∗ ) threatens. Summing up some of the above, (2∗ ) and (N2∗ ) are (complex) non-KEDs. And therefore (N2∗ ) is too strong for the anti-realist’s purpose of ruling out counter-examples to (16∗ ). The impression that this must be so is reinforced by the following consideration. On their own, neither (2∗ ) nor (N2∗ ) is intuitionistically incoherent. This is easily shown by running classical S4-trees on their classical S4-maps. There must be something wrong with the strategy of trying to frame an anti-realist principle along the lines of (1), which is supposed to capture Heyting’s informal semantical account of the intuitionistic constants, and yet is such as to contradict (2). Once again, (N2∗ ) is too strong for the anti-realist’s purpose. Note the surviving point that the antirealist will wish to assert (N2∗∗ ), on grounds not connected with the paradox. This means that if a paradox is still in the ofﬁng, it must have among its assumptions (2∗ ). ¹⁹ Cf. Dummett 1969 and McDowell 1978.

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IX Turning now to the restriction strategy, we shall brieﬂy consider two brands of this broad approach. The two agree in putting restrictions on the antirealist’s chosen variety of assumption (1)—(1∗ ), say—in such a way that the Fitch substitution is inadmissible. To consider ﬁrst Neil Tennant’s proposal, his restriction is to the effect that (1∗ ) holds anti-realistically only for so-called Cartesian ‘p’. A sentence, ‘p’, is Cartesian, just in case it is not anti-realistically incoherent to assume that ‘p’ is assertible. Crucially, (2∗ ) is not Cartesian. Note, ﬁrst, that Tennant’s position is of course vulnerable to the mapping objection, just as was Williamson’s. In addition, the position has speciﬁc weak points of its own. Tennant’s claim that (2∗ ) is not Cartesian seems compatible with the above discussion: (2∗ ) really is not anti-realistically assertible. However, Tennant is concerned to reﬂect a kind of intuitionistic assertibility in the object-language, and therefore proposes the following as a formal rendering of Cartesianism in sentences (Tennant 2002: 136) (T): A Cartesian proposition is a proposition p—of any syntactic complexity—such that Kp is [intuitionistically, SAR] consistent. But ‘K(p & ¬Kto p)’ does not appear to generate a formal contradiction, unless the preﬁxed ‘K’ is construed as ‘Kto ’. The intended reading, however, ought to be ‘(∃t)♦Kt ’, since otherwise Tennant fails to engage with the ﬁrst assumption in the paradox-generating reasoning, which we agreed to construe as in (1∗ ). For this is the version of the assumption which occurs conditionally asserted in the ﬁrst proof-line. But is not all of this refuted by Tennant’s explicit formal derivation of a contradiction from ‘K(p & ¬Kto p)’ (Tennant 1997/2002: 259–60)? No. Tennant ﬁrst derives a contradiction from the joint assumption of ‘p & ¬Kp’ and ‘K(p & ¬Kp)’. He then applies his rule (I) to the effect that ‘, Kp ⊥’ is derivable from ‘, p ⊥’, to arrive at ‘¬K(p & ¬Kp)’. Tennant’s particular way of displaying this within the framework of Prawitz-style natural deduction may cause some confusion. For clarity, here is what happens, written out in full sequents: (S1) K(p & ¬Kp), p & ¬Kp ⊥ (D) and & E (S2) K(p & ¬Kp), K(p & ¬Kp) ⊥ (S1) (I), with = {K(p & ¬Kp)} (S3) K(p & ¬Kp) ⊥ (S2) Deletion of repetitions This seems somewhat roundabout, but at least the proof nowhere commits Tennant to claiming that ‘p & Kp’ itself is contradictory; and he (perhaps deliberately) avoids appeal to the factivity of knowledge, (F). The latter is appealed to in the more direct derivation of a contradiction from ‘K(p & ¬Kp)’

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by (D), &E, and (F). In any case, suppose someone were to apply Tennant’s medicine to his own case. His rule (I) seems plausible enough: surely, we cannot know a contradiction. (By the way, this is probably because of factivity of knowledge.) It might be argued, however, that the rule should be restricted to Cartesian assumptions, ‘p’. The rationale behind this is a wish to eliminate trivial applications of (I). Obviously, any contradiction derivable from the joint assumption of ‘Kp’ and ‘p’ must be ascribable to ‘Kp’. It is not apparent why this line of thinking should be less plausible than Tennant’s restriction on (1). Indeed, since the latter would be plausible only as a general restriction on admissible assumptions in proofs to Cartesian ones, perhaps we are dealing with the very same restriction in an only slightly different proof-context. If this were the only worry raised by Tennant’s proof, it would be alleviated by the fact already adverted to that a contradiction is derivable from ‘K(p & ¬Kp)’ in ways not vulnerable to the objection just ﬁelded. The salient point is that Tennant fails to take due account of the fact that the fate of ‘K(p & ¬Kp)’ is irrelevant to the paradox, unless it is construed as ‘(∃t)♦Kt (p & ¬Kto p)’, in which case it does not entail a contradiction. Impressed by the above, Tennant might rescind from the attempt to reﬂect ‘assertibility’ in the object-language and make do with the informal (C): A Cartesian proposition is a proposition p, such that it is intuitionistically coherent to assert that p. The proposal has a certain air of ad hocness about it. This would have to be remedied by a case being made for the claim that, quite generally, an assumption, ‘p’, is inadmissible in intuitionistic natural deduction, if the assertion of ‘p’ is intuitionistically incoherent. This must be so even in cases, such as the present one, where the assumption is possibly to be discharged by reductio ad absurdum. Since RAA is perfectly respectable in intuitionistic logic, an assuption is of course not to be dismissed as inadmissible, on the sole ground that it gives rise to a contradiction, either in isolation or in the context of the surrounding argument. It is, rather, the intuitionistic non-assertibility of the assumption that is responsible for its inadmissibility. However, this is a puzzling suggestion. As the Gödel-mapping vividly reveals, the intuitionistic derivation of a contradiction from assumption ‘p’ corresponds exactly to the classical S4-derivation of a contradiction from the assumption that it is intuitionistically provable that p. And, surely, provability is the strongest form of assertibility in the market. Hence, if—as is the case—the intuitionist cannot derive a contradiction from (2∗ ), he seems to be excused for regarding that assumption as perfectly Cartesian, in any sense congenial to Tennant’s proposal. This bit of reasoning suggests that, to give the requirement of Cartesianism the right sort of bite in the case under consideration, there is a need to represent the notion of intuitionistic provability, or assertibility, in the intuitionistic

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object-language. Indeed, Tennant’s (T) might be regarded as an attempt to do something of the sort. However, as pointed out, (T) does not appear to be equal to the task. Finally, we should not let drop out of sight that the time-indexing of the K-operator seems to remove altogether the need for Tennant’s restriction on (1); or that, give or take the restriction, (1) and (1∗ ) are seen to be suspect in the light of the mapping objection.

X Dummett’s brand of restrictionism is more subtle (Dummett 2001). I guess his point of departure is that just as a semantical realist is under no compulsion to frame his theory of meaning in the straightforward manner of imposing the Tarskian T-scheme universally on every sentence of the object language; so the anti-realist is not forced to adopt the principle encapsulating his epistemic constraint on truth directly on each and every such sentence. On the contrary: pursuing this strategy is apt to ensue in a rather uninformative account of meaning (Dummett 1991: 25–7). We may, however, assume something akin to (1) for basic sentences. The generalization to all meaningful sentences would have to be earned recursively. Now, as has been pointed out by Tennant, Dummett is not clear about what is meant by a ‘basic’ sentence (Tennant 2002). His invocation of the recursiveness of the expected overall theory of meaning strongly suggests that he means basic sentences to be atomic. It is however not clear whether all atomic sentences are to count as basic. However that may be, nothing with a structure akin to (2) may be counted as basic. Consequently, Fitch’s derivation fails to take off. Dummett’s account clearly does not fall foul of the requirement of non-ad hocness. However, it is not clear that his strategy will be effective against what is assumed to be the otherwise paradox-generating assumption (1). The anti-realist seems to face a dilemma. Either he thinks he can recursively earn an entitlement to the fully general ﬁrst assumption in the paradox-generating argument, in which case he is back where he started. Or he envisages the possibility that (1), or some suitable descendant of (1), will not be forthcoming. This would have dramatic repercussions for the entire positive anti-realist programme in the theory of meaning (Wright 1993a: Section V), as well as for the signiﬁcance of that programme for metaphysics. The former concerns how to systematically implement in a theory of meaning the idea that truth is epistemically constrained. Concerning the latter, the anti-realist programme for the theory of meaning has no immediate metaphysical bearing, save through the conception that the meaning of non-KED sentences is fully determinate, despite the fact that such sentences are not determinate as to truth-value because of epistemic constraints on the concept of truth.

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XI The upshot, then, is this. The realist, restricted or not, can hold on to there being hitherto unknown truths. But he can make his position clear only by using timeindexing or some device tantamount to this. It is tempting for the anti-realist to attempt a similar ploy. This just might work, along the lines suggested by Williamson. But all such attempts must face the charge of implausibility derived from the negative half of the mapping objection. The restriction strategy, in the manner of Tennant or Dummett, turns out to be most likely a distraction, from the anti-realist point of view. Put differently, the anti-realist status of (2∗ ) is that of being a non-KED sentence, i.e., (2∗ ) is not assertible, but nor is (N2∗ ). On the other hand, (N2∗∗ ) is intuitionistically assertible. In view of how the Gödel-mapping works, whatever exact assumption one puts for (1), the reading of it must be intuitionistic. But the mapping objection presents an obstacle to some proposed versions of that assumption, although the objection does not in itself prove these incoherent. Weiss’s suggestion, as improved to (1W∗ ), is not vulnerable to this particular objection and in fact remains a principle that the intuitionist would wish to adopt, quite independently of the intricacies surrounding the Paradox of Knowability. The relevant reading of (N2) does not now follow; but this is just as well. Moreover, the specimens of the restriction strategy brieﬂy considered in the above display weaknesses of their own. Consequently, (1W∗ ) and the rightly construed (1∗ ) survive as at least coherent candidates for the status as the best, sc. strongest, anti-realist descendant of (1). Notably, both crucially involve time-indexing of the K-operator. The dialectical ﬂow in the above, especially as it concerns the mapping objection, has been as follows. The realist, in the guise of Fitch and others, originally presented the anti-realist with the obstacle of somehow overcoming the apparent paradox resulting from assumptions (1) and (2). The anti-realist was supposed to share common ground with the realist over (2), that is, over the assumption of there being currently unknown truths. The trouble for the anti-realist must then arise over assumption (1), which is in any case supposed to encapsulate the gist of anti-realism. The mapping objection is presented as part of the anti-realist’s defence against the challenge that (1) gives rise to a paradox. The anti-realist thought is that (i) either (1) or some suitable descendant in the vein of (1∗ ), with or without restrictions, is supposed to be couched in classical terms. But if so, the challenge has no bite, unless the classical formula translates something he wishes to claim intuitionististically. The positive part of the mapping objection points out that (1), and its cognates, map nothing the anti-realist would wish to claim. So far, then, the realist is arguing beside the point by attempting to saddle the anti-realist with adherence to a thesis he does not hold.

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Or (ii), the realist intends the anti-realist to accept (1), as construed intuitionistically. Since it will have to be agreed on all hands that time-indexing is needed in any case, the realist is now imputing to the anti-realist adherence to the intuitionistically construed (1∗ ), with or without restrictions. This is where the negative part of the mapping objection comes to the fore. The anti-realist will quite naturally claim that in adopting intuitionistic, as opposed to classical, logic as his base logic, he has already intensionalized logic to the extent necessary to state any claim central to his concerns. He does not, that is, wish to be forced to introduce additional modal machinery for the purposes of meting the realist’s challenge to his central tenets. He is not, therefore, forced to accept (1) or (1∗ ), even when construed intuitionistically. He would not, then, face the paradoxical challenge, even supposing that the paradox survives when transposed in a time-indexed, intuitionistic version. As Williamson has pointed out, and as transpires from the above considerations concerning the anti-realist status of (2), it is even doubtful that the paradox does survive, when thus transposed. The point, as far as the negative part of the mapping objection is concerned, is that even if the paradox did survive, the anti-realist should not worry unduly over this: he most likely does not wish to endorse (1∗ ) anyway. So, the anti-realist has successfully rejected the challenge posed by the apparent paradox.²⁰ The question remains as to how, then, he should characterize his position, in his own terms. I have offered no reason to think that he could not coherently do so by means of Williamson’s (1∗ ). But the negative part of the mapping objection strongly suggests that he should not wish to do so. An anternative way would be Weiss’s amended (1W∗ ). The anti-realist is committed to this, certainly. However, (1W∗ ) seems to be too weak to capture all the anti-realist is out to say about the epistemic constraint on truth. The suggestion, in the above, is that whatever else is to be said is best left to meta-linguistic sayings. ²⁰ There is another way of taking the proposals of Williamson, Tennant, and Dummett, as seen in the context of the mapping objection. I suspect many will see them as attempted straightforward anti-realist replies to the challenge raised by the paradox. This point of view is not altogether implausible, on the assumption that the anti-realist is in any case committed to something like (1). But is he? Be that as it may, the overarching conclusion will stand up: the restriction strategies are distractions, and Williamson’s proposal is a non-mandatory, albeit a seemingly coherent, anti-realist response to the original challenge. Furthermore, the latter proposal ought not to hold much prima facie attraction to the anti-realist, for the reason encapsulated in the negative half of the mapping objection. It is however true that, from the perspective of this overall view of the debate, the negative half of the mapping objection will be seen as a part of the realist challenge to the anti-realist’s proposals for straight solutions, rather than as a contribution, along with the positive half of the objection, to the anti-realist’s defence against the realist’s onslaughts.

6 Truth, Indeﬁnite Extensibility, and Fitch’s Paradox Jos´e Luis Berm´udez

I Fitch’s original presentation in Fitch (1963) of the line of argument that has come to be known as Fitch’s paradox begins with the notion of a truth class of propositions. A class α of propositions is a truth class just if, as a matter of necessity, every member of α is true. That is, (1) ∀p[p ∈ α ⇒ p] Suppose that α is a truth class closed under conjunction elimination and consider the proposition (2) p & ¬(p ∈ α) asserting that p is a true proposition that is not a member of α. Assume, for reductio, that (2) is included in α: (3)

[p & ¬(p ∈ α) ] ∈ α

Since α is closed under conjunction elimination we have (4) p ∈ α and (5)

[¬(p ∈ α) ] ∈ α.

Since α is a truth class, (5) gives (6) ¬(p ∈ α) Thanks to two anonymous referees for comments on an earlier draft.

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which contradicts (4) and shows that, as a matter of necessity, (3) cannot be true. That is (7) ∀p ¬♦[ (p ∧ ¬(p ∈ α) ) ∈ α] This is Fitch’s Theorem 2. Now, let α be the class of known truths, where q is a member of the class of known truths just if there is a time at which q is known by somebody. Plainly α is a truth class, so that (1) holds. On this interpretation (2) states that p is an unknown truth—i.e. that there is no time at which someone knows p. Suppose we assume, as seems highly plausible, that there is at least one unknown truth. Let that be p. It follows that it is true that [p & ¬(p ∈ α)], which is our (2). But Theorem 2 shows that it is contradictory to suppose that there is a time at which someone knows that [p & ¬(p ∈ α)]. So, not only is there at least one truth that is unknown, but at least one proposition that is unknowable. What has come to be known as Fitch’s paradox derives essentially from Theorem 5 in the same paper. Theorem 5 states that, provided we accept the existence of an unknown truth, it cannot be the case that all truths are knowable. This is supposed to be paradoxical because there are well-established philosophical positions that maintain precisely the claim that Fitch shows to be incoherent, namely, that it is true both that there is at least one unknown truth and that all truths are knowable. Any form of veriﬁcationism or semantic anti-realism appears to be committed to the general principle that all truths are knowable, while no anti-realist or veriﬁcationist is likely to accept that all truths are known. However, accepting that all truths are known seems to be the only alternative to denying that all truths are knowable. There is a familiar response to Fitch’s paradox. It has been pointed out by a number of authors (originally in Williamson 1982) that the argument from the knowability principle that all truths are knowable to the omniscience principle that all truths are known is not intuitionistically valid. Suppose we formulate the knowability principle as (8) ∀p [p ⇒ ♦(p ∈ α) ]. We can substitute the assumption that there is at least one unknown truth into (8) to give (9) ∀p [ [p ∧ ¬(p ∈ α) ] ⇒ ♦( [p ∧ ¬(p ∈ α) ] ∈ α) ]. We recall Theorem 2 (7) ∀p ¬♦[ (p ∧ ¬(p ∈ α) ) ∈ α] Trivially, (7) and (9) jointly yield (10) ∀p ¬[p ∧ ¬(p ∈ α) ].

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The inference from (10) to the omniscience principle (11) ∀p [p ⇒ (p ∈ α) ] is classically, but not intuitionistically, valid. From an intuitionistic point of view we are entitled only to move from (10) to (12) ∀p [p ⇒ ¬¬(p ∈ α) ]. Nonetheless, as it stands this response is hardly satisfying. Although the appeal to intuitionistic logic may well block the move from the knowability principle to the omniscience principle, one can plausibly ask why we should adopt an intuitionistic logic at all. It is true that anti-realists such as Dummett have argued that anti-realism stands or falls with an intuitionistic revision of classical logic. But at the very least, if the appeal to intuitionism is not to be question-begging, we need some independent reason for thinking that classical logic should be revised in the way the intuititionist suggests. The aim of this paper is to take a step back from the details of Fitch’s argument and the particular rules of inference on which it depends in order to explore a line of argument that holds the promise both of undercutting Fitch’s enterprise as a whole (as opposed to simply the deployment of Theorem 5 against anti-realism) and, as a corollary, of explaining why intuitionistic logic is appropriate in this context. This argument has its roots in the notion of indeﬁnite extensibility, as discussed by Michael Dummett in a number of writings (most extensively in chapter 24 of Dummett 1990). Dummett uses the argument to try to motivate anti-realism about mathematics. In particular, he deploys the putative indeﬁnite extensibility of such concepts as set, natural number, and real number to argue for the rejection of classical logic in the relevant domains. Only an intuitionistic logic, he thinks, can do justice to indeﬁnite extensibility. The problem arises when we try to quantify over indeﬁnitely extensible domains. Quantiﬁcation over indeﬁnitely extensible domains does not always, Dummett thinks, yield statements with a determinate truth-value. When we make an existential quantiﬁcation over an indeﬁnitely extensible domain what we are really doing is claiming to be able to cite an instance, and when we make a universal quantiﬁcation we are claiming to have an effective operation that is universally applicable. And this requires that the quantiﬁers be understood intuitionistically rather than classically. How might the notion of indeﬁnite extensibility be applied to the issues about knowability raised by Fitch? Suppose it is the case that the concepts proposition and true proposition are indeﬁnitely extensible, so that there is no deﬁnite totality of (true) propositions of which we have a deﬁnite grasp. If Dummett’s claims about indeﬁnitely extensible concepts are along the right lines then at the very least we need to inquire into the status of the universal quantiﬁcations that are at the heart of Fitch’s reasoning. What is the status of the claim that all truths are knowable, which is required to derive the so-called paradox from Theorem 5?

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And, for that matter, what is the status of the claim that deﬁnes the notion of a truth class? It may turn out that the indeﬁnite extensibility of the concept proposition renders these universal quantiﬁcations problematic in ways that blunt the force of Fitch’s work and the conclusions that have been drawn from it. II In order to explore this terrain, however, we need to begin by clarifying the basic notion of indeﬁnite extensibility and the connection that Dummett sees between indeﬁnite extensibility and intuitionistic logic. Dummett gives the following characterization of an indeﬁnitely extensible concept in ‘What is Mathematics About?’. An indeﬁnitely extensible concept is one such that, if we can form a deﬁnite conception of a totality all of whose members fall under that concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it. (Dummett 1996: 441)

Elsewhere he mentions as an antecedent Russell’s diagnosis of the set-theoretic paradoxes in terms of what he (Russell) terms self-reproductive classes. Russell states: The paradoxes result from the fact that there are what we may term self-reproductive processes and classes. That is, there are some properties such that, given any class of terms having such a property, we can always deﬁne a new term also having the same property. (Russell 1906: 144)

Indeﬁnite extensibility is a property of concepts, while it is classes that are selfreproductive. In both cases the problematic phenomena emerge from features of (certain) inﬁnite totalities. The totalities of which we have indeﬁnitely extensible concepts are all inﬁnite and the properties that generate self-reproductive classes are properties deﬁning inﬁnite collections. For Dummett, the indeﬁnitely extensible concepts include the concepts set, natural number, real number, and ordinal. Russell’s list of self-reproductive classes would no doubt be similar, although I doubt that he would have counted the property of being a natural number as generating a self-reproductive class. The fact that, for Dummett, indeﬁnitely extensible concepts invariably characterize inﬁnite totalities does not mean either that indeﬁnite extensibility is really a property of inﬁnite totalities (so that we can distinguish deﬁnite totalities from indeﬁnitely extensible totalities) or that every concept that characterizes an inﬁnite totality is thereby indeﬁnitely extensible. We can appreciate both these points, together with the general deﬁnition of indeﬁnite extensibility, through examples. One example, which Dummett discusses frequently, is the concept of an ordinal number. One interesting feature of the indeﬁnite extensibility of the

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concept ordinal is that it reveals itself in paradox (in this case the Burali-Forti paradox). Suppose that there is a set α of all ordinal numbers. This set is transitive (since every member of it is an ordinal and every member of an ordinal number is itself an ordinal number) and it is well-ordered by ∈.¹ Since every transitive set well-ordered by ∈ is an ordinal number (Enderton 1977: 191), it follows that α is an ordinal number. But then α would be a member of itself, which no ordinal can be. What generates the paradox is the simple fact that any collection of ordinal numbers gives rise to an ordinal number that is not in that collection (viz. the order-type of the collection). This ﬁts Dummett’s description perfectly. For any collection of ordinals α we can characterize a larger collection, which is the union of α and the order-type of α. The concept set is itself indeﬁnitely extensible. This can be seen in a number of ways. For any given collection β of sets we can characterize a larger collection γ ⊃ β, all of whose members are sets. This is ℘ (β), the power set of β—since, by Cantor’s theorem, card (℘ (β) ) = 2 card (β) . But the indeﬁnite extensibility of the concept set can be shown with far less machinery. For any set δ we can construct a set that is not a member of δ. Let A = {x ∈ δ : ¬(x ∈ x)}. By construction we have A ∈ A ⇔ A ∈ δ ∧ ¬(A ∈ A). Hence, if A ∈ δ, then we have the obviously contradictory A ∈ A ⇔ ¬(A ∈ A). So we can characterize our larger totality by taking δ ∪ {A}—in fact, in a set theory with no self-membered sets, δ = A and the larger totality is δ ∪ {δ}. This is sufﬁcient to show that assuming a set containing all sets leads to paradox. But indeﬁnite extensibility does not entail or require paradox. The indeﬁnite extensibility of the concept real number is revealed by Cantor’s proof of the nondenumerability of the set of real numbers, but there is nothing paradoxical about the fact that the real numbers cannot be put into a one–one correspondence with the set of natural numbers. What Cantor’s proof shows is that any putative enumeration of the set of real numbers will yield a real number that does not feature in the enumeration. Adding that new real number to the original enumeration will give the ‘larger totality’, all of whose members fall under the concept real number. In fact, according to Dummett, the domains of all fundamental mathematical theories are given by indeﬁnitely extensible concepts because, in a claim that has puzzled many commentators, he holds that the concept natural number is indeﬁnitely extensible.² The indeﬁnite extensibility of the concept natural number derives from what Dummett terms the intrinsic inﬁnity of the totality of ¹ A set A is transitive if it is closed under set membership—i.e. if, whenever x ∈ A and y ∈ x, then y ∈ A. ² Dummett’s thinking on this developed signiﬁcantly between ‘The philosophical signiﬁcance of Gödel’s theorem’ (1978) and Frege’s Philosophy of Mathematics (1991). In the earlier article Dummett stopped short of claiming that the concept natural number is indeﬁnitely extensible. There he placed the burden of indeﬁnite extensibility with respect to our understanding of the natural numbers at the door of the Gödel phenomenon. What he says there is that the concept of a

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natural numbers. The totality of natural numbers is intrinsically inﬁnite because, whatever totality of natural numbers we are given, we always have a means of ﬁnding another element of the totality—an element characterized in terms of the elements we already have. Indeﬁnitely extensible concepts are problematic, according to Dummett, because their extensions do not form deﬁnite totalities. This is not because their extensions are in some sense hazy or vague. There is no vagueness in the concept set. There are no entities that we would place on the borderline between things that are sets and things that are not sets. Nor is there any indeterminacy about what is to count as an ordinal number or a real number. The problem comes, Dummett thinks, because it is, strictly speaking, misleading to think of them having extensions in any straightforward sense at all. As he evocatively puts it, indeﬁnitely extensible concepts have ‘an increasing sequence of extensions: what is hazy is the length of the sequence, which vanishes in the indiscernible distance’ (Dummett 1990: 317). Each member of the sequence (each putative extension of the concept natural number or real number is perfectly deﬁnite. But the sequence can be indeﬁnitely extended. This means that we do not have determinate conceptions of the relevant domains of quantiﬁcation for statements about objects falling under those concepts. In fact we cannot have determinate conceptions of the totality of mathematical objects (be they sets, ordinal numbers, or real numbers) over which we are quantifying—as soon as we try to form such a determinate conception we are led inexorably to the conception of a totality that is a superset of the totality with which we began. The problem is not to be avoided by familiar strategies such as the distinction between sets and proper classes. If, as in von Neumann-Bernays set theory, we allow there to be collections that are not sets (because they cannot be members of any collection), then this gives us the means to name the extensions of indeﬁnitely extensible concepts. We are in a position to include terms such as ‘On’ (denoting the proper class of all ordinals) in our set theory, but the totalities thereby denoted are no less indeﬁnitely extensible. The power to name the extension of an indeﬁnitely extensible concept can hardly be thought to eliminate its indeﬁnite extensibility. In any event, we do not need proper classes to deﬁne the extension of the concept real number, which is a perfectly respectable set (on one way of constructing the real numbers it is the set of Dedekind cuts)—and we already have the name ‘ω’ for the extension of the concept natural number, indeﬁnitely extensible concept though it is (according to Dummett). We can now see how there can be concepts of inﬁnite totalities that are not indeﬁnitely extensible—and, indeed, how the same inﬁnite totality can be characterized both by an indeﬁnitely extensible concept and by a perfectly property well-deﬁned over the natural numbers is indeﬁnitely extensible. The argument in the text follows the presentation in the later book at pp. 318–19.

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deﬁnite one. The set of natural numbers is a good example. We can consider the set of natural numbers either as the extension of the concept natural number or as the extension of the concept member of the ﬁrst limit ordinal (since ω, the ﬁrst limit ordinal, has as members all the ﬁnite ordinals). The ﬁrst way of thinking about the set of natural numbers yields an indeﬁnitely extensible concept, for the reasons sketched out earlier. The second way does not, however. It is certainly true that if I form a conception of the totality of members of the ﬁrst limit ordinal then I can characterize a larger totality. I can, in the standard manner, extend the totality by taking the union of all the members of the totality. But the number that I thereby generate, ω, is not the extension of the concept member of the ﬁrst limit ordinal. It is the extension of the concept member of the successor of the ﬁrst limit ordinal. Dummett’s interest, then, is not with inﬁnite totalities per se, but rather with inﬁnite totalities given by indeﬁnitely extensible concepts. These include, he thinks, the domains of the basic mathematical theories, such as number theory and analysis. The fact that these domains are given by indeﬁnitely extensible concepts has deep implications for our understanding of those mathematical domains. Of course, we do manage to quantify meaningfully over mathematical domains that are given by indeﬁnitely extensible concepts—and we do, correspondingly, have some sort of a grasp of the relevant domains of quantiﬁcation. But this is very different from our grasp of totalities given by deﬁnite concepts. In both cases we have, as mentioned earlier, clear and unequivocal criteria for determining, of any particular object, whether it falls under the relevant concept—and for determining when an object falling under the concept but given in one way is identical to an object given in another way. But only for deﬁnite concepts is this enough to ﬁx a determinate totality as the extension of the concept—and hence enough to give us a clear understanding of the extension of the concept. For indeﬁnitely extensible concepts we need something more. We need, ﬁrst, a clear collection of objects that canonically satisfy the relevant criteria, and, second, a principle of extendibility that shows us how the domain is to be extended beyond the canonical base. In the case of the indeﬁnitely extensible concept natural number the principle of extendibility is the fact that every number has a successor. Things are slightly more complicated for the concept ordinal number. We have, of course, the analogous principle that the successor of every ordinal number is an ordinal number, but we also have the further principle that, if A is a set of ordinals, then the least upper bound of A is also an ordinal. The ﬁrst principle gives us the successor ordinals, while the second gives the limit ordinals. The crucial step in Dummett’s argument, and the one that has bafﬂed most commentators (e.g. Clark 1998: 61; and see Potter 2004: 29–30 for further references), is the argument from the indeﬁnite extensibility of key mathematical concepts to the rejection of classical quantiﬁcation. It is, Dummett maintains, quite simply not the case that every quantiﬁcation over a mathematical domain

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given by an indeﬁnitely extensible concept has a determinate truth-value. In trying to understand this we would do well to begin with those quantiﬁcations that Dummett does think acceptable. Any deﬁnite totality of ordinals must therefore be so circumscribed as to foreswear comprehensiveness, renouncing any claim to cover all that we might intuitively recognize as being an ordinal. It does not follow that quantiﬁcation over the intuitive totality of all ordinals is unintelligible. A universally quantiﬁed statement that would be true in any deﬁnite totality of ordinals must be admitted as true of all ordinals whatever, and there is a plethora of such statements, beginning with ‘every ordinal has a successor’. Equally, any statement asserting the existence of an ordinal can be understood, without prior circumscription of the domain of quantiﬁcation, as vindicated by the existence of an instance, no matter how large. (Dummett 1990: 316)

It must be recognized that here, as in many places, Dummett is talking about what it is to understand particular statements—as opposed, for example, to what might make them true. The question of what makes a universal quantiﬁcation over all ordinals true has a simple and uninformative answer, namely, that the statement hold true of every ordinal. The question of understanding is, unfortunately, rather trickier—although of course we cannot understand a statement without understanding what it would be for that statement to be true. What we understand when we understand a universal quantiﬁcation over all the ordinals is the fact that the statement holds true of every deﬁnite totality of ordinals—a fact that we grasp by grasping that the statement holds true of any arbitrarily chosen collection of ordinals. Dummett is somewhat elliptical here, but we can reconstruct his reasoning with the example that he himself gives—the statement that every ordinal has a successor. Let α be an arbitrary ordinal number and α+ the successor of α. By deﬁnition, α+ = α ∪ {α}. The statement says, then, that α ∪ {α} is an ordinal number. Plainly α ∪ {α} is a deﬁnite totality of ordinals (composed of α together with all the members of the members of . . . α). So, for Dummett, what we understand when we understand the statement that every ordinal has a successor is the claim that every deﬁnite totality of ordinals of the form α ∪ {α} is an ordinal. This claim is perfectly intelligible, Dummett claims (on my reconstruction), because we can understand it as the assertion that there is a procedure for showing that any such deﬁnite totality is an ordinal (namely, by noting that α+ is a transitive set all of whose members are ordinals). We can, I think, put Dummett’s point in a more general way as follows. When we are dealing with a universal quantiﬁcation of the form ∀x ϕx claimed to hold over an indeﬁnitely extensible totality, there is (by assumption) no deﬁnite domain of which we can say ∀x ϕx is true just if ϕ holds for every object in the domain. Instead, there is an indeﬁnite sequence of domains and ∀x ϕx is true just if ϕ holds of every object in every domain. But of course the indeﬁnite sequence is not itself a deﬁnite totality, which means that we cannot understand this implicit quantiﬁcation over the indeﬁnite sequence of domains in the standard manner.

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What we are really doing when we assert ∀x ϕx is asserting that the fact that ϕ holds of every object in a given domain is transmitted across the principle of extendibility that creates a new and more inclusive totality from any given deﬁnite totality. But what is it to make such a claim? According to Dummett, when we claim that universal ϕ-ness is transmitted across the principle of extendibility what we are really claiming is that there is a way of showing that, if ϕ holds of all the members of any particular deﬁnite totality in the sequence, then it holds of all the members of the totality to which that deﬁnite totality might be extended Matters are somewhat obscured here by the fact that our example is itself one of the principles of extendibility governing ordinal numbers, but we can still see what is going on by considering the other principle of extendibility. Let ω be the least upper bound of the ﬁnite ordinals. Our second principle of extendibility tells us that ω is itself an ordinal. Part of what is asserted when we assert that ∀x ∈ On ∃y (y = x + ) is that the very same means by which we show that any ﬁnite ordinal has a successor can be extended to the inﬁnite ordinal that is the least upper bound of the set of ﬁnite ordinals. And this in fact is the case. The very same line of reasoning that shows that α ∪ {α} is an ordinal when α is a ﬁnite ordinal will equally show that ω ∪ {ω} is an ordinal. An opponent of Dummett will most likely object at this point that Dummett is confusing proof and truth. On this view, what we assert when we assert ∀x ∈ On ∃y (y = x + ) is simply that every ordinal, be it zero, a successor ordinal, or a limit ordinal, has a successor. Although a proof of this claim will no doubt have to cite just such an operation, its truth depends simply upon there being a successor for every ordinal. We should understand assertion in terms of truth-conditions, not in terms of the operations discovery of which will convince us that the truth-condition holds. Here we come to the nub of the issue, because Dummett’s point (as I understand it) is precisely that we cannot grasp the truth-conditions for quantiﬁcations over indeﬁnitely extensible totalities except in terms of the type of transmissibility sketched out in the previous paragraph. And we can only grasp the possibility of such transmissibility through the idea that there is something that secures it—there is no such thing as transmissibility simpliciter. Of course, there would be no need for this were we dealing with deﬁnite totalities, where the truth-conditions for universally quantiﬁed statements can be understood in the normal manner. But we cannot treat domains deﬁned by indeﬁnitely extensible concepts as if they were deﬁnite totalities. Dummett’s position, then, is that the truth-conditions for universal quantiﬁcations over domains given by indeﬁnitely extensible concepts must be understood in terms of operations that secure transmissibility in the manner discussed. To assert such a universally quantiﬁed statement is to assert that such operations exist. But this has the inevitable consequence that we must abandon classical logic. From the fact, for example, that it is not the case that it is not the case that every x in some indeﬁnitely extensible domain is F it by no means follows that there is an operation that will secure the transmissibility of ϕ-ness throughout

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the increasing sequence of extendible totalities. We might be able, for example, to give a reductio of the thesis that ∀x Fx can be reduced to absurdity. But this would hardly give us the required operation. Nor will the standard quantiﬁer interchange rule ¬∀x ¬Fx ⇔ ∃x Fx be valid. Even if it is absurd to suppose that there is an operation securing the transmissibility of F not holding in an appropriate domain, this by no means provides an instance of something in the domain that is F. Dummett is surely right that if the truth-conditions of universal quantiﬁcations over indeﬁnitely extensible totalities make ineliminable reference to operations securing transmissibility, then we cannot understand the logic of such statements classically. III We return now to Fitch’s paradox. The excursion into the philosophy of mathematics in the previous section has shown that it is possible to argue (with some plausibility, in my opinion) for the thesis that quantiﬁcation over mathematical domains given by indeﬁnitely extensible concepts should be understood intuitionistically rather than classically. Plainly, applying this to Fitch’s paradox depends upon construing Fitch’s paradox as making ineliminable reference to indeﬁnitely extensible totalities in a way that will support the type of argument canvassed in the previous section. Exploring whether this is indeed the case is the task of this section. But suppose for the moment that we can apply a Dummett-style argument in this domain. There are two ways in which this holds promise for dealing with Fitch’s paradox. Most obviously, as we saw earlier, the omniscience principle can only be derived by an inference that is classically but not intuitionistically valid. If it can be shown that we are dealing with an indeﬁnitely extensible totality over which quantiﬁcation must be understood intuitionistically then we have a plausible case for denying the validity of this inference. The appeal to intuitionistic logic becomes more than simply a technical ﬁx. But there is a more subtle way in which an argument from indeﬁnite extensibility might get a grip here. The logic governing statements that quantify universally over indeﬁnitely extensible domains is intuitionistic because those quantiﬁcations have to be understood in a constructivist manner—that is, in a manner that appeals to the existence of effective operations securing transmissibility across increasing sequences of domains. To assert a universal quantiﬁcation is essentially to assert that there is such an operation. But then it is very natural to wonder whether a prudent anti-realist really ought to commit themselves to the knowability principle in the form that we have given it (that is, as the universal quantiﬁcation (8)). In any event, let us begin at the beginning. Are there good reasons for thinking that the collection of all propositions forms an indeﬁnitely extensible totality?

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A straightforward argument shows that there can be no set S of all true propositions (cf. Grim 1991: Ch. 4). We take a proposition to be an abstract entity, such that S can be non-denumerably inﬁnite, and we assume that S does not contain the proposition corresponding to the sentence ‘0 = 0’. Since S is a set it has a cardinality κ. Consider the power set ℘(S). Any arbitrary member si of ℘(S) is a set of true propositions. As such there is a true proposition corresponding to the sentence ‘0 = 0 ∈ / si ’. Let that proposition be pi . Since S is the set of all true propositions, a subset axiom permits us to form the set P of all such pi . Since P can be put into one–one correspondence with ℘(S) we have card (P) = card (℘ (S) ) = 2 κ > κ = card (S). Since it is impossible for a set to have a subset of greater cardinality there can be no set of all true propositions. With minor alterations this argument can be used to show that there is no set of all propositions. (For each si we take pi as either ‘0 = 0 ∈ / si ’ or ‘0 = 0 ∈ si ’, as appropriate.) Of course, the argument just sketched out contains many hostages to fortune. Without a precise understanding of what a proposition is it is hard to know how to understand sets of propositions. Nor do we have a proper deﬁnition of set P. But we can put these problems to one side. We have enough to go on to see how the case might be made for the indeﬁnite extensibility of the concept proposition. Certainly, Dummett’s deﬁnition seems to be satisﬁed. Given any deﬁnite totality S of propositions we can deﬁne a totality S∗ = S ∪ P such that S ⊂ S∗ and S∗ is itself a totality of propositions. The real question is not whether the concept proposition is indeﬁnitely extensible in this sense, but whether it is indeﬁnitely extensible in a way that permits the type of argument for intuitionistic logic sketched out in the previous section? That line of argument rests upon certain claims about what it is to grasp the truth-conditions of quantiﬁcation over totalities given by indeﬁnitely extensible concepts. The argument, in essence, is that universal quantiﬁcation over a totality given by an indeﬁnitely extensible concept must be understood in terms of principles of transmissibility that secure the holding of the relevant property across an increasing sequence of domains. We know, when we are dealing with indeﬁnitely extensible concepts in the mathematical sphere, that any given deﬁnite totality of a given type of object will generate a larger totality of the same type that includes it. So, what we assert when we assert some statement to be universally true within the domain given by such an indeﬁnitely extensible concept is that, if the statement holds true of any given deﬁnite totality, it will hold true of the larger totality to which the original totality can be expanded—and so on through the increasing sequence of domains. It is for this reason, Dummett thinks, that universal quantiﬁcation over indeﬁnitely extensible domains incorporates ineliminable commitment to the existence of effective operations, thereby requiring an intuitionistic logic. We saw how this way of thinking about universal quantiﬁcation makes sense in the context of the ordinals. But is it mandated by quantiﬁcation over the intuitive

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totality of propositions? In one sense it is hard to see how it could not be. After all, the argument just canvassed shows that there is no deﬁnite set of all propositions and so, even though we are assuming that we have clear criteria of identity and individuation for propositions (which may, of course, be a vague hope), we cannot assume that those criteria will determine a truth-value within that deﬁnite totality. And so one might well feel justiﬁed in arguing with Dummett that we can only make sense of the truth-conditions for statements quantifying over all propositions in terms of operations that secure the transmissibility of the relevant property. But this does not really get to the heart of the matter. What we really want to know is what those operations would look like. In the case of quantiﬁcation over the ordinals we saw an example of how transmissibility across principles of extendibility might be achieved. In order to see how something comparable might work in the case of quantiﬁcation over all propositions we need a clear idea of what the relevant principle of extendibility might be. We need to know what, in the case of propositions, plays the role that is played for ordinals by the twin principles that every ordinal has a successor that is an ordinal and that the least upper bound of any collection of ordinals is itself an ordinal. Let us revert to Dummett’s original characterization of how we grasp domains given by indeﬁnitely extensible concepts. A necessary but not sufﬁcient condition is that we have clear and unequivocal criteria for determining, of any particular object, whether it falls under the relevant concept—and for determining when an object falling under the concept but given in one way is identical to an object given in another way. When we are dealing with indeﬁnitely extensible concepts we also need, ﬁrst, a clear collection of objects that canonically satisfy the relevant criteria, and, second, a principle (or principles) of extendibility that shows us how the domain is to be extended beyond the canonical base. The domain given by an indeﬁnitely extensible concept is closed under the relevant principle(s) of extendibility. It is straightforward to apply this general model to quantiﬁcation over all propositions. We begin with the distinction between simple and compound propositions, where a simple proposition is one that does not contain any quantiﬁers or truth-functional connectives and a compound proposition is constructed from simple propositions with quantiﬁers, truth-functional connectives or other operators. Plainly the totality of all propositions is the closure of the set of simple propositions under the operations of conjunction, disjunction, and so on—just as the totality of all ordinals is the closure of the empty set under the successor and limit operations. So, we may conclude that the principles of extendibility for the totality of propositions are, in effect, the rules governing the truth-functional connectives, quantiﬁers, and other operators. We can now see what form must be taken by the principles of transmissibility for quantiﬁcation over the totality of all propositions. The principles of transmissibility for a universal quantiﬁcation of the form ∀p Fp must show that

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the property of F-ness is transmitted across the logical operations whose closure gives the totality of propositions. By the same token, the truth-condition for the statement ∀p Fp is, in essence, that all simple propositions are F and that F-ness is transmitted in the appropriate manner across the relevant logical operations. To assert that ∀p Fp is to claim that F-ness is transmitted, and it is this claim that one understands when one understands the statement that ∀p Fp. It appears, therefore, that the concept proposition qualiﬁes as indeﬁnitely extensible by Dummett’s lights, even though it is not (in any obvious sense) mathematical. How does this affect the reasoning that leads to Fitch’s paradox?

IV We note ﬁrst that the set of true simple propositions is a deﬁnite totality. We can run arguments such as that canvassed above on many different proper sub-totalities of the totality of all propositions. If we assume, for example, that ‘0 = 0 ∈ / si ’ expresses a proposition of mathematics (where si is a set of mathematical truths), then a parallel line of reasoning shows that there is no set of true mathematical propositions—and hence that the concept mathematical proposition is an indeﬁnitely extensible concept. No such argument can work, however, for the totality of true simple propositions, since the propositions required to run the argument are not simple propositions. Arguments of this type contain an ineliminable use of negation. They also hinge upon propositions that have simple propositions as constituents. Since the concept true simple proposition is deﬁnite we can quantify over simple propositions unproblematically. We can say, for example, that every true simple proposition is knowable. This might be formulated using restricted quantiﬁcation as follows (where S is the set of true simple propositions): (13) ∀p ∈ S [p ⇒ ♦(p ∈ α) ] By Dummett’s lights this principle has a determinate truth-value and can be asserted and understood in the standard manner. The antecedent effectively restricts us to the deﬁnite totality of simple propositions. The same does not hold, however, for what we can term the unrestricted knowability principle—i.e. (8) ∀p [p ⇒ ♦(p ∈ α) ] Here the quantiﬁer ranges without restriction over an indeﬁnitely extensible totality. By the earlier arguments it appears that the truth-condition for (8) rests upon the transmissibility of the property of knowability across the principles of extendibility that collectively yield the indeﬁnitely extensible totality of all propositions. We can certainly take as our ‘base’ in spelling out how this might work the restricted principle of knowability for simple propositions. But the real

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work in spelling out the truth-condition for (8) comes with what we might term the transmissibility principles—those principles that ensure that the totality of knowable propositions extends and expands at exactly the same rate as the totality of propositions. Since the extendibility principles for the totality of propositions are precisely those governing the truth-functional connectives and the quantiﬁers, we can expect the transmissibility principles to track the principles laying down the truth-conditions for compound propositions. In fact, it is possible to argue that the transmissibility principles just are the principles specifying truth-conditions for compound propositions. If we concede that intuitionistic logic is the appropriate logic for indeﬁnitely extensible totalities then the truth-conditions for compound propositions will have to be given in intuitionistic terms. They will, in fact, take something like the following form, where in each clause the connective/quantiﬁer/operator on the right is to be understood intuitionistically: (14) (15) (16) (17) (18) (19) (20) (21)

‘not ϕ’ is true ⇔ ¬ ϕ ‘ϕ or ψ’ is true ⇔ ϕ ∨ ψ ‘if ϕ then ψ’ is true ⇔ (ϕ ⇒ ψ) ‘ϕ and ψ’ is true ⇔ ϕ ∧ ψ ‘For some x ϕ’ is true ⇔ ∃x ϕ ‘For all x ϕ’ is true ⇔ ∀x ϕ ‘Possibly ϕ’ is true ⇔ ♦ϕ ‘Necessarily ϕ’ is true ⇔ ϕ

The key point here is that, precisely because the connectives/quantiﬁers/operators are being understood intuitionistically, principles (14) through (21) secure the transmissibility of the property of knowability across the indeﬁnitely extensible totality of all (true) propositions. Suppose we take an implication of the form ‘if ϕ then ψ’ where ϕ but not ψ is a simple proposition. If we understand ‘if . . . then . . . ’ intuitionistically then ‘if ϕ then ψ’ is true iff there is some form of procedure that will transform a proof of ϕ into a proof of ψ (we assume that we can formulate an analog of the proof-conditional interpretation of intuitionistic logic when we ﬁnd ourselves outside the domain of mathematical proof ). If we have such a procedure then the conditional is knowable and, by assumption, ϕ is knowable. Hence ψ is knowable. Pari passu for the other logical constants—and, of course, for quantiﬁcation (since ‘∀x’ and ‘∃x’ make ineliminable reference to effective operations when understood intuitionistically). In fact, Dummett himself proposes (Dummett 2001) that the correct response to Fitch’s paradox is an inductively speciﬁed theory of truth of the type given by principles (14) through (19), coupled with a basic principle akin to our (13) to the effect that simple propositions are knowable. Although Dummett (in what can only be described as a rather elliptical paper) does not put the matter in quite these terms, the inductive speciﬁcation secures the knowability of the totality

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of all true propositions without a global quantiﬁcation such as (8)—and hence without allowing the type of substitution that yields Fitch’s paradox. We can see how this works as follows. Suppose that p is a true simple proposition that has never been and never will be known. From (13) we still have that [p ∧ ♦(p ∈ α)]. But this, after all, simply tells us what we already knew, which is that p is a true simple proposition that could be known at some time. Since [p ∧ ♦(p ∈ α)] is a compound proposition we cannot substitute it back into (13) to yield the paradox. The basic point, then, is that (8) is not the correct way to express the basic principle that all truths are knowable. Because we are dealing with a domain given by an indeﬁnitely extensible concept our expression of the basic principle needs to reﬂect the principles of extendibility in terms of which we grasp that domain. In this case these are the principles governing the logical constants and quantiﬁers. By spelling out those principles in an intuitionistic metalanguage, as in principles (14) through (19) we effectively give expression to the basic principle that all truths are knowable without stating it explicitly—and so without either compromising the indeﬁnite extensibility of the concept proposition or opening the door to Fitch’s paradox. The ﬁnal step in the argument is plainly in view in Dummett 2001 where he sets out the inductive speciﬁcation as a way of blocking Fitch’s paradox. Few commentators, however, have seen the rationale for the inductive speciﬁcation (and Dummett himself has nothing to say about it). This paper has tried to make that rationale explicit. As I have argued, the motivation for the inductive speciﬁcation is to be found in the indeﬁnite extensibility of the concept proposition. The inductively speciﬁed theory of truth gives the principles of extendibility that we must grasp if we are to have a grasp of the indeﬁnitely extensible totality of all propositions. If Dummett is right, moreover, that the correct logic to use over a domain given by an indeﬁnitely extensible concept is intuitionistic, then the inductive speciﬁcation must itself be intuitionistic, which is sufﬁcient to ensure that the principles of extendibility are also principles of transmissibility in the sense we have discussed.

Pa r t I I I Pa r a c o n s i s t e n c y a n d Pa r a c o m p l e t e n e s s

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7 Beyond the Limits of Knowledge Graham Priest

1 . In t ro d u c t i o n Are there limits to knowledge? Well, there are certainly many things that we do not, as a matter of fact, know. We do not know (at the moment) whether Iraq will continue its downward spiral into anarchy. We will know in due course. We do not know how to make the Theory of Relativity and Quantum mechanics consistent with each other. Maybe we will in due course. More interesting is the question of whether there are things that it is not possible to know. Perhaps there are things that are so difﬁcult, remote, or recondite, that they transcend anything we could ﬁnd out. If this is the case, there are even limits to what it is possible to know. Whether or not this is so is the main topic of this paper. ‘Possible’ is a highly ambiguous word in philosophy. It can mean ‘logically possible’, ‘physically possible’, ‘epistemically possible’, and doubtless many other things. It may therefore reasonably be asked what sense of possibility is at issue here. The answer is that it doesn’t really matter. For most of the purposes of this paper, it can mean any sense of possibility one likes. One group of people who assert that all truths are knowable (in some appropriate sense) comprises veriﬁcationists, including mathematical intuitionists. For them, this is a constraint on truth itself (or maybe on meaning): everything that is true is such that it is possible (at least in principle) to know it. What sense of ‘possible’ veriﬁcationists have in mind here, I leave them to explain. But at least in their honour, I call the principle that all truths are possibly known the Veriﬁcation Principle. This Principle settles the matter at issue in one way. Ancestors of this paper were given under the title ‘The Limits of Knowledge’ at Mt Holyoke College, the Graduate Center, City University of New York, the University of California at San Diego, the University of Melbourne; and at the conferences The Philosophy of Uncertainty: Epistemic Limits, Probability, and Decision, held at the University of Tokyo, and Logica 2005, held in the Czech Republic. I am grateful to the audiences present for their helpful discussions, and to Masake Ichinose, Tim Childers and Vladimir Svoboda, for organizing the conferences.

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On the hand, there is a well known argument, usually attributed to Fitch, to the effect that the Veriﬁcation Principle is false. If this is the case, then there are some truths that it is impossible to know. This resolves the issue in the opposite direction. We may therefore approach the matter by considering the tenability of the Veriﬁcation Principle in the light of the Fitch argument.

2 . Se t t i n g u p t h e Is s u e Let us start by getting the geography of the issue straight. Some notation: I will use lower case Greek letters for sentences of whatever language is at issue. ♦ and are the usual modal operators of possibility and necessity. Kx will be the predicate ‘is known’, and . is a name-forming device. Now, let T be the set of truths. The question is how what we know relates to this. There are two relevant subsets. The ﬁrst comprises the truths that are (actually) known, K = {x : Kx}. The second comprises the truths that it is possible to know, P = {x : x ∈ T ∧ ♦Kx}. (Note that K α entails that α is true; but ♦K α does not—only that α is possibly true.) Since what is known is possibly known (in any normal sense of possibility), the general relationship between the three sets is as shown in Figure 7.1. T

P

K

Figure 7.1.

K is certainly non-empty. Melbourne, for example, is known to be in Australia. P − K is also non-empty. As I have already observed, there are things about the future that we do not know, but will; so that knowledge is certainly possible. Similarly, the Ancient Greeks did not know that there was a planet beyond Uranus; but it is possible to know this: we do. The status of T − P is less clear. The Veriﬁcation Principle says that α → ♦K α. If this is true, T − P is empty; if there is a counter-example to the Principle then there are truths that it is not possible to know.

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Now to the Fitch argument.¹ This is to the effect that if it is possible to know whatever is true, then everything true is not just possibly known, but actually known. That would appear to be a reductio ad absurdum of the view. It is clear that not everything is actually known—even if one is a veriﬁcationist. A priori, there is something highly suspect about the argument, however. Surely one cannot get from the mere fact that it is possible to know something to the fact that it is known? Informally, Fitch’s argument goes as follows. Suppose that everything true is knowable, and suppose for reductio that there is something, α, which is true but not known, α ∧ ¬K α. Then it must be possible to know this, ♦K α ∧ ¬K α. By a few straightforward inferences concerning knowledge, it follows that it is possible to both know α and not know it, ♦(K α ∧ ¬K α), which it isn’t. 3 . T h e Fi t c h A r g u m e n t

3.1. Stage 1: Knowledge Let me spell out the argument in detail (in natural deduction form), so that we may look at the moves in it more carefully. For the purpose of discussing the argument, and in the cause of simplicity, I will write K α as K α, effectively turning the predicate K into the more usual operator. (As long as we are not quantifying-in, there is no real difference.) The part of the proof concerning knowledge goes as follows. Call it 1 . K ( ∧¬K ) K

[ ∧¬K ]

K ( ∧ ¬K ) ∧ ¬K ¬K

K ∧ ¬K

1 uses four inferences: [ ] ∧ ∧

K

K K

In the fourth of these, the column from β to γ represents an argument with premise β (and only β), and conclusion γ. The square brackets represent the fact that the inference discharges β, so that the ﬁnal argument no longer depends on it. ¹ Fitch (1963). Fitch himself attributes the argument to an anonymous source. It lay dormant for some time, but was published again by Hart and McGinn (1976), whose attention was drawn to it, again, anonymously.

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The ﬁrst two inferences require no comment; nor does the third: what is known is true. The fourth says that knowledge is closed under entailment. This is certainly not correct. What is actually known is not closed under entailment. For example, medieval monks knew that Aristotle was Greek. They did not know that (Aristotle was Greek or the formalism of quantum mechanics deploys Hilbert spaces), even though this entails it. Or consider the Peano postulates. I know all these. But I do not know all their consequences (amongst which are probably the solutions to some famous unsolved problems in number theory). But the Fitch argument cannot be dismantled by simply rejecting this principle of inference. This is because the only use made of the principle in the argument is to infer a special case: that the knowledge of a conjunct follows from the knowledge of a conjunction. Hence, the rule could be replaced by the much simpler: K (β ∧ γ) Kγ This seems much harder to contest.² In particular, the sorts of counter-example just mentioned relevant to the failure of the closure of knowledge under entailment (in general) seem to get very little grip on it. The knowledge of a conjunct seems implicit in the knowledge of a conjunction. There is therefore little scope for faulting this part of the argument.

3.2. Stage 2: Possibility The second part of the argument embeds 1 in an argument concerning possibility. This is as follows, where the right-hand column represents 1 . Call this part 2 . [K (α ∧ ¬K α) ] α ∧ ¬K α ∇ K α ∧ ¬K α ♦K (α ∧ ¬K α) ♦(K α ∧ ¬K α) 2 applies two new rules, which are as follows: [β] .. . β ♦K β

♦β γ ♦γ

² Harder, but not impossible. Connexivist logicians (including some medievals) held that β ∧ γ does not entail γ—for example, if β is ¬γ, this simply cancels out the γ. Such a logician could know β ∧ γ, but not believe, and a fortiori know, γ. To avoid this kind of problem we can just restrict the class of knowers in question to those who have the normal beliefs about the validity of inferences concerning conjunction—which includes us.

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The ﬁrst of these is simply the Veriﬁcation Principle, which is what the argument assumes (for the sake of reductio). The second says that possibility is closed under entailment. This seems to hold for any notion of possibility. If α is true in a possible world (of any appropriate kind), and α entails β, then β is true in that world, and so possible (in the same sense). There is little in this stage of the inference that one can balk at, then.

3.3. Stage 3: Contraposition The third part of the argument embeds 2 in an argument deploying negation. This is as follows, where the left-hand column represents 2 . Call this 3 . [α ∧ ¬K α] ∇ ♦(K α ∧ ¬K α) ¬♦(K α ∧ ¬K α) ¬(α ∧ ¬K α) 3 employs one premise and one further rule of inference. The premise is ¬♦(β ∧ ¬β), or equivalently, given the usual connections between and ♦: ¬(β ∧ ¬β) The inference is contraposition: [β] .. . γ

¬γ ¬β

The only plausible way to contest these steps is to suppose that contradictions may be true. The rationale for contraposition is that if β delivers something that is not true, γ, it must be false. This rationale collapses if γ can be true despite the truth of ¬γ. Unsurprisingly, then, the inference fails in many paraconsistent logics (including the one whose semantics I will describe below). Suppose, for example, that the logic contains the Law of Excluded Middle (LEM), β ∨ ¬β. Then we have γ β ∨ ¬β. Contraposing, ¬(β ∨ ¬β) ¬γ, that is (assuming De Morgan Laws), β ∨ ¬β ¬γ—which fails, since γ was arbitrary. This stage of the argument may therefore be broken by appealing to dialetheism. It might be thought that dialetheism would invalidate the new premise of the argument as well: if contradictions may be true, one might expect ¬(β ∧ ¬β), and so its necessitation, to fail. Surprising as it might be to those meeting paraconsistency for the ﬁrst time, it does not. There are many paraconsistent logics where the law holds (including the one whose semantics I will describe below). Of course, any contradiction, β ∧ ¬β, will then generate a secondary

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contradiction, (β ∧ ¬β) ∧ ¬(β ∧ ¬β), but there is nothing in a paraconsistent logic to rule this out. Actually, the simplest way of avoiding ¬(β ∧ ¬β) (and so its necessitation) is to appeal, not to truth-value gluts, but to truth-value gaps. If β is neither true nor false, so (given the natural semantics for the connectives) is ¬(β ∧ ¬β). Appealing to truth-value gaps also invalidates contraposition unless the logic is paraconsistent. If the logic is not paraconsistent, we have β ∧ ¬β γ, and so ¬γ ¬(β ∧ ¬β), i.e., ¬γ β ∨ ¬β, which is not the case if we do not have the LEM. It might therefore be thought that appealing to truth-value gaps is a way of avoiding the argument without an appeal to gluts. Unfortunately (for the friends of consistency) it is not. As 2 shows, given the Veriﬁcation Principle, α ∧ ¬K α already leads to ♦(K α ∧ ¬K α), and thus to the possibility of true contradictions. Moreover, if the logic is not paraconsistent, we have, for an arbitrary β, K α ∧ ¬K α β. By the closure of possibility under entailment, we have ♦(K α ∧ ¬K α) ♦β. Given that ♦(K α ∧ ¬K α), everything is possible—not an enticing conclusion (for the friends of consistency). One way or another, then, true contradictions are required to break this step of the argument.

3.4. Stage 4: Double negation There is one ﬁnal part of the argument. This embeds 3 in the argument which actually takes us from α to K α. This goes as follows, where the right-hand column represents 3 . ¬♦(K α ∧ ¬K α) ∇ ¬(α ∧ ¬K α) ¬¬K α Kα This stage of the argument uses contraposition again, discharging ¬K α. (And in this application, there is also another assumption in the sub-proof. As is to be expected, this does nothing to restore validity in a paraconsistent logic. It just makes matters worse.) It uses one further rule, double negation: α [¬K α] α ∧ ¬K α

¬¬β β Double negation fails in intuitionist logic, which is intimately connected with veriﬁcationism. Hence, breaking the argument by denying this step is a very plausible move. If we do, we can get from α only to ¬¬K α, which is not so bad. Well, not really. Given α → ¬¬K α, we obtain ¬¬¬K α → ¬α by a

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form of contraposition that is intuitionistically valid. And in intuitionist logic, ¬β ↔ ¬¬¬β. So by transitivity, ¬K α → ¬α. Even intuitionists cannot accept this in general. Let α be ‘Alpha Centauri has a planetary system’. I do not know that α; I do not know that ¬α. (Nor does anybody else—maybe for ever.) It cannot follow that ¬α and ¬¬α.³ 4 . A Si m p l e Mo d e l We have seen that appealing to dialetheism breaks the Fitch argument against veriﬁcationism. We can do more than this, however. It can be shown that once contraposition (and only contraposition) is removed from the principles employed, the inference from α to K α is not forthcoming. I demonstrate this with a counter-model based on the semantics for a simple paraconsistent modal/epistemic logic.⁴ Interpretations are of the form W , ∞, R, S , ν. W is a set of worlds. ∞ is a distinguished member of W. R is the modal binary accessibility relation, and we require that for every w ∈ W , wR∞. S is the epistemic binary accessibility relation, which is at least reﬂexive. ν maps every world and propositional parameter to {1}, {0} or {1 , 0} (true, false, both). I write the value of α at w as νw (α). Truth-conditions at worlds, w, other than ∞ are as follows: 1 0 1 0 1 0 1 0

∈ νw (α ∧ β) iff 1 ∈ νw (α) and 1 ∈ νw (β) ∈ νw (α ∧ β) iff 0 ∈ νw (α) or 0 ∈ νw (β) ∈ νw (¬α) iff 0 ∈ νw (α) ∈ νw (¬α) iff 1 ∈ νw (α) ∈ νw (♦α) iff for some w such that wRw , 1 ∈ νw (α) ∈ νw (♦α) iff for all w such that wRw , 0 ∈ νw (α) ∈ νw (K α) iff for all w such that wSw , 1 ∈ νw (α) ∈ νw (K α) iff for some w such that wSw , 0 ∈ νw (α)

∞ is the trivial world. That is, for every α: ν∞ (α) = {1 , 0} Validity is deﬁned in terms of truth-preservation at all worlds. Leaving aside the Veriﬁcation Principle for the moment, it is not difﬁcult to check that the semantics verify all the inferences involved in the Fitch argument (including the closure of knowledge under entailment, and the premise ¬♦(β ∧ ¬β) ) except contraposition. ³ On this and related objections, see Percival (1990). ⁴ This is an extension of the propositional paraconsistent logic LP (see Priest (1987), ch. 5). The existence of the trivial world, ∞, does not affect the logic of the extensional connectives.

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For the veriﬁcationist inference: for any α, 1 ∈ ν∞ (K α), and so for every w (including ∞), 1 ∈ νw (♦K α). The inference α ♦K α is therefore (vacuously) valid. To ﬁnish the job, we just need an interpretation where there are worlds, w 0 and w 1 , such that 1 ∈ νw0 (p) , w 0 Sw 1 , but 1 ∈ / νw1 (p). Then 1 ∈ / νw0 (Kp). We can depict the simplest interpretation of this kind as shown in Figure 7.2 (+ indicates that a formula holds; − indicates that it fails; square brackets indicate things that hold at worlds, other than what is part of the speciﬁcation). Notice that R and S can be made as strong as one likes without ruining the argument. In other words, the modal logic of K and ♦() can be beefed up to S5 without affecting the result. ∞

S w0

p+

[◊Ka+, Kp −]

R R S

[Ka+, ◊Ka+] S w1

p−

[◊Ka+]

Figure 7.2.

Note, also, that we may take the language to contain a conditional operator, →, with strict truth/falsity conditions as follows.⁵ At any world, w, other than ∞: 1 ∈ νw (α → β) iff for all w such that wRw , if 1 ∈ νw (α) then 1 ∈ νw (β) 0 ∈ νw (α → β) iff for some w such that wRw , 1 ∈ νw (α) and 0 ∈ νw (β) Assuming that R is reﬂexive, these semantics verify (at least) the inferences: [α] .. . α

α→β β β α→β (where α is the only undischarged assumption in the second inference). In the above model (with the additional proviso that R is reﬂexive), α → ♦K α holds for all α at w 0 , but p → Kp fails. 5 . E n t e r t h e K n owe r We have seen that the Fitch argument may be blocked by an appeal to dialetheism. Moreover, it is the only way that we have found in which the argument may ⁵ See Priest (1987), ch. 6.

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be blocked.⁶ But—it might well be argued—an appeal to dialetheism in this context is extreme and unmotivated. Better to take the argument to be a simple reductio of the Veriﬁcation Principle. Matters are not that simple, though. First, there are situations in which the Veriﬁcation Principle appears to hold (at least in some sense of possibility) and where the agent in question does not know everything true. It is coherent, I take it, to suppose the existence of an omniscient (and omnipotent) being. Let us call them ‘God’. Everything true it is possible for God to know; indeed, everything true God actually does know. But God has a friend; call him ‘Gabriel’. Gabriel is not omniscient. There are many things that Gabriel doesn’t know, and doesn’t care about—such as who won the 4.30 at Flemington. But Gabriel knows at least that God is omniscient. Moreover, he knows that he can always ask God if he wants to know something; God, being a decent and trustworthy fellow, will tell him. Hence, anything that is true, it is possible for Gabriel to know—just by asking. Yet Gabriel does not know everything true. The Fitch argument must therefore fail. The Fitch argument itself suggests an objection to this. Let us suppose that Red King Hit won the 4.30 at Flemington—call this κ—and that, as a matter of fact, Gabriel does not know this, since he never bothers to ask. Then: (∗ )

κ and Gabriel does not know (at any time) that κ

is true. God knows it. It might be argued that it is, none the less, not possible for Gabriel to know it. To do so, he would have to know κ and know that he does not know κ (at any time), which is impossible. But could he not ask God whether (∗ ) is true, and get an answer? Of course he could. If, as we suppose, (∗ ) is true, God will tell him so. Hence, Gabriel will know κ, and (∗ ) is false. Suppose, on the other hand, that (∗ ) is false. Then God will tell him so. At this point Gabriel still does not know whether κ is true or false. Suppose we then shoot him; he never will. So (∗ ) is true. None of this shows that Gabriel cannot know (∗ ); all it shows is that, if he does ask the question, the situation is a paradoxical one. In fact, the paradox is a version of a well known one—the Bridge. A person has to cross a bridge; on the other side there is a bridge-keeper who asks a question. If the person answers truly, they are allowed to pass; if not, the bridge-keeper hangs them. The bridge-keeper asks ‘what will you do when you get to the other side of the bridge?’ The person answers ‘I will be hanged by you’.⁷ Again, the question forces a paradoxical situation. ⁶ Human ingenuity being what it is, there may, of course, be other suggestions. A number of these are discussed (and rejected) in Williamson (2000a), Ch. 12. The chapter also contains references to other discussions of the argument in the literature. ⁷ The paradox is one of Buridan’s sophismata but, according to Sorensen, it probably goes back to Chrysippus. A version of it is told by Cervantes in Don Quixote. See Sorensen (2003), pp. 207–9.

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A much simpler version of the paradox is forthcoming by just letting κ be the sentence ‘Gabriel (or even God) does not know κ’. Let us make this more precise. By applying techniques of self-reference, we can construct a sentence, κ, that says of itself that it is not known. That is, κ is of the form ¬K κ. (I now revert to writing K as a predicate. Self-referential constructions require this.⁸) Suppose that K κ; then κ is true, so ¬K κ. Hence, ¬K κ. That is, κ, but we have just demonstrated this, so it is known to be true, K κ. (This is the Knower paradox.) We have demonstrated K κ ∧ ¬K κ. This is therefore necessarily true (in whatever sense of necessity one cares for); a fortiori, ♦(K κ ∧ ¬K κ). And the Veriﬁcation Principle ﬁgures nowhere in the argument for this. We see, in particular, that quite independently of the Fitch argument there are sentences of the form required to invalidate the contraposition in 3 . Appealing to dialetheism to break the Fitch argument is, therefore, not at all ad hoc or unmotivated. In the context, it is very natural.⁹

6 . C o n t r a d i c t i o n a n d t h e L i m i t s o f K n ow l e d g e We can bring this to bear explicitly on the question of the limits of knowledge as follows. Let X ⊆ K. Provided that X has a name, and given appropriate techniques of self-reference, we can form a sentence that says of itself that it is not in X ; that is, a sentence, αX , of the form αX ∈ / X . We can show that αX ∈ /X but that αX ∈ K as follows: αX ∈ X ⇒ αX ∈ K ⇒ αX ⇒ αX ∈ /X Hence, αX ∈ / X . But this is αX , and we have just established this, so it is known to be true; that is, αX ∈ K. The situation may be depicted as shown in Figure 7.3. When X is the empty set, αX can be located anywhere in K − X (= K). As X gets bigger and bigger,¹⁰ there is less and less space in which αX can be consistently located; until, at the limit, when X coincides with K there is nowhere consistent for αX to go. αK ∈ K ∧ αK ∈ / K. (This is the ⁸ In fact, we can maintain K as an operator provided that we have a truth-predicate, T , in the language. We can then deﬁne an appropriate predicate, K x, as KTx. (Thanks to Jc Beall for this observation.) ⁹ The ﬁrst person to moot the possibility of a connection between the Fitch argument and the Knower was Routley (1981) (see esp. p. 112, n. 26). The connection was made more robustly by Beall (2000). ¹⁰ Of course, X does not literally grow. In particular, we are not considering the case where more and more is known. (That would be a case of K growing.) This is just a picturesque way of saying that for larger and larger X . . .

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Knower paradox. κ is just αK .) The limit of what is known is dialetheic. That is, there are certain truths that are both within the known and without it. K X × aX

Figure 7.3.

Exactly the same is true of P. Let X ⊆ P. As before, we can construct a sentence, αX , of the form αX ∈ / X. αX ∈ X ⇒ αX ∈ P ⇒ αX ∈ T ∧ ♦K αX ⇒ αX ⇒ αX ∈ /X Hence, αX ∈ / X . But this is αX , and we have just established this, so it is true and known to be so, K αX . A fortiori, it is possible to know it, ♦K αX . Thus, αX ∈ T ∧ ♦K αX . That is, αX ∈ P. Just as with K, when X is small, there is plenty of room for αX to reside, consistently, outside it but inside P. As X gets bigger and bigger, there is less and less room, until when X is P, a contradiction arises: αP ∈ P ∧ αP ∈ / P. The boundary of possible knowledge is inconsistent too. An Inclosure involving a set, , a predicate, ψ, and a function, δ, is a structure satisfying the following conditions: 1. ψ() 2. if X ⊆ and ψ(X ) (a) δ(X ) ∈ / X (Transcendence) (b) δ(X ) ∈ (Closure) Whenever we have an Inclosure, a contradiction arises at the limit, when X = . For we then have δ() ∈ / ∧ δ() ∈ . All the standard paradoxes of self-reference are limit-paradoxes of this kind.¹¹ The two contradictions we have just looked at are of this form. In the ﬁrst, is K; in the second, is P. In both, ψ(X ) is ‘X is deﬁnable (has a name)’, and δ(X ) is αX . Hence, both are inclosure contradictions. ¹¹ See Priest (1995), part 3. For the Knower paradox, see 10.2. There, is deﬁned as {x : ϕ(x)}, where ϕ is the appropriate predicate.

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7. Conclusion Let us recall our original diagram, and take stock (Figure 7.4). K, we know, is non-empty, as is P − K. And this may be so even if the Veriﬁcation Principle is correct, and so T − P is empty, since the Fitch argument fails. We have also learned that the boundaries between K and T − K, and P and T − P are dialetheic. That is, there is a true sentence, αK , such that αK ∈ K and αK ∈ / K, and a true sentence, αP , such that αP ∈ P and αP ∈ / P. (This is what the ‘×’s on the new version of the diagram indicate.) And since αP is true, αP ∈ T ∧ αP ∈ / P, so T − P is also non-empty. For all I have said, this might be its only denizen. It cannot, therefore, be ruled out that T − P is empty as well. Whether or not this is so might well depend on the sense of possibility at issue. It is, at any rate, a matter for another occasion. T

P

K

Figure 7.4.

×

×

8 Knowability and Possible Epistemic Oddities Jc Beall

1 . No n - o m n i s c i e n ce a n d t h e K n ow a b i l i t y Ru l e Our world is non-omniscient. Nobody knows all truths, and nobody ever will. Does it follow that there are unknowable truths? Frederic Fitch (1963) ‘proved’ the afﬁrmative. In short, if some truth is unknown, then that it is unknown is itself unknowable; hence, given non-omniscience, there is some unknowable truth. Veriﬁcationists, who tie truth to veriﬁability, are committed to the so-called knowability rule (henceforth, KP).¹ Let K be the epistemic operator it is known by someone at some time that . . . , and ♦ the aletheic it is possible that. . . . KP is the following rule. α ♦K α Non-omniscience gives us α ∧ ¬K α, for some α. KP, in turn, gives us ♦K (α ∧ ¬K α). A few related rules governing ♦ and K quickly yield ♦(K α ∧ ¬K α), the possibility of ‘true contradictions’. (For the relevant rules, see Section 2) Fitch’s proof qua reductio makes the ﬁnal step: KP is unsound. In this paper, my concern is not so much with Fitch’s ‘proof’ against KP (or the conditional version). The proof is blocked in familiar ‘paraconsistent’ and ‘paracomplete’ logics (see Sections 2 and 3), both of which are independently motivated and, hence, available to veriﬁcationists. Rather, my concern is with the apparent commitment to ‘possibly true contradictions’. For discussion I am grateful to Colin Caret, Carrie Jenkins, Graham Priest, Greg Restall, and various members of the AHRC Arch´e Centre for Logic, Language, Mathematics, and Mind. Thanks also to Joe Salerno for editing the volume. ¹ The key idea is often given as a (universally quantiﬁed) conditional principle, but, to simplify current discussion, I focus on the rule form. ( The conditional version is often called ‘KP’, short for the knowability principle, but little confusion should result.)

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Regardless of its effect on veriﬁcationism, Fitch’s ‘proof’ highlights the oddity of epistemic optimism in a non-omniscient world. Given non-omniscience, KP involves something out of the ordinary. My main aim is to brieﬂy explore a few options for cashing out the given oddity. The paper runs as follows. In Section 2, I brieﬂy set out the relevant rules (and one corresponding premise) on which the Fitch argument relies. Section 3, in turn, brieﬂy reviews the main point of Beall, which suggests a paraconsistent response to Fitch’s ‘proof’ qua reductio of KP; however, the same considerations also motivate a similar paracomplete, non-paraconsistent, response, on which I will focus.² In Section 4 I set out the main focus: what to make of Fitch’s (initial) argument for the ‘possibility of gluts’. While the paracomplete response undercuts Fitch’s ‘proof ’ qua reductio, the issue of ‘possible, true contradictions’ remains open—especially in a nonparaconsistent framework, which is the target. I explore two options. In Section 5 I suggest but reject a ﬂat-footed option: living with possible—but merely possible—inconsistency. Section 6, in turn, explores the other option: avoiding even the ‘mere possibility’ of ‘true inconsistencies’. As a sort of synthesis, Section 7 brieﬂy sketches another option: a paracomplete and paraconsistent framework. Section 8 closes with some general comments and (brief ) responses to objections. 2 . Fi t c h’s Pro o f, i n Sh o r t For present purposes, the basic rules, involved in Fitch’s proof, may be divided into four categories: epistemic, (aletheic) modal, modal–epistemic (viz., KP), and ‘background’. The rules run as follows.³

1. Epistemic rules Veridicality (KV). The idea is that ‘knowledge implies truth’. Kα α Distribution (KC). That a conjunction is known implies that its conjuncts are known. K (α ∧ β) Kα∧Kβ ² See Priest’s (Ch. 7, this volume) for a development of the LP-based paraconsistent (indeed, dialetheic) position, and Section 7 for an alternative paraconsistent framework. ³ To facilitate comparison with Priest’s (Ch. 7, this volume), which discusses the related paraconsistent—indeed, dialetheic—response, I use a natural deduction version of the rules. As in Priest’s paper, [α], in the context of rule (or proof ), indicates a discharged assumption.

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2. Aletheic modal rules Non-contradiction (LNC). It is false that it’s possible that α ∧ ¬α is true, for any α.⁴ ¬♦(α ∧ ¬α) Closure (CP). That α is possible and that α implies β implies that β is possible. [α] .. . β

♦α ♦β

3. Modal–Epistemic rule Knowability (KP). The idea is that ‘truth implies knowability’, the key veriﬁcationist position. α ♦K α

4. Background Rules Adjunction and Simpliﬁcation. Conjunction behaves normally. α β α∧β

α∧β α β

Contraposition. That β is not true and that α implies β implies that α is not true. [α] .. . β

¬β ¬α

Fitch’s Proof, in short: Suppose, for reductio, α ∧ ¬K α, for some α. KP yields ♦K (α ∧ ¬K α). Given KC, we have that K (α ∧ ¬K α) K α ∧ K ¬K α. Simpliﬁcation, VK, and Adjunction (and transitivity of implication), yield that K (α ∧ ¬K α) K α ∧ ¬K α. But, then, CP gives ♦(K α ∧ ¬K α). LNC, in turn, delivers ¬♦(K α ∧ ¬K α). Contradiction. ⁴ Given the inter-deﬁnability of α and ¬♦¬α, which is assumed, LNC (as here put) amounts to the validity of ‘inferring’ ¬(α ∧ ¬α), for all α, from no premises—i.e., as theorem.

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3 . Ve r i ﬁ c a t i o n i sm , t h e K n owe r, a n d Fi t c h’s ‘ Pro o f ’ While I am not a veriﬁcationist, I agree with those who think that veriﬁcationism is not undermined by the ‘proof’. The result shows only that, given nonomniscience, veriﬁcationists cannot consistently endorse all of the rules involved in Fitch’s ‘proof ’. Since the rules in question are largely ‘classical’ (e.g., LNC, Contraposition), veriﬁcationism is best understood in a non-classical framework, one in which some of the given rules are invalid. One might worry that going non-classical is ad hoc. Were there no independent reason to reject some of the given rules, the worry would be warranted. But there are independent reasons to reject some of the given rules. Familiar semantic paradoxes, cases of vagueness, or other commonly ‘deviant’ phenomena, motivate familiar logics in which some of the given rules fail—notably, the LNC or Contraposition. Consider, for example, the Knower paradox, which involves a sentence κ that says of itself (only) that it is not known.⁵ Given LEM, κ is either known or not. In the latter case, κ is not known, and hence true. In the former case, KV gives us that κ is true, in which case κ is not known. Either way, κ is not known, and, so, κ is true. But, now, we have a proof that κ is true, and hence—on the basis of our proof—we know that κ is true. The upshot: there is a sentence, namely, κ, such that we know that κ is true but, as κ says, do not know that κ is true. In response to the Knower (or many such paradoxes), one might, as in Beall (2000), take the Knower to independently motivate ‘dialetheism’ with respect to knowledge—that K α ∧ ¬K α is true, for some α. On such a line, a paraconsistent logic, in which such inconsistency is ‘harnessed’, is motivated. But, then, at least Contraposition is invalid—and, hence, Fitch’s ‘proof ’ fails. Graham Priest (Ch. 7, this volume) advocates just such a line in the context of LP, a paraconsistent logic in which both LEM and LNC are valid. On the other hand, a paracomplete (and non-paraconsistent) response to the Knower is equally natural. A paracomplete logic is one in which LEM is invalid.⁶ Unlike the dialetheic response, a paracomplete theorist rejects the Knower instances of LEM. One familiar paracomplete framework is K 3 , the Strong Kleene framework.⁷ In such a logic, not only is Contraposition invalid, ⁵ Here, I simply use ‘is known’ rather than the operator. In a suitably non-classical framework, we can enjoy a genuine (intersubstitutable) truth predicate that, in turn, diminishes the importance of distinguishing between operators and predicates. ⁶ Accordingly, Intuitionistic logic counts as paracomplete. I will not discuss the Intuitionistic options, as these are well known. (Besides, Intuitionistic logic does not afford viable options for the broader class of paradoxes—e.g., Liars, etc.) ⁷ I set issues of a suitable conditional aside, and concentrate mostly on the given ‘rules’. A suitable conditional is an important issue, but it is one that would take the discussion too far aﬁeld.

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but LNC is also invalid.⁸ Accordingly, a veriﬁcationist who, for independent reasons, endorses K 3 (or some suitable extension), need not worry about Fitch’s proof qua reductio of KP. While I am (very) sympathetic with the paraconsistent—indeed, dialetheic—framework, I will focus on non-dialetheic and, except, for Section 7, non-paraconsistent but paracomplete responses. Either way, veriﬁcationists have independent reason—e.g., Knower or the like—to endorse a non-classical logic in which Fitch’s ‘proof’, qua reductio of KP, fails.

4 . T h e Re a l Is s u e : Po s s i b l y Tr u e Gl u t s ? Though veriﬁcationists needn’t worry about Fitch’s proof qua reductio of KP, there is more to Fitch’s argument than the ﬁnal few (reductio) steps. As in Section 1, Fitch’s argument highlights the oddity of epistemic optimism in a non-omniscient world. Contraposition and LNC aside, the remaining rules (see Section 2) still leave a curiosity: the apparent commitment to ‘possibly true contradictions’. To see the issue, we assume an extension of K 3 in which the relevant rules remain, except, of course, for Contraposition and LNC. (See Section 5 for the natural semantics.) The initial steps of Fitch’s proof still go through: nonomniscience gives us α ∧ ¬K α, for some α. KP gives us ♦K (α ∧ ¬K α). But K (α ∧ ¬K α) K α ∧ K ¬K α from KC. Simpliﬁcation, VK, and Adjunction (and transitivity of implication), yield that K (α ∧ ¬K α) K α ∧ ¬K α. But, then, CP gives ♦(K α ∧ ¬K α). So, losing Contraposition or LNC still leaves the noted oddity. In a non-paraconsistent setting, ‘possibly true contradictions’ are at least curious. The real issue, then, is what to make of the given oddity in a non-paraconsistent, paracomplete setting. How should a paracomplete, nonparaconsistent veriﬁcationist—or KP theorist, in general—respond to the apparent ‘possibly true contradictions’? As in Section 3, I will explore two salient options. The ﬁrst option is to simply live with the given ‘oddity’. The second option, rejecting even the ‘mere possibility’ of ‘true inconsistency’, involves expanding one’s space of possibilities while restricting one’s account of validity. After discussing such (non-paraconsistent) options, I turn to a brief sketch of a ‘compromise’, a non-dialetheic but nonetheless paraconsistent and paracomplete framework. I will then close (in Section 8) with general comments, brieﬂy answering two objections. ⁸ According to K 3 , ‘Explosion’ is valid: α ∧ ¬α β. But LEM is invalid: ¬(α ∧ ¬α). Were Contraposition valid, we’d immediately have ¬β ¬α ∨ α, for any β and α. But we don’t have that in K 3 , since, as said, we do not have LEM.

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5 . L i v i n g w i t h Me re l y Po s s i b l e Gl u t s Given non-omniscience, veriﬁcationists—and KP theorists, in general—are apparently committed to the possibility of ‘true contradictions’. In a nonparaconsistent context, which is the chief concern in this paper, such a commitment is curious. The question is: what to make of it? The ﬂat-footed option is to just live with it. On the surface, the ‘possibility of true contradictions’ is startling. Upon inspection, though, the situation is in many respects mundane, especially if there’s exactly one—non-actual, merely possible—such ‘possibility’. The ﬂat-footed response acknowledges a (unique) trivial world, and she learns to live with it. To make the idea clearer, I brieﬂy sketch a basic—paracomplete but nonparaconsistent—semantics. I then return to the ﬂat-footed response.

5.1. Paracomplete semantics with the trivial world We are considering a paracomplete and non-paraconsistent framework for veriﬁcationism (or KP theorist, in general), one in which there’s a unique possibility of ‘true contradictions’. By way of contrast with the natural LP-based paraconsistent framework,⁹ I focus on an extension of K 3 , the Strong Kleene framework. Our set of semantic values, namely, V = {1 , .5 , 0}, is ordered in the standard way. D, our designated values, comprises exactly 1. In addition to our usual extensional connectives, we add two unary connectives, the epistemic K and the aletheic ♦. (We deﬁne as ¬♦¬.) K and ♦ are intended to be modal connectives. Accordingly, we pursue a modal extension of K 3 . Interpretations are structures W , R, E , v, w⊥ , where W ∩ {w⊥ } comprises ‘worlds’, with w⊥ ∈ / W the trivial world. R and E are binary relations on W ∪ {w⊥ } (each at least reﬂexive), and v : A × W → V is a valuation from atomics and worlds into {1 , .5 , 0}. For convenience, we let vw (α) = v(α, w), this being the value of α at w. The value of any sentence at any w ∈ W is achieved via the following clauses.¹⁰ vw (¬α) vw (α ∧ β) vw (α ∨ β) vw (♦α) vw (K α)

= = = = =

1 − vw (α) m i n{vw (α) , vw (β)} m a x{vw (α) , vw (β)} m a x{vw (α) : wRw for any w ∈ W ∪ {w⊥ }} m i n{vw (α) : wEw for any w ∈ W ∪ {w⊥ }}

⁹ See Priest (Ch. 7, this volume). ¹⁰ Except for the clause concerning w⊥ , the following are the standard Kleene clauses for modal connectives.

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With respect to w⊥ , the trivial world, the clause for any interpretation is the obvious (trivial!) one: v⊥ (α) = 1 Finally, we deﬁne validity as ‘truth-preservation’ over all worlds of all interpretations. A few features of the framework are notable. To begin, the semantics is clearly paracomplete in that α ∨ ¬α is invalid. Just consider a model in which vw (α) = 0.5, for some w ∈ W. Since, as one may verify, α ∨ ¬α and ¬(α ∧ ¬α) are equivalent in the semantics, the same (counter-) model serves to invalidate LNC (see Section 3). Similarly for Contraposition.¹¹ On the other hand, it is clear that Adjunction and Simpliﬁcation are valid. Moreover, and more to the current point, the remaining epistemic and aletheicmodal rules are all validated. (See Section 2.) KV. Suppose that vw (K α) = 1, for some w ∈ W and α. Since R is reﬂexive, we have it that vw (α) = 1. KC. Suppose that vw (K (α ∧ β) ) = 1. Then vw (α ∧ β) = 1 = vw (α) = vw (β) for all w ∈ W such that wRw . But, then, vw (K α) = 1 = vw (K β). CP. Suppose that α β and, for some interpretation, vw (♦α) = 1. Then vw (α) = 1, for some w such that wRw . But, by supposition, there’s no world, in any interpretation, at which α is true and β not true. Hence, vw (β) = 1, and so vw (♦β) = 1. The question, of course, turns to our essential modal–epistemic rule KP, which is not valid on the current semantics. Can the semantics be tweaked to ensure the validity of KP? Yes. Indeed, the whole point of invoking w⊥ , which has thus far played no role, is to ensure the validity of KP. To achieve KP we stipulate that wRw⊥ , for all worlds w (including w⊥ ). That KP is now valid is obvious; it is vacuously so.¹² So, except for Contraposition and LNC—which, as in Section 3, are suspect for independent reasons—the current framework preserves all of the key rules, including KP. Because KP is preserved, (actual) non-omniscience forces an oddity: the trivial world. Indeed, so long as validity is deﬁned as ‘all points validity’ (e.g., truth-preservation at all worlds), then, unless one goes with a paraconsistent framework, I see no way to avoid the trivial world without giving up KP.¹³ The issue, to which I now return, is whether such oddity is too odd. ¹¹ The corresponding LP-based paraconsistent framework validates LNC but, as here, not Contraposition. See Priest’s (Ch. 7, this volume). ¹² Compare the LP-based dialetheic model (Ibid.), which likewise invokes w⊥ for the same job. (One difference, of course, is that the trivial world naturally falls out of the LP framework, whereas here it is at least curious.) ¹³ If one endorses a paraconsistent logic, a more natural paracomplete framework might be had. See Section 7 for a sketch.

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5.2. The ﬂat-footed response KP, the reﬂection of high epistemic optimism, produces an oddity in a non-omniscient world. The apparent oddity, given the going rules (except Contraposition and LNC), is the possibility of true contradictions. The ﬂatfooted response to such oddity is to accept it, but accept it as merely possible and, importantly, a unique case. Given non-omniscience, KP is preserved in virtue of the unique trivial world—the possibility in which ‘true contradictions’ occur. Is the trivial world too high a price to pay for KP? The answer is not obvious. Admittedly, it may be very difﬁcult to fully understand the trivial world. While one can easily understand that the trivial world is the world at which every sentence is true, it is not easy to understand what such a world is like. Still, there are a few things that can be said on the trivial world’s behalf. 1. KP! As in Section 5.1, if validity is to be understood as all points validity (e.g., truth-preservation at all worlds), it is difﬁcult to retain KP without the trivial world—unless one goes with a paraconsistent logic, which is set aside at this stage. (See Section 7.) So, one virtue of the trivial world is that it affords the chief desideratum for a non-classical veriﬁcationism: it preserves KP. 2. Concrete Explosion! In the current semantics we have ‘explosion’, that is, α, ¬α β. In many (most) non-paraconsistent logics, explosion itself is vacuously achieved: it is valid in virtue of no interpretation in which the premises are true. Here, we have ‘concrete evidence’ of explosion: any world in which α ∧ ¬α is true is the explosive one in which everything is true. There is something to say for such ‘concrete evidence’ (although I wouldn’t put too much weight on this). 3. Merely possible! Similarly, while the possibility of ‘true contradictions’ sounds startling at ﬁrst, the current proposal is rather mundane. After all, in discussing the possibility of ‘true contradictions’, one may quickly point out that we’re talking about a unique and limit case—the merely possible trivial world. In addition to (1)–(3), there is another—perhaps the strongest—point to consider. As throughout, KP’s validity in a non-omniscient world is indeed odd. One reason we might think it odd is that it clashes against the ‘normal’ behaviour of our connectives—which behaviour, perhaps, is by and large classical. The trivial world, which, on the current proposal, is the result of KP’s validity (and a non-omniscient world), might best be seen as a world in which our connectives ‘go on holiday’. Clearly, the connectives are not behaving normally at w⊥ . Perhaps such abnormal behaviour is the price of KP’s validity, given non-omniscience.

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5.3. Trouble with ﬂat-footedness Despite its virtues (if virtues they be), the trivial world is nonetheless disappointing in the current context. While I do not think the trivial world itself is terribly objectionable, its role in the current context is prima facie problematic. The heart of veriﬁcationism is KP, a rule that, at least traditionally, has served to distinguish veriﬁcationists from non-veriﬁcationists. On the current proposal, KP is preserved—indeed, its validity achieved—solely in virtue of the trivial world. But, now, the traditional role of KP cannot be served. After all, it is obvious that anyone—even a classical logician—could acknowledge the trivial world, at least in the fashion in Section 5.1. But if anyone can have the trivial world, anyone can have KP. Surely veriﬁcationism is more demanding than that.¹⁴ The ﬂat-footed response, then, is ultimately unsatisfactory. Unfortunately, without going paraconsistent (though not necessarily dialetheic), there is no obvious way to preserve KP without the trivial world, at least if validity remains ‘all points validity’. Giving up such a notion of validity provides an alternative paracomplete approach, to which I now brieﬂy turn. 6 . Ab n o r m a l Ep i s t e m i c Po s s i b i l i t i e s In Section 5, I suggested—but found wanting—the ‘ﬂat-footed’ paracomplete response to the veriﬁcationist’s apparent commitment to possibly true contradictions. Might an alternative paracomplete response do away with the ‘possibly true contradictions’ altogether? In this section, I brieﬂy explore one route towards doing as much.¹⁵ On this approach, the oddity of KP in a non-omniscient world is not ‘possibly true contradictions’, but rather the sheer oddity of possibly knowing an unknown truth. I will ﬁrst give a philosophical sketch of the idea, followed by a slightly more formal sketch, and then offer a few comments on the overall framework.

6.1. The philosophical story Veriﬁcationists tie truth to veriﬁcation. An essential ingredient of the connection is reﬂected in (at least) KP. What Fitch seemed to show is that, given nonomniscience, KP leads to possibly true contradictions. But perhaps another ¹⁴ I should say that this point might affect Priest’s LP-based proposal (Ch. 7, this volume). A more natural (paraconsistent) approach, not subject to the same problem, is brieﬂy sketched in Section 7. ¹⁵ Other options are available, of course, if the extensional connectives (negation, conjunction, disjunction) behave non-standardly, but I am chieﬂy interested in ‘normal’ behaviour for such (extensional) connectives—i.e., classical input, classical output. (One could go weaker than Strong Kleene, but independent motivation for such logics is more difﬁcult to ﬁnd than for the K 3 case.)

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lesson may be drawn. In particular, what veriﬁcationists are committed to is not some possibly true contradiction; rather, they’re committed to epistemically abnormal—but none the less entirely (aletheically) possible—worlds, worlds in which, for example, knowing an unknown truth happens. Veriﬁcationists are committed to ♦(α ∧ ¬K α), for some α, but the possibility in question is epistemically abnormal, a world, perhaps, in which ‘epistemic ﬁctions’ transpire.¹⁶ At such worlds, the normal behaviour of K breaks down in various respects. In particular, given that possibilities are one and all consistent (though not necessarily complete), the normal distributive behaviour of K breaks down. Such abnormal worlds are precisely where the oddity—but not inconsistency—of KP’s clash with non-omniscience emerges.¹⁷ The proposal, then, is to avoid ‘possibly true gluts’ via expanding one’s range of possibilities. Speciﬁcally, the veriﬁcationist acknowledges epistemically abnormal possibilities in which K is deviant. At the same time, the veriﬁcationist is committed, on the whole, to the validity of standard K -rules. While knowledge might deviate from its normal behaviour at odd points, the validity of standard K -rules ought to remain intact. Accordingly, in addition to expanding her range of possibilities, the veriﬁcationist narrows her account of validity—or, what comes to much the same, keeps her account of validity focused on the non-deviant, normal possibilities. A formal—and, in some respects, familiar—picture will be helpful. I will return to philosophical discussion in Section 6.3.

6.2. A formal picture The basic idea can be modelled along ‘non-normal lines’.¹⁸ We make a distinction among worlds—the normal and non-normal (or abnormal, as I will say). In turn, we deﬁne validity as ‘truth-preservation’ over only one sort of world, not as ‘all points (worlds) validity’. The behaviour of target operators at the abnormal worlds is recognized, but such behaviour is (in effect) ignored for purposes of deﬁning validity. A simple account is as follows. Let V and D be as in Section 6 (Strong Kleene base). Our interpretations are structures W , N , N ∗ , R, E, v, ε, where W = N ∪ N ∗ , with N (normal worlds) and N ∗ (abnormal) non-empty, and N ∩ N ∗ = ∅. R and E are as before, each being at least reﬂexive on W. ε, to which I’ll return, has the job ¹⁶ Compare Priest (1992). ¹⁷ Admittedly, if one acknowledges ‘abnormal epistemic worlds’ in which, e.g., K ’s normal distributive behaviour breaks down, there may be no strong reason to reject other such abnormal worlds in which more radical deviance occurs (such as knowing a contradiction!). Even so, the current proposal aims at avoiding ‘possibly true inconsistency’ altogether. ¹⁸ The idea behind ‘non-normal semantics’ comes from Kripke (1965), wherein the aim was to model Lewis systems weaker than S4. Arguably more signiﬁcant philosophical use of non-normal semantics has emerged in literature on ‘relevant logics’. See Dunn and Restall (2002) and references therein.

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of evaluating K -claims at abnormal worlds. v : A × W −→ V assigns values to all atomics at all worlds, normal and not. Interpretations are extended to all sentences at all worlds via the following clauses. 1. Extensional. For any w ∈ W, vw (¬α) = 1 − vw (α) vw (α ∧ β) = m i n{vw (α) , vw (β)} vw (α ∨ β) = m a x{vw (α) , vw (β)} 2. Possibility. For any w ∈ W, vw (♦α) = m a x{vw (α) : wRw for any w ∈ W} 3. Knowledge. (a) Normal worlds. For any w ∈ N vw (K α) = m i n{vw (α) : wEw for any w ∈ W} (b) Abnormal worlds. For any w ∈ N ∗ vw (K α) = εw (K α) The job of ε, as above, is to give values to K -claims at our abnormal worlds. ε may be viewed as an ‘arbitrary evaluator’ of K -claims at abnormal worlds, though the arbitrariness, to avoid inconsistency at abnormal worlds, is subject to the following constraint. εw (K α) = 1 ⇒ vw (α) = 1 Finally, validity is deﬁned as ‘truth-preservation’ over all normal worlds of all interpretations. So given, the semantics delivers some, but not all, of the target principles. Importantly, we do not get KP. For example, consider an interpretation in which N = {w}, N ∗ = {w∗ }, and, in addition to reﬂexivity, we have only wEw∗ . Now let vw (α) = 1 and vw∗ (α) = 0 = εw∗ (K α). Figure 8.1 shows diagram of the counter-example. R-accessibility R

w

w*

E-accessibility E

w

Values at worlds

w*

α

Kα

◊Kα

w

w

w

1

0

0

w*

w*

w*

0

0

0

Figure 8.1.

The trouble, of course, is that our class of interpretations is too big.

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Towards narrowing our class of interpretations, let a narcissistic world—an n-world, for short—be any world w (normal or abnormal) such that, for any w ∈ W, wRw or wEw ⇒ w = w N-worlds see only themselves, in either relevant sense of ‘see’. Now, deﬁne a V∗ -model to be any interpretation (as above) such that the following holds. V∗ . For any normal w, if vw (α) = 1, then there is some abnormal n-world w∗ such that εw∗ (K α) = 1 and wRw∗ . In turn, validity is deﬁned as ‘truth-preservation’ over all normal worlds of all V∗ -models. That there are V∗ -models may be seen by tweaking the previous counter-example to get the results shown in Figure 8.2 (where ‘starred’ worlds are abnormal).¹⁹ R-accessibility R

w

E-accessibility

w1* w2* w3*

E

Values at worlds α

w1* w2* w3*

w

Kα K(α

Kα)

◊Kα

w

w

w

1

0

0

1

w1*

w1*

w1* 1

0

1

0

w2*

w2*

w2* 0

.5

.5

.5

w3*

w3*

w3* 1

1

.5

.5

Figure 8.2.

The corresponding picture is shown in Figure 8.3.²⁰ R R,E

R,E

R,E

w3*

w2*

α

α

Kα

E

w α Kα

R,E R

w1* α K(α

Kα Kα)

Figure 8.3. ¹⁹ .5 is not forced at w∗2 . One could also give K α the value 0. ²⁰ In general, a doubly squared world is abnormal.

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Notice that each of the abnormal worlds except for w∗2 serves as an n-world for w. The V∗ -model above also serves to invalidate Fitch’s chief inference—from knowability to known. In the model, w is a non-omniscient (normal) world with respect to α, but—thanks to the abnormal worlds—it is possible to know α.

6.3. Comments I turn to a few comments about the ‘abnormal’ approach. I begin with a few salient virtues of the semantics, and then brieﬂy turn to the broader, philosophical picture (returning to the topic in Section 8). As expected, Contraposition and LNC are invalidated, and the regular extensional connective remain normal (as in Strong Kleene). More importantly, the semantics validate each of the standard K -rules, including the essential KP. KV. Let vw (K α) = 1 for some w ∈ N . Then vw (α) = 1 for all w ∈ W such that wEw . Since E is reﬂexive, vw (α) = 1. KC. Let vw (K (α ∧ β) ) = 1 for some w ∈ N . Then vw (α ∧ β) = 1 = vw (α) = 1 = vw (β) for all w ∈ W such that wEw . Hence, as E is reﬂexive, vw (K α) = 1 = vw (K β).²¹ KP. Let vw (α) = 1. Then, by V∗ , there’s some abnormal n-world w∗ such that wRw∗ and vw∗ (K α) = εw∗ (K α) = 1. On the other hand, not everything is retained. Not surprisingly, the deviation from ‘all points validity’ to ‘all normal points’ invalidates certain inferences, notably, CP (see Section 2). For example, as above, KC is valid, and so K (α ∧ ¬K α) implies K α ∧ K ¬K α. Moreover, KP is valid (as above). Yet, ♦K (α ∧ ¬K α) ♦(K α ∧ K ¬K α), since the relevant world—the world at which K (α ∧ ¬K α) is true—might be abnormal.²² In abnormal worlds, K can deviate from its normal behaviour. One might think of such worlds not only as ‘odd epistemic possibilities’ but, further, as worlds at which valid K -behaviour breaks down. Turning to the broader philosophical picture, a few virtues of the current account may be noted. 1. Consistency. A main motivation behind the ‘abnormal’ approach was to avoid even the possibility of ‘true contradictions’. While achieving as much requires constraints on ε, the aim seems to be realized, for what that is worth. ²¹ But see further discussion below! ²² This is not surprising given non-normal semantics. Indeed, as mentioned, Kripke’s original motivation beyond non-normal worlds semantics was to model Lewis systems weaker than S4, systems in which Necessitation fails. Moreover, in subsequent non-normal approaches to conditionals, the aim is often to model conditionals for which there is no (standard) deduction theorem.

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2. Fitch’s Lesson. Fitch’s ‘proof’, as in Section 3, points to a genuine oddity in combining KP and non-omniscience. In the current ‘abnormal’ case, the oddity is fully acknowledged; it shows up as ‘abnormal possibilities’, possibilities that seem inconsistent but, in the end, avoid outright inconsistency via deviant K behaviour. 3. Failure of CP. While CP’s failure is odd, the current story comes with an explanation: distribution of K fails inside (aletheic) modal contexts because such contexts are pointing to epistemically deviant worlds. By my lights, there is a coherent story along the ‘abnormal’ lines—odd, but coherent. Some oddness, as Fitch highlighted, is inevitable, at least given KP and non-omniscience. The question, of course, is whether the ‘abnormal’ approach to the inevitable oddness is overly odd. Ultimately, that is an issue for veriﬁcationists. As far as I can see, there is nothing in veriﬁcationism that either rules out or implausibly conﬂicts with (something like) the foregoing ‘abnormal’ approach. Whether, in the end, the abnormal approach is ultimately viable is something that I leave for debate. Doing away with even the possibility of ‘true contradictions’ is difﬁcult. Perhaps, ultimately, veriﬁcationists are better off accepting Fitch’s argument for apparently possible ‘true contradictions’ in a broader paraconsistent (but non-dialetheic) framework. I turn now to a brief sketch of such an approach. In Section 8 I (very brieﬂy) return to the overall philosophical viability of the canvassed approaches.

7 . Sy n t h e s i s : Ga p s a n d Me re l y Po s s i b l e Gl u t s If, as in Section 5, one goes with ‘all points validity’ in a (normal) paracomplete but non-paraconsistent framework, the veriﬁcationist seems to be stuck with the trivial world—and a vacuous KP. Dropping ‘all points validity’, as in Section 6, affords more options, but one is forced to give up a few more rules (e.g., KC in the context of aletheic modalities). While each option may hold promise (especially the second), a further option is worth noting. In this section, I brieﬂy sketch—without arguing for—another option: an ‘all points validity’ approach that is both paracomplete and paraconsistent but nonetheless non-dialetheic. The paraconsistent veriﬁcationist blocks Fitch’s ‘proof ’ at the same place(s) that K 3 does—either LNC or Contraposition. With respect to the ‘oddity’ of KP in a non-omniscient world, the paraconsistent response is straightforward: non-omniscience and KP generate an inconsistent possibility—knowing an unknown truth. But such possible inconsistency needn’t generate actual inconsistency, at least in a paracomplete paraconsistent framework. In a paracomplete framework, the

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paraconsistent veriﬁcationist may acknowledge ‘possible gluts’ without thereby accepting dialetheism—the view according to which there is actual inconsistency.²³ Here, I sketch a basic four-valued framework for veriﬁcationists, an extension of the familiar Anderson–Belnap framework (1992).

7.1. The basic model Interpretations are structures W , R, E , v, where W , R, and E are as before (with R and E at least reﬂexive). Here, it is convenient to let V, our semantic values, be P ({1 , 0}).²⁴ Then v is any function from S × W into V subject to the following constraints. 1. Negation (a) 1 ∈ vw (¬α) iff 0 ∈ vw (α) (b) 0 ∈ vw (¬α) iff 1 ∈ vw (α) 2. Conjunction (a) 1 ∈ vw (α ∧ β) iff 1 ∈ vw (α) and 1 ∈ vw (β) (b) 0 ∈ vw (α ∧ β) iff 0 ∈ vw (α) or 0 ∈ vw (β) 3. Disjunction (a) 1 ∈ vw (α ∨ β) iff 1 ∈ vw (α) or 1 ∈ vw (β) (b) 0 ∈ vw (α ∨ β) iff 0 ∈ vw (α) and 0 ∈ vw (β) 4. Possibility (a) 1 ∈ vw (♦α) iff 1 ∈ vw (α) for some w such that wRw . (b) 0 ∈ vw (♦α) iff 0 ∈ vw (α) for all w such that wRw . 5. Knowledge (a) 1 ∈ vw (K α) iff 1 ∈ vw (α) for all w such that wEw . (b) 0 ∈ vw (K α) iff 0 ∈ vw (α) for some w such that wEw . Validity is deﬁned as ‘truth preservation’ over all worlds of all interpretations. With the expected exception of Contraposition and LNC, the semantics, with validity so deﬁned, preserve most of the target principles: KV, KC, CP, Adjunction, etc. The question, of course, concerns KP. ²³ In Priest’s alternative LP (dialetheic) setting, which is not paracomplete, the ‘mere possibility’ of ‘true contradictions’ immediately generates actual inconsistency. In LP (or the target extension), we have ¬♦(α ∧ ¬α). Hence, given any β such that ♦(β ∧ ¬β) is actually true, we immediately have actual inconsistency. For further discussion, see Restall (1997), wherein Restall ﬁrst discussed the point regarding the LP situation, and Beall and Restall (2006) for broader discussion [3]. ²⁴ This idea is due to Dunn (1966, 1976).

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Alas, KP is invalid. Just consider an interpretation in which W = {w, w } and, in addition to the required reﬂexivity of R and E, we have wRw and wEw , but also vw (α) = {1} and vw (α) = {0}.²⁵ This serves as a counter-example to KP. The diagram shown in Figure 8.4 may be useful. R-accessibility R

w w′

E-accessibility E

Values at worlds

w w′

α

Kα

◊Kα

◊K(α

w

w

w

{1}

{0}

{0}

{0}

w′

w′

w′

{0}

{0}

{0}

{0}

Kα)

Figure 8.4.

A picture of the counter-example is shown in Figure 8.5. I give only the value of α. R,E

R,E R,E w

w′

α

¬α

Figure 8.5.

So, KP fails. The trouble, of course, is that our class of interpretations is too big. To get the target interpretations, we need to pare down our class of interpretations.

7.2. The target: V-models The natural remedy is to invoke n-worlds, as in Section 6 (but now without abnormal worlds). Let an epistemically narcissistic world—an n-world, for short—be a world w such that, for any w ∈ W, wEw ⇒ w = w Since E is reﬂexive, every world epistemically sees itself; n-worlds (epistemically) see only themselves. In turn, we deﬁne a V-model (for Veriﬁcationism model) to be any interpretation (as above) that conforms to the following. ²⁵ For that matter, you could let vw (α) = ∅.

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V. If 1 ∈ vw (α) then there’s some n-world w such that wRw and 1 ∈ vw (α ∧ ¬α).²⁶ Validity is deﬁned as before, but now only over V-models. And with that we get KP. That there are V-models is clear. In particular, we have V-models that invalidate Fitch’s chief inference—from knowability to knowledge. A simple V-model—perhaps the simplest—in which the Fitch inference fails (viz., from knowable to known) is shown in Figure 8.6. R-accessibility R

w0 w1 w2

E-accessibility

Values at worlds

w0 w1 w2

E

α

Kα

◊Kα

◊K(α

w0

w0

w0

{1}

{0}

{1,0}

{1,0}

w1

w1

w1

{1,0}

{1,0}

{1,0}

{1,0}

w2

w2

w2

{0}

{0}

{1,0}

{1,0}

Kα)

Figure 8.6.

A picture of the model is shown in Figure 8.7. R R,E

R,E w2

α

E

w0

α

R,E w1

R

α

α

Figure 8.7.

This is a model in which the given Fitch inference fails, since α is knowable at w 0 but not thereby known. The trouble, of course, is that the paracomplete (but paraconsistent) V-models were supposed to afford an entirely consistent actual world while allowing for ‘merely possible inconsistency’. In the simple model above, such a promise does not show up. After all, w 1 serves as an n-world for ²⁶ A simpler, perhaps more natural, route would be to add a distinguished actual world and impose V only on that, but I will go with the ‘all worlds of all models’ approach.

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both w 0 and w 2 (and itself). Since there are only three worlds, the above model, in effect, is basically an LP-based model. (If we demanded LEM for all atomics, it would be an LP-based model.) The upshot is that, in the above model, the possibility of α-inconsistency—and, in particular, K α-inconsistency—trickles back into actual inconsistency: ♦K α is true and false at all worlds, and hence the actual. To get a consistent but non-omniscient ‘actual world’ (say, w 0 ), we simply add more worlds. The simplest addition is the null world w∅ , shown in Figure 8.8.²⁷ Values at worlds

R-accessibility

E-accessibility

R w0 w1 w2 w0

E w0 w1 w2 w0

w0

w0

α

Kα

◊Kα

◊K(α

Kα)

w0

{1}

{0}

{1}

{1}

{1,0}

{1,0}

{1,0}

w1

w1

w1

{1,0}

w2

w2

w2

{0}

{0}

{1}

{1}

w0

0

0

0

0

w0

w0

Figure 8.8.

The corresponding picture can be seen in Figure 8.9. R R,E

R

R,E w0

R

R,E w2 α

E

w0 α

R,E w1

R α

α

Figure 8.9.

This model is more attractive than the former, simpler model, as it leaves ‘the actual world’ consistent while nonetheless refuting the Fitch inference. I move to a few general comments.

7.3. Comments There are various virtues of the V-models over the LP-based paraconsistent approach. For present purposes, I list the salient ones. ²⁷ Note that the null world is not essential; one merely needs the appropriate ‘incompleteness’.

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1. Merely Possible Inconsistency. The foregoing approach shares the basic response common to any paraconsistent veriﬁcationism: namely, that KP (and the other set of K-rules) forces inconsistency in the face of non-omniscience. But since the current approach is also paracomplete, there’s no threat that ‘merely possible inconsistency’ implies actual inconsistency—as is the case in an LP-based approach.²⁸ The upshot is that a veriﬁcationist can admit that the possibility of knowing unknown truths forces inconsistency; however, it need only force inconsistency ‘elsewhere’ and only elsewhere—some merely possible world. 2. Trivial world. While the trivial world, without further constraints, certainly shows up in V-models, it isn’t required to ensure KP (or, as discussed below, the countermodel to Fitch’s basic inference). There are V-models, of course, in which LEM holds among all worlds of the given models (viz., LP-models!); however, being based on a broader four-valued framework, LEM certainly isn’t valid. In short, V-models allow for ‘incomplete worlds’, worlds in which neither α nor ¬α show up (as it were). 3. Not entirely inconsistent. Moreover, while KP is fully ensured, as above, by inconsistency ‘elsewhere’, V-models allow for ‘local inconsistency’ to do the work. In particular, the n-worlds, into which knowing ‘non-omniscience truths’ (e.g., α ∧ ¬K α) forces inconsistency, need not themselves be entirely inconsistent. Because of incompleteness, there can be many n-worlds throughout which the given inconsistency is distributed, and many of them can be perfectly consistent in proper quarters. There are probably other notable virtues vis-`a-vis the LP-based (paraconsistent) approach, but I turn to one ﬁnal matter. One might think that V, which invokes suitable en-worlds to ensure KP, is ad hoc. Such charges are notoriously difﬁcult to adjudicate, and I won’t pursue the issue in any depth here. By my lights, V is not at all ad hoc. After all, V reﬂects the (paraconsistent and paracomplete) veriﬁcationist’s chief tenet: that all truths are knowable—even those that reﬂect non-omniscience, and hence generate inconsistency elsewhere. Rather than being some ad hoc posit, the relevant en-worlds that V invokes might best be seen as an implicit feature of veriﬁcationism. One thing is uncontroversial about Fitch’s argument: veriﬁcationism’s commitment to KP makes for some oddity in its confrontation with our actual non-omniscience. The paraconsistent-cum-paracomplete framework accepts that the given oddity is indeed as it appears: possibly true inconsistency. But veriﬁcationists are not thereby dialetheists; such inconsistency, in virtue of incompleteness, is harnessed at the merely possible. ²⁸ Of course, enriching the language might raise further problems, but the aim here is merely to sketch a beginning option.

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8 . C l o s i n g Re m a r k s I would like to end this paper by arguing for the supremacy of one of the canvassed options, but I cannot. As above, I am not a veriﬁcationist, and so not committed to KP via a prior theory of truth (or meaning, or etc.). Moreover, I know of no good arguments for KP.²⁹ Still, I ﬁnd KP plausible and think that each of the canvassed approaches has merit. Instead of trying to settle which, if any, of the given approaches is best, I will close by answering the most salient worries that confront each of the two chief options—setting aside the ‘ﬂat-footed response’ (see Section 6).

8.1. Abnormal epistemic possibilities? The suggestion, here, is that KP holds in virtue of abnormal epistemic possibilities, where these are possibilities in which normal K behaviour breaks down. The chief worry about such a picture is that we are no longer talking about knowledge when we are talking about ‘abnormal K behaviour’. Put differently (with echoes of Quine), the charge is that necessarily, K behaves like such and so—in particular, distributes over conjunction (and is such that, e.g., CP is valid). Hence, the ‘abnormal epistemic possibility’ framework is really one in which we are introducing two distinct epistemic operators, one reﬂecting our ‘real knowledge operator/predicate’, the other some ‘deviant’ (but distinct) operator/predicate. As such, veriﬁcationists—and KP theorists, in general—are still stuck with the original problems confronting our ‘real’ item.³⁰ By way of reply, the way I look at the situation is (brieﬂy) as follows. Veriﬁcationists are committed to some sort of oddity. If veriﬁcationists are likewise committed to the bulk of the given rules (see Section 3) and ‘no possibly true inconsistency’, then a natural suggestion, as in Section 6, is that K behaves differently at different sorts of worlds. Now, the charge, as above, maintains that we have two different K s, rather than a single K that, as said, behaves differently at different points. I’m not sure how to adjudicate this. If the project is to give the veriﬁcationist—or KP theorist, in general—entirely consistent worlds across the board, while also retaining the bulk of the given rules, then it’s unlikely that there’s a distinct K of the sort presupposed in the charge (as opposed to a single K that behaves differently at different points, as per the proposal). After all, if ‘the real K ’ is like that (e.g., supports distribution inside the diamond), then the veriﬁcationist is stuck with inconsistent worlds. Again, I don’t know how to ultimately adjudicate the matter. I’m not a veriﬁcationist, but I think it worthwhile to see how the veriﬁcationist might ²⁹ I do not consider appeals to ‘intuition’ good arguments. ³⁰ I am grateful to Carrie Jenkins for pushing this point.

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enjoy entirely consistent worlds and the bulk of the rules. Of course, she has to give up something, and the Section 6 proposal gives up ‘all points validity’ (and, in turn, CP). In the end, perhaps the resulting picture is too implausible to suit veriﬁcationists. I don’t know. But I do not see why they can’t have a single K that behaves differently at different points. Indeed, veriﬁcationists—or, again, KP theorists, in general—can take the lesson of Fitch’s ‘proof’ to be that we were ignoring various possibilities, namely, ones in which our unique K behaves in very abnormal ways.³¹

8.2. Paraconsistent but non-dialetheic V-models There may well be various worries about this approach, many of which might spring from general worries about ‘possibly true contradictions’. This paper is not the place to address such broad worries.³² Instead, I will assume a general openness to the idea of (merely) ‘possibly true gluts’. There remains a salient worry for the V-model approach.³³ The worry, in short, is that the proposal calls for too much. In particular, the proposal commits us to the possibility of α ∧ ¬α for every true α. Even if one is prepared to acknowledge merely possible gluts, it is hard to accept that for every truth α, it is possible that α is true and false! I think that, by way of reply, one needn’t quite accept as much as the V-model approach yields. The V-models were so given as a simple example, but one should be able to restrict matters further so as to avoid the going worry. For example, one approach might be to restrict condition V to any ‘non-omniscience truth’, any truth of the form α ∧ ¬K α.³⁴ Whether this would immediately yield KP is not obvious, but it would at least deal with the main worry over KP—namely, the sort of ‘non-omniscience’ claims involved in Fitch’s argument. ³¹ One might also argue that the veriﬁcationist—or KP theorist, in general—ought to acknowledge possibilities in which the constraints on K (on knowledge, in general) vary. In the case of veriﬁcationism, it is not implausible to think that knowledge might be achieved in some (admittedly, abnormal or remote) possibilities in which veriﬁcation criteria are weaker than normal. I think that this line is worth exploring, but for space reasons I omit further discussion. ³² For discussion of such broader issues, see Priest, Beall, and Armour-Garb (2004). ³³ I am grateful to Greg Restall for pushing this concern. ³⁴ In this case, it might be easier to add a distinguished ‘actual world’ to the models, and deﬁne validity as ‘truth-preservation’ over actual worlds (of all such models), but I will leave details for another occasion.

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Pa r t I V Ep i s t e m i c a n d Te m p o r a l Op e r a t o r s : Ac t i o n s , Ti m e s a n d Ty p e s

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9 Actions That Make Us Know Johan van Benthem

1 . T h e Pr o b l e m : Ve r i ﬁ c a t i o n i sm In c u r s t h e Fi t c h Pa r a d o x Veriﬁcationism is an account of meaning and truth whose origins lie in logical proof theory, especially, in its constructivist versions. The idea is that ‘truth’ can only be assigned to propositions for which we have evidence. This view can be found with logical authors like Dummett and Martin-L¨of from the 1970s onwards, but it has also penetrated since into general philosophy. Stated as a sweeping claim, this take on truth implies the general veriﬁcationist thesis that what is true can be known: ϕ → ♦K ϕ

VT

Here the K can be taken as a relatively unproblematic knowledge modality, while the ♦ is an as yet unspeciﬁed modality ‘‘can’’ of ‘feasibility’ in some relevant sense. Now, a surprising argument by Fitch trivializes this principle. It uses just a weak modal epistemic logic to show that VT collapses the notions of truth and knowledge, by taking the following clever substitution instance for the schematic formula ϕ, like elsewhere: q ∧ ¬Kq → ♦K (q ∧ ¬Kq) Then we have the following chain of three conditionals—which works in quite weak and apparently unproblematic modal logics: ♦K (q ∧ ¬Kq) → ♦(Kq ∧ K ¬Kq) → ♦(Kq ∧ ¬Kq) → ♦ ⊥ → ⊥ Thus, a contradiction follows from the assumption q ∧ ¬Kq, and we have shown overall that q implies Kq, making truth and knowledge equivalent. Is there a real problem here? How plausible was Veriﬁcationism anyway? There can be legitimate doubt on this score—but all the same, looking at ‘paradoxes’ like Fitch’s can be worthwhile. Of course, not every

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paradoxical argument points at a genuine problem. Some are just spats on the Apple of Knowledge, which can be removed with a damp cloth. But others are the telltale brown spots of worm rot inside, and deep surgery is needed—and the Apple may not even remain in one piece when restoring consistency. Professional paradox hunters and puzzle-driven researchers always claim the ‘deep trouble’ diagnosis—and sometimes they are right. Proposed remedies for the Paradox fall mainly into two kinds (cf. Brogaard and Salerno 2002; van Benthem 2004). Some solutions weaken the logic in the argument still further. This is like tuning down the volume on your radio so as not to hear the bad news. You will not hear much good news either. Other remedies leave the logic untouched, but weaken the veriﬁcationist principle itself. This is like censoring the news: you hear things loud and clear, but they may not be so interesting. Some choice between these strategies is inevitable. But what one really wants is a new systematic viewpoint beyond plugging holes, and opening up a new line of thinking with beneﬁts elsewhere. In our view, the locus for this is not Fitch’s proof as such, but rather our understanding of the two key modalities involved, either the modal K or the epistemic , or both.

2 . A Fi r s t Q u i c k A n a l y s i s : Ep i s t e m i c L o g i c a n d Ev i d e n c e Let us ﬁrst get to the essence of Fitch’s argument. The above substitution instance exempliﬁes a much older problem called Moore’s Paradox. Originally stated about belief, it consists in the observation that the statement ‘‘P, but I don’t believe it’’ can be true, whereas it cannot be consistently believed. Transposed to knowledge, this same problem was discussed by Hintikka in the 1960s, using the inconsistency of the formula K (q & ¬Kq) in epistemic logic. So, it is easy to understand the issue. Some truths are ‘fragile’ whereas knowledge is ‘robust’: and hence the former need not support the burden of the latter. Thus, one sensible and straightforward approach to the paradox weakens the scope of applicability of VT as follows (Tennant 2002): Claim VT only for propositions ϕ such that K ϕ is consistent CK CK has clear merits, but it fails our more general desideratum: it provides no exciting new account of either knowledge K or feasibility . We have put our ﬁnger in the dike, but no larger polder management system has emerged. Indeed, there seems even an obvious missing link in CK , reﬂecting one’s intuitive semantic understanding of the setting for VT . We have the truth of ϕ in some epistemic model M with actual world s, representing our current information

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state. But consistency of K ϕ per se gives us only the truth of K ϕ in some possibly quite different epistemic model (N, t). The real issue is rather: What natural step of ‘coming to know’ would take us from (M, s) to (N, t)? One could see this as asking for a principled account of the above operator , while the K can retain its standard meaning from epistemic logic. One way in which the has been unpacked in the literature goes back to the proof-theoretic origins of VT . In well-established type-theoretic approaches to provability, the evidence for a conclusion is displayed and manipulated in binary assertions of the form p: ϕ, where p is a proof for ϕ, or a piece of evidence in a more general sense. Type theory seems the most sophisticated underpinning of Veriﬁcationism to date. Van Benthem (1993) took this idea to standard epistemic logic, and proposed an explicit calculus of evidence for its K -assertions. One striking modern realization of this is the ‘logic of proofs’ of Artemov (1994, 2005), which replaces the box ϕ of the usual modal provability logic by operators [p] ϕ ‘p is a proof for ϕ’. Indeed, labels p of many sorts appear in the ‘labeled deductive systems’ of Gabbay (1996). This ‘evidence parameter’ for logical investigation seems a deep response to any paradox—but I am not aware of an inspiring solution to Fitch-style problems in this proof-theoretic setting. Thus, I take a different tack in this essay, in terms of dynamic semantic actions that produce knowledge. 3 . D y n a m i c s o f In f o r m a t i o n a n d C o m i n g t o K n ow Broadening our view of what a feasibility modality might stand for, van Benthem (2004, 2006a) looks at general mechanisms producing knowledge. Mathematical proof, no matter how liberally construed, is not the best paradigm for understanding how we come to know things, since it does not add new truths beyond our premises. Genuine actions by which we come to know new things seem much more domestic: we observe, or we ask some expert who knows! The latter actions involve a notion of change beyond proof steps: new information changes the current epistemic model—and in the process our knowledge changes, too. The simplest mechanism achieving this reﬂects the folklore sense in which ‘new information shrinks the current range of possibilities’: An announcement of some proposition P changes the current range of possible worlds, leaving those where P holds, while removing all others. More precisely, consider an epistemic model (M, s), with designated actual world s. What can be known in this setting seems restricted to what might be known correctly about that actual situation s. We know already that it is one of the worlds in M . What we might learn is that this model can be shrunk further, zooming

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in on the location of s. In this dynamic epistemic setting, we can recast the Veriﬁcationist Thesis as follows. Saying that every true statement may be known amounts to stating that: What is true in the current setting may come to be known there

VT-dyn

What this means in a simplest scenario is that some authoritative true statement could be made which changes the current model (M, s) to some submodel (M |ϕ, s) where the relevant proposition ϕ is known. Indeed, announcing ϕ itself seems an obvious and infallible candidate for this purpose, but more on this in a moment. The dynamic turn toward knowledge-producing actions involves some delicate issues. A ﬁrst thing to note is that making announcements is not just a matter of accumulating knowledge. This is true for atomic facts—but truth-values of more complex epistemic assertions can change in the process. When I tell you that p, which you did not know, the statement Kyou p changes its truth-value from false to true. But at the same time, the iterated knowledge statement Kyou ¬Kyou p goes from true to false—and so on upward, with changes in iterated statements of epistemic reﬂection. Thus, one single action !ϕ of publicly announcing ϕ can have repercussions for truth-values across the epistemic language. In particular, the Moore sentence shows that some propositions ϕ have the ‘self-afﬂicting’ property of changing their own truth-value when they are announced: A true public announcement !(q & ¬Kq) of q & ¬Kq makes the fact q into common knowledge, thereby invalidating the conjunct ¬Kq. Thus, announcing a truth is not an infallible way of turning it into knowledge. We will investigate the subtleties of epistemic update in the next section. For now, we contrast our new dynamic view with the earlier consistency requirement on CK. Here is the connection between our new proposal VT-dyn and the earlier CT : Fact

(a) VT-dyn implies CK (b) CK does not imply VT-dyn for all propositions ϕ

Proof Implication (a) is obvious. Its converse (b) is not, as we need truth of K ϕ not in just in any model (which would sufﬁce for consistency), but in some submodel of the current one. Here is a counter-example. Not surprisingly by now, it works with a relative of the Moore-type assertion q & ¬Kq: ϕ = (q & ¬Kq) ∨ K ¬q,

where q is a proposition letter.

This is knowable in the sense of CK , since K ( (q & ¬Kq) ∨ K ¬q) is consistent. For instance, this formula holds in a model consisting of just one world with ¬q. Indeed, in S5, the statement K ϕ is equivalent to K ¬q. But now consider the following two-world epistemic S5-model M with an actual world s and an

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epistemically indistinguishable world t, where the atomic formula q holds at s but not at t. In this situation, no truthful announcement would ever make us learn the above ϕ:

s

t

q

q

Figure 9.1.

In the actual world, (q & ¬Kq) ∨ K ¬q holds, but it fails in the other one. Hence, K ( (q & ¬Kq) ∨ K ¬q) fails in the actual world. Now, there is only one truthful proper update of this epistemic model M , which just retains its actual world with q: the actual world q

Figure 9.2.

But in this one-world model, the formula K ( (q & ¬Kq) ∨ K ¬q) fails. The preceding example suggests that CT, though correct in spirit, is still too weak in a dynamic setting. This point is somewhat technical, but telling all the same. It shows how, in a natural semantic scenario of coming to know things, the Veriﬁcationist Thesis places stronger requirements on propositions than those found in the literature so far. How can this happen? Why does not a true assertion (q & ¬Kq) ∨ K ¬q stay true when we ‘learn more’? Once again, the learning intuition behind world elimination is only valid for factual propositions. But epistemic propositions involving K -modalities may change their truth-value when a model contracts, as ignorance has now turned into knowledge. To understand this better, let us now look in more detail at logical mechanisms for epistemic change and learning (van Benthem 2002, 2006a, 2006b). 4 . Ep i s t e m i c L o g i c D y n a m i ﬁ e d

4.1. Static epistemic logic The basic language of epistemic logic and its semantics are well-known, with the individual knowledge modality Ki ϕ interpreted as follows: Ki ϕ is true at a world s iff ϕ is true in all worlds t with s ∼i t, where ∼i is the epistemic accessibility relation for agent i.

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In what follows, for convenience of exposition, we use an S5 version, where world accessibility is an equivalence relation. This simple semantics of knowledge has inspired much philosophical discussion, partly by the logical precision that it offers, but also, it has to be said, by its perceived deﬁciencies. Hotly debated until today are ‘logical omniscience’ (closure of knowledge under valid implications), and ‘introspection’ (automatically knowing that one knows or does not know a proposition): cf. van Benthem (2006a). Moving beyond single agents, epistemic logic can also analyze new forms of ‘social’ knowledge in groups. In particular, common knowledge. CG ϕ for a group G says intuitively that everyone knows that ϕ, they also know that the others know, and so on to any ﬁnite depth of iteration of mutual knowledge operators. Semantically, the corresponding epistemic modality CG ϕ is true at a world s whenever ϕ is true in the whole ‘component’ of the model consisting of all worlds accessible from s by some ﬁnite sequence of agent accessibility steps. In scenarios with just a single agent 1, common knowledge C{1} ϕ is just the same as knowledge K1 ϕ. (This would not work in weaker epistemic semantics than that for S5.) One can read the following discussion up to Section 7 either way, as being about knowledge of a single agent, or about common knowledge in a group. As for epistemic inference, well-known complete axiom systems exist for the valid laws in this language over standard model classes, such as multi-agent S5 (plus common knowledge) for models where the accessibilities are equivalence relations. Finally, as to computational complexity, most current versions of epistemic logic are decidable.

4.2. Dynamic epistemic logic To deal with the dynamics of Section 4, we need to add epistemic actions to this framework. Here, the driving engine for update of agents’ information is model change. The simplest case described earlier is that of a truthful public announcement !ϕ of an assertion ϕ. This does not just evaluate ϕ truth-conditionally in the current model (M, s). It rather updates that model to a new model M |ϕ , s, a submodel of (M , s)—and it does so by eliminating all those worlds from it which fail to satisfy ϕ: from

s

to

(M, s)

f

(M|f, s)

f

Figure 9.3.

s

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This update scenario can analyze questions and answers producing new information, and it even works for much more intricate puzzles involving knowledge and ignorance (van Benthem 2002). Thus, we get a dynamic-epistemic logic, as a general semantic setting for information ﬂow and learning. There is a family of epistemic models: the relevant information states, and a repertoire of announcement actions, which increase information by moving from one model to another. Full-ﬂedged dynamic-epistemic logics arise from standard epistemic ones by adding an action modality from dynamic logics of computation. It expresses what holds after an action was performed: M , s |= [ !ϕ ]ψ

iff

if M , s |= ϕ , then M |ϕ , s |= ψ

Thus, the dynamic modality [!ϕ]ψ says that ‘‘after ϕ has been truthfully announced, ψ. holds at the current world.’’ With this language, one can express systematic effects of communication, using combined knowledge-action statements such as [!ϕ]Kj ψ: after a public announcement of ϕ, agent j knows that ψ There are complete and decidable logical calculi for this richer language, too. Their key axioms systematically analyze the result of an epistemic action in terms of things that were true before. Dynamic epistemic logic does not magically solve the problems of static epistemic logic, as perfect reasoning with logical omniscience, and perfect reﬂection with epistemic introspection are still assumed. But our new logics do help analyze and even high-light further issues of potential philosophical interest. Sometimes, it is just liberating to move to new problems instead of remaining stuck with old ones. In particular, one additional idealization of the dynamic setting seems worth pondering. The central valid law of the logical calculus of public announcement reduces knowledge resulting from communication to relativized knowledge that was true before: [!A] Ki ϕ ↔ (A → Ki (A → [ !A] ϕ)) The semantic soundness of this principle has its own further presuppositions, including perfect memory of agents (Liu 2006). This idealization has been called into question in game theory and cognitive psychology under the heading of ‘bounded rationality’. Moving beyond single knowers, however, the most exciting applications of epistemic logics today emphasize the multi-agent character of speakers, hearers, and audiences. In particular, even Hintikka’s original language can iterate knowledge assertions, as in K1 ¬K2 P

‘‘1 knows that 2 does not know that P.’’

Also, common knowledge was a group phenomenon par excellence. ‘Social’ epistemic notions are crucial to information ﬂow and communication. Some

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philosophers think such issues are not profound, having to do with gossip, ICT, and other shallow necessities of living with a lot of people on one small planet. But the pursuit of knowledge and rational behavior consists to a large extent of intelligent interaction with others—and we need to understand that success. This point will return below, as so-called paradoxes afﬂicting lonesome knowers may look brighter in groups.

4.3. Interaction, partial observation, and event update Public announcement is a basic mode of transmitting information. But information can ﬂow in many more subtle ways. For example, we observe informative events without overt linguistic aspects. And, crucially, observation can then be different for different observers. I see which card I am drawing from the current stack; you only see that I am drawing one. By now, sophisticated event update mechanisms exist for such phenomena, far beyond simple world elimination (Baltag, Moss, and Solecki 1998). These can model complex multi-agent forms of communication mixing public actions and information hiding. Think of whispering to your colleagues during a seminar, or sending an email using the button bcc. In cases like these, the current epistemic model need not shrink: it may even grow in size. Example: Reading a Letter You have taken an exam, but neither you nor your friend knows the outcome yet. Here is a simple epistemic model, where in fact (viz. the bold-face actual world to the left), you passed: me pass

fail you

Figure 9.4.

Now you receive a letter in the presence of your friend, and read that you have passed. If this were a case of public announcement, the model would just shrink to the left-hand world as before. But this time, you cannot tell whether your friend has seen the content of the letter, though she does know it is an ofﬁcial notiﬁcation. She might, and she might not have seen what you read—and so, as far as she is concerned, you might also have been reading a letter which says that you failed. In this case, taking both your situations into account, there are three relevant possible pairs of simultaneous events: (you read Pass, she reads Pass) (you read Pass, she sees nothing) (you read Fail, she sees nothing)

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The ﬁrst two joint events can only occur if you have passed, the third if you failed. Note also that these events themselves have epistemic relations. For example, you cannot distinguish the ﬁrst from the second, and she cannot distinguish the second from the third. Next, as for update, the new epistemic model resulting from incorporating the new information in the reading/observing event into the preceding two-world model has three instead of two worlds now, with the relevant pairs (old world, new event) as depicted here: pass, (you read Pass, she sees nothing) you

she

pass, (you read Pass, she reads Pass) fail, (you read Fail, she sees nothing)

Figure 9.5.

Here the new epistemic relations arise as follows. Agents cannot distinguish two pairs (s, e) and (t, f ) if they can distinguish neither the old worlds s, t nor the new events e, ϕ. Suppose that in fact your friend read what was in the letter. Then the actual world is pass, (you read Pass, she reads Pass) In that world, by standard evaluation in terms of epistemic logic, you know that you passed, she knows it, too, but you do not know that she knows. These are typical asymmetries of information that may arise between players in the course of a card game. General event update takes a model M for the current information of a group of agents, plus some event model A modeling all relevant events, and then compute a new ‘product model’ MxA. This construction covers much of the information ﬂow in communication, games and other more realistic activities. Again, there are complete and decidable dynamic-epistemic logics dealing with what agents know stage-by-stage as general actions of this sort take place (van Ditmarsch, van der Hoek, and Kooi 2006). 5 . L e a r n i n g by Up d a t e The self-refuting nature of true Moore-type assertions noted in Section 4 shows that Fitch-style issues about Veriﬁcationism reﬂect core phenomena in information update. Indeed, this analogy is the main point of this paper. But

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it is of interest to see how these issues play in dynamic epistemic logic. They do not have the same doom-laden atmosphere. Informal studies of speech acts sometimes state that the generic effect of a public announcement !ϕ is simply that ϕ becomes common knowledge. The neat corresponding axiom in the earlier calculus would read as follows: [!ϕ]CG ϕ But this principle is not valid in general, witness the earlier-mentioned Moore sentence ϕ = q & ¬Kq. The latter assertion, once announced, cannot be true any more—and it even makes its own negation common knowledge! The reason is that announcing ϕ makes q common knowledge, and hence also Kq, but Kq implies ¬(q & ¬Kq). This is not an isolated curiosity. Gerbrandy (2007) gives a new analysis of the well-known Paradox of the Surprise Examination, which revolves around a teacher’s problematic assertion that some upcoming exam in the following week will take place on a day ‘when the student does not expect it’. Gerbrandy shows how the usual perplexity dissolves once we see that the teacher’s assertion can be of the above true-but-self-refuting type. For example, with a two-day time span, the formula for the teacher’s statement in our dynamic-epistemic logic is this (writing Ei for ‘the exam is on day i’): (E1& ¬Kyou E1) ∨ (E2 & [ !¬E1]¬Kyou E2) This says that the exam is on Day 1, and you do not know that now, or it will be on Day 2, and even learning that it is not on Day 1, you will not know that it is on Day 2. For details and a further defense of this analysis, we refer to the cited publication. Simple epistemic models of the above sort then clarify various surprise exam scenarios.

5.1. From paradox to typology These observations do not suggest at all that one must ban self-refuting assertions—as has been proposed in some remedies to the Fitch Paradox. To the contrary, they rather bring to light a rich diversity of types of behavior which calls for a dynamic typology of epistemic assertions. For example, we can investigate which precise forms of assertion are ‘self-fulﬁlling’, in that they do become common knowledge upon announcement. For instance, all universal modal formulas are self-fulﬁlling in this sense. These are the ones constructed using atoms and their negations, conjunction, disjunction, Ki and CG . But there are other self-fulﬁlling types of statement, and a complete syntactic characterization has been an interesting open model-theoretic problem since the late 1990s (Gerbrandy 1999; van Benthem 2002; van Ditmarsch and Kooi 2006).

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Logical studies in this vein have brought to light further delicate phenomena. In particular, some epistemic assertions ϕ are only self-fulﬁlling or ‘self-refuting’ in the long run. When announced truly for as long as possible, they either result in common knowledge CG ϕ, or the opposite: CG ¬ϕ. Van Benthem (2002) applies this insight to game theory, and shows how well-known game solution procedures may be analyzed in terms of repeated announcement of formulas ϕ expressing the ‘rationality’ of all players. Such statements are informative in general, and remove possible strategic equilibria, but at the ﬁrst stage where they no longer shrink the model, common knowledge of rationality sets in. Thus, instead of exorcizing paradox, we chart the diversity of epistemic behavior. This turn may be compared to that in Kripke’s theory of truth, where self-reference of propositions became an object of study, rather than a taboo. Another interesting typology goes back to the ‘coming to know’ of Section 4, our dynamic setting for learning true propositions. Indeed, van Benthem (2004) deﬁnes three possible types of learnability for propositions ϕ, using an existential action modality ψ: one can truly announce A and then get ψ true. |= ϕ → ∃AK ϕ Local Learnability ∃A :|= ϕ → K ϕ Uniform Learnability |= ϕ → K ϕ Autodidactics He shows that each successive type is more demanding than the preceding. Moreover, at least on epistemic S5-models, all three notions of learnability are decidable. Further notions of learning arise with iteration of true assertions, perhaps even the same one. Baltag, van Ditmarsch, Herzig, Hoshi, and de Lima (2006) present sophisticated update calculi of this sort, and they prove in particular that, when added to our basic logic of public announcement, the logic of ‘truth after some announcement’ stays axiomatizable. Thus, once again, the ‘paradox of knowability’ turns from a nuisance into an interesting phenomenon, and a source of intriguing new logical questions.

5.2. Digression: reachability with event updates The event updates of Section 5 took an epistemic model M and an event model A, and computed a new product model MxA. Many more statements may be made true by such drastic changes. Call a model N ‘reachable’ from M , if, for some event model A, N is equal (or better: ‘epistemically bisimilar’) to MxA. Could this approach rescue CT ? We gave a model (M , w) and a statement ϕ true in it—but, even though K ϕ was consistent, having a model N with ϕ true throughout, no eliminative update took M to such a model. But could some more general event update do that job? For the single-agent case, this is not so.

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Every model MxA is then bisimilar to some submodel of M . But, there might be another way of saving CT. To link with the current (M , w), we might require that K ϕ be consistent with a description of the current world in (M , w). But, if we make K ϕ consistent with the state description of w (its true and false atomic propositions), update may still be impossible. If we make K ϕ consistent with the complete modal theory of w, we do get a model bisimilar to (M , w) (van Benthem 2002), but this seems a trivial victory.

5.3. Explicit temporal perspectives on knowledge We conclude with a common criticism of the ‘learning problem’ in the dynamic epistemic setting. Self-refuting Moore-type assertions evoke strong responses. One either loves this sort of subtlety, or one thinks it fundamentally misguided, disregarding the role of time. And, indeed, there is a sense in which announcing any true proposition should always lead to common knowledge. When I say that ϕ is true right now, at time t0 , immediately afterwards, it becomes common knowledge that ϕ was true then at time t0 ! This insight is not in conﬂict with the type of logic we have used. We can add explicit temporal operators to the dynamic epistemic framework, say a Y for yesterday in the time of our epistemic process. Then we get a complete update logic again, including the following attractive validity: ϕ → [ !ϕ ]CG Y ϕ This says that, if ϕ is true now, announcing it makes it common knowledge that it was true at the preceding stage. Some conversational moves work in just this way—like when people say in response to some assertion that ‘‘I knew already what you told me.’’ One might see this as one plausible sense in which the Veriﬁcationist thesis does hold: Every local truth right now can come to be known as being true now at some later stage of investigation. Indeed, analyzing the Paradox of Knowability in an explicit temporal epistemic logic has been proposed before, e.g., in Edgington (1985). In such a formalism, all the above issues still make sense. In particular, we now want to know precisely which assertions will persist over time, from Y ϕ to ϕ. For some further explorations at the border-line with dynamic-epistemic logic, cf. Sack (2006), Yap (2006), van Benthem and Pacuit (2006). It has to be said that this greater expressive power also has its price. In particular, statements of valid ‘learning

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principles’, and complexity of epistemic-temporal logics, depend in subtle ways on which precise strength we give to the temporal operators. This section has presented a number of technicalities that may seem nongermane to our general discussion. But the way we see it, these demonstrate that any ‘banning’ response to the Fitch paradox would be a bad idea, as it would deprive us of a rich area of investigation offering a lot of genuine insight into how we come to know things.

6 . Ma n y A g e n t s , C o m m u n i c a t i o n , a n d In t e r a c t i o n O ve r Ti m e Modern epistemic logic is no longer about lonely knowers in rickety armchairs in leaking attics. It unfolds its true attractions in multi-agent settings, analyzing what agents know about each other, and how they interact: in communication, games, or any other social activities where information ﬂows. The earlier-mentioned notion of common knowledge is crucial then. Here, self-refuting assertions come up naturally without any Moore-like paradoxical ﬂavor, witness the following evergreen from the literature.

6.1. Puzzles of repeated announcement Like other areas of logic, dynamic epistemic logic has its ‘icons’. In the well-known puzzle of the Muddy Children, whose epistemic importance was recognized in Fagin, Halpern, Moses, and Vardi (1995), it is successive public announcements of ignorance which drive the solution process toward common knowledge of the true state of affairs. In a simple version, the story runs as follows: After playing outside, two of three children have mud on their foreheads. They all see the others, but not themselves, so they do not know their own status. Now their father comes and says: ‘‘At least one of you is dirty.’’ He then asks: ‘‘Does anyone know if he is dirty?’’ The children answer truthfully. As questions and answers repeat, what will happen?

Nobody knows in the ﬁrst round. But in the second round, each muddy child can ﬁgure out her status, by explicit reasoning, or by updates. To display these, draw an epistemic model whose worlds assign D or C to each child. The actual world is DDC : that is, child 1 and 2 are dirty, while child 3 is clean. Initially, a child knows only the status of the others’ faces, but not her own. The corresponding epistemic uncertainty relations are indicated by the labeled lines

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in the following diagrams. Epistemic updates start with the father’s elimination of the world CCC: from DDD 1

3 DDC∗

2

CDD

DCD

3

1 2

2 1

CDC

3

2

CCD

DCC

3

1

CCC

Figure 9.6.

to DDD 3

1

DDC∗

2

CDD

DCD

3

1 2

2 1 CCD

CDC

3 DCC

Figure 9.7.

One can see this as a simple ‘symmetry breaking’ of the original pattern which will have startling consequences—like the way, say, a professional starts a snooker game. Next, when it turns out that no one knows his status, the bottom worlds disappear as shown in Figure 9.8:

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DDD 1 CDD

2

3 DDC ∗

DCD

Figure 9.8.

Finally, when the muddy children 1 and 2 say simultaneously that they know their status, all worlds where at least one of them still has an uncertainty line left disappear. Thus, this statement, too, was highly informative-and the ﬁnal update is to: DDC ∗ With k muddy children, k rounds of public ignorance assertions achieve common knowledge about who is dirty, while the announcement that the muddy children know their status achieves common knowledge of the whole situation. Thus, public assertions of ignorance can drive a positive process of gathering information, and their ability to eventually invalidate themselves (the earlier-mentioned phenomenon of ‘self-defeating’ assertions) may even be the crowning event. The last announcement of ignorance for the muddy children led to their knowing the actual world. This puzzle highlights the interplay of many agents, and also the passage of time. We consider both in turn.

6.2. Multi-agent learning Our scenario suggests that learning becomes more interesting, and less ‘paradoxical’ in a multi-agent setting. Indeed, we do not need Muddy Children to make this point. Much simpler epistemic models represent interesting scenarios of communication which might be hard to keep straight just in words. Consider the example in Figure 9.9, with three worlds and two agents: p

2

p

1 p

Figure 9.9.

In the actual world to the top left, indicated in black, p is in fact the case, but neither agent knows if p. Now what can the agents learn by internal

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communication? First, neither can tell the other something factual about p. And yet the agents can discover where they are by communicating their epistemic state. First 1 says ‘‘I don’t know if p.’’ This rules out the right-most world, where K1 ¬p holds. After the update, only the left-hand worlds remain, and so 2 now knows that p. Saying that will then also inform 1, and the agents have achieved common knowledge of p. This is just one case: three-world models support a range of communication scenarios. Thus, epistemic models with different accessibility structure for agents encode useful information exchange, either of ground facts, or of epistemic attitudes. The general issue here is what agents can learn if they communicate what they know, and keep doing so until the model no longer shrinks. Van Benthem (2002) describes such scenarios completely, showing they lead to a unique submodel (with the technical proviso of ‘modulo bisimulation’). Essentially, internal communication turns the ‘implicit knowledge’ of a group into common knowledge. Similar scenarios have been studied in game theory when making hidden correlations in information explicit between players. But there may be other, more complex sorts of speciﬁcation for a communication process. For example, we may want only some group members to learn that ϕ, keeping the others in the dark. This can also happen with Moore-type statements. Here is one more scenario showing such multi-agent phenomena. In Figure 9.10, consider the following model M with actual world p, q: p, q 1

p, q

2 p, q

Figure 9.10.

Announcing q will make 2 know the Moore statement that ‘‘p and 1 does not know it.’’ But this can never become common knowledge in the group {1, 2}. What can become common knowledge, however, is p & q, when 1 announces that q, and 2 then says p.

6.3. Fitch paradoxes for plural knowers? Next, consider the Paradox of Knowability once more. When q is true and you don’t know it, there is nothing problematic with others knowing both those facts.

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Indeed, general communicative actions—though not full public announcement to the whole group—can ensure the truth of: K2 (q & ¬K1 q) ! Thus we might amend the Veriﬁcationist Thesis VT once more, and recast it as: If ϕ is true, then someone could come to know it.

VTmulti-agent

This principle is true, at least in some construals! In any model (M , s) where ϕ holds, adding a perfectly informed agent whose epistemic accessibility relation is identity between worlds is consistent, and in the expanded model that new agent knows that ϕ. More interesting is the issue of coming to know facts about the whole group. Here are two new possibilities. First, let ϕ be true but not common knowledge: ϕ & ¬CG ϕ This cannot be common knowledge in the group G, as the old Fitch argument still applies. But ϕ & ¬CG ϕ can be known by individual agents, and even whole subgroups. Next, consider a stronger case: ϕ is true, but there is a false common belief that it is not: ϕ & CBG ¬ϕ This time, no agent in the group can come to know this—at least in a very plausible epistemic-doxastic logic. For if agent i were to know ϕ & CBG ¬ϕ, we would have (a) Ki ϕ → Ki Ki ϕ → Ki Bi ϕ, (b) Ki CBG ¬ϕ → Ki Bi ¬ϕ, and so (c) Ki (Bi ¬ϕ & Bi ¬ϕ ), and Ki Bi ⊥, and hence a contradiction, at least, if our logic does not allow belief in contradictions. Veriﬁcationism becomes a more varied issue in communities of epistemic agents.

6.4. Temporal perspective once more: game theory and learning theory Dynamic epistemic logics describe single steps in larger processes where information ﬂows. There seems to be a growing consensus that such long-term procedures are crucial to ‘coming to know’. Our concerns so far then merge into larger issues about interactive agents with goals and strategies for achieving them. Thus, dynamic-epistemic logic meets game theory (Osborne and Rubinstein 1994) and learning theory (Kelly 1996), including strategic equilibria and convergent learning procedures in both ﬁnite and inﬁnite settings. These links go beyond the present paper, but their import is clear. In the ﬁnal analysis, what one can come to know is intimately intertwined with the how!

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7. Conclusion We have looked at the Paradox of the Knower in a dynamic-epistemic perspective where learning means changing the current epistemic model. The problematic Moore sentence driving the paradox turns out to be the typical ‘probe’ for investigating the sometimes surprising, but always useful, effects of successive assertions. Moreover, the multi-agent setting of epistemic logic places Veriﬁcationism in a richer interactive setting. This change in perspective trades the atmosphere of paradox and disaster for one of free exploration of dynamic typology of epistemic assertions, learning and reachability, and many further surprising twists in the logic of communication. Even so, we do not claim the last word on Veriﬁcationism, the origin of the Fitch puzzle. The proof-theoretic paradigm of evidence for what we know also has a ring of truth. And, indeed, the dynamic approach so far has no insightful take on the ‘information’ that comes to us via deduction (cf. Egr´e 2004; Jago 2006). Updating with logical consequences of what we know does not change any of the models used here. A uniﬁed account of learning from deduction and from observation is a long way off. For one recent attempt at merging the relevant kinds of information: observational ad inferential, in a dynamic logic setting, cf. van Benthem 2008. And even our own semantic perspective has told only half of the story. In a truly multi-agent setting, learning is not a single-agent matter, and the basic paradigm should have at least two roles: the Learner and the Teacher. And then, the issue with learning is not just what information we get when updated by some given assertion—say, an answer—or some more general observation of an event. It is just as much the other side of the coin: what we ask of others, and how we enquire. Veriﬁcation and veriﬁcationism seems really about both seeking and ﬁnding intertwined: a point made long ago in Hintikka (1973). In that light, our story so far has only addressed half of the real topic.

10 Can Truth Out? John Burgess

. . . truth will come to light; murder cannot be hid long; a man’s son may, but at the length truth will out. The Merchant of Venice, II.ii.73

1 It is rather discouraging that forty years have passed since Frederic Fitch ﬁrst propounded his paradox of knowability without philosophers having achieved agreement on a solution (1963: 135–42).¹ As a general rule, when modal phenomena prove puzzling, it is a good idea to look at the corresponding temporal phenomena, and accordingly I propose to examine here not the knowability principle that whatever is true can be known, but rather the discovery principle that whatever is true will be known. As Fitch’s modal paradox attacks the knowability principle, so an analogous temporal paradox threatens the discovery principle. The formulation of the paradox is as follows. Start with the minimal tense logic with G and H for ‘‘it is always going to be . . .’’ and ‘‘it always has been . . .’’ as primitive, and F and P for ‘‘it sometime will be . . .’’ and ‘‘it once was . . .’’ deﬁned as ∼ G ∼ and ∼ H ∼ (see Burgess 1984).² Add a one-place epistemic operator K for ‘‘it is known that,’’ and add as axioms minimal assumptions for this new operator, expressing that anything known is true, and that if a conjunction is known, so are both conjuncts: (1) Kp → p (2) K(p & q) → Kp & Kq Acknowledgment: Thanks to Michael Fara, Helge Rückert, and Timothy Williamson for perceptive comments on earlier drafts of this paper. ¹ For a summary of recent debates, see Brogaard and Salerno (2004). ² The various theorems of tense logic cited below can all be found in this source.

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In an attempt to formalize the discovery principle, add one further axiom: (3) p → FKp The paradox is that one can then derive the following: (4) p → Kp The derivation of (4) using (3) is, apart from replacing and ♦ by G and F, the same as Fitch’s derivation, which is too well known to bear repeating here. The operator K is intended to indicate human knowledge, not divine omniscience. The grounds for belief in the discovery principle have indeed traditionally involved a belief in divine omniscience, but it is not this belief alone that supports the principle, but rather this belief plus a further belief that on some future day God will bring it about that whatever is hidden is made manifest (quidquid latet apparebit). Obviously that day has not yet come, and the conclusion (4), that everything true is already humanly known, is an absurdity, and so we have a reductio of the principle (3). The ‘‘dialethists’’ and other proponents of radical revisions of classical logic can be counted on to tout their proposed revisions as solutions to this paradox, as they have touted them as the solutions to so many others. But a priori it is overwhelmingly more likely that the problem lies not in the underlying classical logic, but in the least familiar element, the axiom (3), the only axiom in which temporal and epistemic operators interact. And, indeed, that is where the problem lies. One has to be careful in going back and forth between symbolism and English prose, and Fitch, or rather, his hypothetical temporal analogue, wasn’t careful enough. In tense logic p, q, r , . . . are supposed to stand for tensed sentences, whose truth-value may change with time (or if one wants to speak of ‘‘propositions,’’ then they must be propositions in a traditional rather than a contemporary sense, propositions that are themselves tensed, and whose truth-value may change with time). FA is supposed to be true at a given time if A is true at some later time. What (3) actually expresses thus amounts to this: (5) If p is true now, then at some later time it will be known that p is true then. The proposed formalization as (3) has in effect turned the principle that any truth will become known into the principle that any sentence that expresses a truth will come to be known to express a truth. But this last formulation invites the immediate objection that the sentence in question may cease to express a truth before the knowledge of the truth it once expressed is acquired. And so (5) surely does not express what Shakespeare meant in saying ‘‘Truth will out.’’ He meant to imply that if Smith murders Jones secretly, so that no one knows, then it will become known that Smith murdered Jones secretly, so that no one knew. He did not mean to imply that if what the form of words ‘‘Unknown to all, Smith has murdered Jones’’ now expresses is true, then there

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will come a time when what that same form of words then expresses will be known to be true. Thus the temporal analogue of Fitch’s argument does not discredit the discovery principle, because the target of that argument is not a correct expression of that principle.

2 That one particular objection to a principle fails is no proof of the principle itself, and indeed no proof that it may not be open to simple, straightforward objection along other lines. And in fact the discovery principle is open to two kinds of objection, each of which requires us either to impose a restriction on the principle, or to assume charitably that a restriction on the principle was already intended by its advocates. As background to a ﬁrst objection consider the timing of the collision of two ordinary extended material objects. The boundaries of such objects generally are sufﬁciently ill-deﬁned on a scale of nanometers as to make dating their collision on a scale ﬁner than nanoseconds meaningless. If murders, say, are all the events we want to talk about, we do not need to conceive of ‘‘times’’ as durationless ‘‘instants,’’ but may conceive of them as very brief ‘‘moments,’’ of no more than, say, a nanosecond’s duration. In this case, chronometry—by which I here mean no more than our usual ways of dating events by year, month, day, hour, minute, second, and on to milli- or micro- or nanosecond and beyond if one wishes, all tacitly understood relative to some ﬁxed time zone—supplies a term for every time. But it may be otherwise if we wish to speak of point-particles and their collisions. The worry is that there will be truths that can never be known because they can never be stated. Suppose, for instance, that x = 0.1 8 2 5 6 4 7 9 3 . . . is an irrational number, and that exactly x seconds before 12:00 p.m., particle i collided with particle j. Can it ever become known that particles i and j collided at exactly x seconds before 12:00 p.m. on June 1, 2003? According to the discovery principle, all the following will become known: (1) Particles i and j collided at 182 ± milliseconds before 12:00 p.m. on June 1, 2003. (2) Particles i and j collided at 182564 ± microseconds before 12:00 p.m. on June 1, 2003. (3) Particles i and j collided at 182564793 ± nanoseconds before 12:00 p.m. on June 1, 2003. Here ‘‘±’’ abbreviates ‘‘the nearest unit’’, to the nearest milli- or micro- or nanosecond, as the case may be. (For the sake of argument, set aside any quantum-mechanical doubts about whether the series (1)–(3) really could be continued indeﬁnitely.)

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But for it to be knowable that i and j collided at exactly x seconds before 12:00 p.m., would it not have to be sayable that i and j collided at exactly x seconds before 12:00 p.m.? And for this to be sayable, there would have to be some means in language or thought of referring to the irrational number x—I mean, of course, some means other than referring to it as the number of seconds before 12:00 p.m. when i and j collided. √ Mathematics supplies such means for relatively few irrational numbers, such as 2 , π, e, and so forth. Coincidence may supply a few others: the time when i and j collided may be describable also as the time when k and l collide, if the two collisions happen to be simultaneous. But by cardinality considerations we inevitably lack means of reference to most irrational numbers. The discovery principle must be understood to exclude ineffable truths. It must be understood as restricted to truths expressible in our language. Such a restriction will be built into any tense-logical formalization of the principle, if the letters p, q, r , . . . are understood as standing for sentences of our language. Such a restriction seems in one sense not too serious, because the principle still tells us that the true answer to any question we have the language to ask will become known.

3 A second objection to the discovery principle is more subtle. Suppose that as I write it is 12:00 p.m., June 1, 2003. Then the following is true: (1) Now, this moment, it is 12:00 p.m., June 1, 2003. Obviously (1) itself will never be true in the future. And it seems that no sentence of our language will ever express in the future exactly what (1) expresses now. Thus the truth that (1) now expresses seems to be one that will be unknowable in the future because it is unsayable in the future. Moreover, the demonstrative ‘‘this moment’’ and the indexical ‘‘now’’ are both pleonastic, what they indicate being already sufﬁciently indicated by the fact that the verb ‘‘is’’ is in the present tense. Thus what has just been said about (1) is equally true of the following: (2) It is 12:00 p.m., June 1, 2003. And indeed, if now, this minute, Smith is murdering Jones, then the following is another example subject to the same difﬁculty as (1) and (2). (3) Smith is murdering Jones. The truth that Smith is (now, this moment) murdering Jones seems one that will be unsayable and therefore unknowable in the future, even though it is sayable now and even knowable now. The discovery principle must be understood to exclude not only ineffable truths, which are never expressible in our language, but also ephemeral truths,

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which are expressible for a moment, and then never again. Such a restriction seems in one sense not too serious, because it does not leave us with any question that can always be asked and never be answered. The ephemeral will be equally inexpressible interrogatively as assertorically. Such a restriction seems not too serious for another reason, because the truths it excludes from human knowledge in the future are excluded even from divine knowledge in eternity, if one follows those theologians who make the latter knowledge timeless. For (1)–(3) are no more true in a timeless eternity than they will be true in the seconds and minutes and hours and days and months and years to come. The old riddle that suggests an exception to the principle that God can see anything I can see is a joke.³ But the counterexamples (1)–(3) to the principle that God knows anything I can know are not. This point seemed worth digressing to mention, if only because a desire to have a formal apparatus in which such issues could be discussed was an important part of the motivation of the creation of tense logic by Arthur Prior.

4 We have seen that (1.3)—displayed item (3) of §1—is not the right formalization of the discovery principle. What is? It cannot be claimed that a complete solution to the paradox has been obtained until this question is answered. One answer suggests itself at once. Now that we have restricted the principle to truths that will remain expressible in our language in the future, it is tempting to formulate the principle as the principle that any sentence that will continue to express a truth in the future will come to be known to express a truth. This goes over into symbols as follows: (1) Gp → FKp And (1) is, unlike (1.3), immune to Fitch-style paradox, even if one considerably strengthens the background tense logic. For deﬁniteness, let us consider the tense logic, call it Llinear , that is appropriate for linearly ordered time without a last time. Then the immunity of (1) from Fitch-style paradox is the content of the following proposition. Proposition. Let T be Llinear plus (1.1), (1.2), and (1). Then (1.4) is not a theorem of T . ³ I mean the riddle: Q. What is it that God never sees, that the king seldom sees, but that you and I see every day? A. An equal. This seems less a problem for theologians than for partisans of ‘‘substitutional quantiﬁcation.’’

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Proof. Consider an auxiliary theory T ∗ , obtained from Llinear by adding a constant π and the following axiom: (2) Fπ Then π is not a theorem of T ∗ . For if we take any model of Llinear , and let π be true at and only at the times later than the present, then (2) will be true at all times, but π will not, being false at all past times and at the present time. Next assign each formula A of the language of T a translation A∗ into the language of T ∗ , by taking Kp to abbreviate p & π. Thus (1.1), (1.2), (1), and (1.4), respectively, are translated as follows: (3) (4) (5) (6)

p&π → p (p & q) & π → (p & π) & (q & π) Gp → F(p & π) p → p&π

Note that the translation (6) of (1.4) is not a theorem of T ∗ . For, if it were, substituting ∼ π for p and applying truth-functional logic, π would be a theorem, and as we have seen it is not. To show that (1.4) is not a theorem of T , it will sufﬁce to show that the translation of any theorem of T is a theorem of T ∗ . And, to show this, it will sufﬁce to show that the translations (3)–(5) of the three axioms of T are theorems of T ∗ . For the ﬁrst two axioms this is trivial, since (3) and (4) are truth-functional tautologies. For the third axiom, this follows using the following theorem of Llinear : (7) Gp & Fq → F(p & q) And (5) follows by truth-functional logic from (2) and (7), to complete the proof.

5 The formalization (4.1) has several corollaries worth noting. Proposition. Let T be as in §4. Then the following are theorems of T : (1) (2) (3) (4)

Pp → FKPp p → FKPp Fp → FKPp Gp → FKGp

Proof. First note that each of the following is either an axiom or a theorem of Llinear : (5) Pp → GPp (6) p → GPp

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(7) Fp → FGPp (8) FFp → Fp (9) Gp → GGp Also, the following is a derived rule of Llinear : (10) If A → B is a theorem, then FA → FB is a theorem. (For the cognoscenti, the assumption here is that the rule of temporal generalization, on which (10) depends, continues to imply after the formal language has been enriched by the addition of the epistemic operator K.) (1), (2), and (4) are immediate from (5), (6), and (9), respectively. As for (3), it can be derived as follows: (11) FPp → FFKPp (12) FPp → FKPp

from (1) by (10) from (11) and (8) 6

To illustrate these corollaries just derived, if Smith has murdered Jones, or is murdering Jones, or will murder Jones, then according to whichever of (5.1)–(5.3) is applicable, it will become known that Smith has murdered Jones. Let us write brackets around present tense verbs to indicate omnitemporality, so that, for instance (1) Smith [murders] Jones. is to be understood as meaning (2) Smith has murdered, is murdering, or will murder Jones. Then we may say that if Smith [murders] Jones, then it will become known that Smith murdered Jones. And similarly in any other case. Murder cannot be hid—though (4.1) does not go so far as to join the Bard in claiming (unfortunately, erroneously) that murder cannot be hid long. And if the memory of Smith’s victim will never cease to be honored, then according to (5.4) this fact will become known—though there is (again, unfortunately) no guarantee it will become known soon enough to comfort the victim’s grieving friends and relations. And if the universe will be forever expanding, according to (5.4) this fact, too, will eventually become known—though there is (yet again, unfortunately) no guarantee it will become known soon enough to satisfy the curiosity of present-day cosmologists. Still, despite its corollaries, (4.1) may look unsatisfactory for the following reason. Consider what the corollary (5.2) tells us about a present truth: (3) If p is true now, then at some later time it will be known that p was true once.

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The ‘‘once’’ here invites the question, ‘‘When?’’ And (5.2) provides no answer. Or so it may seem. But in a sense (5.2), taken together with chronometry, does provide an answer. If (3.2) and (3.3) are true, then their conjunction is true: (4) It is 12:00 p.m., June 1, 2003, and Smith is murdering Jones. Applying (5.2) not to (3.3) alone, but to this conjunction, we obtain It will become known that it was once 12:00 p.m., June 1, 2003, and Smith was murdering Jones. Or more idiomatically: (5) It will become known that Smith murdered Jones at 12:00 p.m., June 1, 2003. What more could one want by way of answer to a when-question? Quite generally, an event occurs at a given time, one can conjoin to a sentence p asserting the event’s occurrence a sentence q giving the standard chronometric speciﬁcation of the time, and then apply (5.2), not to p alone, but to the conjunction.

7 Nonetheless, it may seem that the most obvious correction of (1.5) would be the following: (1) If p is true now, then at some later time it will be known that p was true now. And (1) seems to tell us more than (4.1) (by way of (6.3)) tells us. It is known that (1) cannot be expressed using just the temporal operators G and H and F and P. But tense logicians have considered other operators. Most to the point in the present context, they have considered a ‘‘now’’ operator J, so interpreted that even within the scope of a past or future operator Jp still expresses the present, not the past or future, truth of p. And with this operator (1) can be symbolized, as follows: (2) p → FKJp One may be tempted to think that (2) would do better as a formalization of the discovery principle than does (4.1). But this is a misleading way of putting the issue. For if the operator J is admitted, subject to its usual laws, then (4.1) implies (2). For one of the usual laws is precisely (3) p → GJp and (2) is immediate from (3) and (4.1). So the temptation here is simply the temptation to add J to the language.⁴ ⁴ I owe this observation to Williamson.

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I think the temptation should be resisted for a double reason. My ﬁrst reason is that introducing the J-operator is unnecessary in order to answer a when-question. For I have just ﬁnished arguing that (4.1) does, after all, provide answers to such questions. Against this it may be said that (2) appears to have the advantage of doing so without depending on chronometry. But my second reason for avoiding the J-operator is that this apparent advantage comes at the cost of involving us with the problematic notion of a de re attitude towards a time. This truth is perhaps most easily brought to light by switching temporarily from regimentations using tense operators to regimentations using explicit quantiﬁcation over times. So let t , u, v, . . . range over times. And let t < u mean that time t is earlier than time u, or equivalently, time u is later than time t. Let each tensed p be replaced by a one-place p∗ (t ) for ‘‘p [is] the case at time t.’’ Every formula A built up from the letters p, q, r , . . . will similarly be replaced by an open formula A∗ (t ). PA and FA, respectively, will be replaced by: (4a) ∃u(u < t & A∗ (u) )

(4b) ∃u(t < u & A∗ (u) )

In a formula A(t ) the parameter t may be thought of as standing for that time which is now present. Leaving open how to symbolize the epistemic operator, (5.2) and (2) above go halfway into symbols as follows: (5) p(t ) → ∃u(t < u & it is known at time u that ∃v(v < u & p(v) )) (6) p(t ) → ∃u(t < u & it is known at time u that p(t )) There is this difference between the two semi-formalizations, that what occurs towards the end of (5) can be understood in a de dicto way, thus: (7) At time u, ‘‘p was true once’’ [is] known to be true. By contrast, what occurs towards the end of (6) must be understood in a de re way, thus: (8) At time u, ‘‘p was true then’’ [is] known to be true of time t. The symbol-complex KJp in (2) above may be pronounced ‘‘it is known that p was true now,’’ but what it really amounts to is more like this: (9) It is known of t that p was true then, where t is that time which is now present. 8 There are (at least) three major difﬁculties in making sense of the notion of a de re knowledge about an object a. Or to put the matter another way, there is only one obvious strategy for making sense of the notion of a de re attitude, namely, reduction to a de dicto attitude, and there are (at least) three major obstacles to this strategy. The strategy is to understand a subject as knowing of an object a

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that F (x) holds of it if and only if the subject knows that F (a) where a is a term denoting a. The three obstacles or problems relate to the choice of term a. A ﬁrst general problem with de re knowledge is that of anonymity. There may simply be no term a denoting a. This problem has been encountered in the case of times in §2, and given the restriction on the discovery principle imposed there, it may be set aside here. A second general problem with de re knowledge, and one relevant to the question whether J should be admitted is the problem of aliases. The problem is that there may be two terms a and b denoting an object a, and it may be that the subject knows that F (a) but does not know that F (b), or the reverse. The star whose common name is ‘‘Aldebaran’’ has also the ofﬁcial name ‘‘Alpha Tauri.’’ It seems that a subject may have been told by different authoritative sources, and hence may know that: (1) Aldebaran is orangish. (2) Alpha Tauri is the thirteenth brightest star. and yet, being in ignorance that the two names are names for one and the same heavenly body, the subject may not know that: (3) Alpha Tauri is orangish. (4) Aldebaran is the thirteenth brightest star. And this makes it hard to answer the question whether the subject knows of the star itself, independently of how it is named, that it is orangish, or the thirteenth brightest. The existence of aliases is a problem insofar as privileging one of them over the other seems arbitrary. The same problem can arise for times. Robinson may know that one rainy day Smith committed murder, and may know that Jones was murdered, and not know that the murder Smith committed was that of Jones. In this case Robinson will know that: (5) At the time when Smith committed murder, it was rainy. But not that: (6) At the time when Jones was murdered, it was rainy. And this makes it hard to answer the question whether Robinson knows of the time itself, independently of how it is described, that it was rainy then. Where there exists some standard term for each object of a given kind, one can always stipulate that a subject is to be credited with de re knowledge about the object a that F (x) holds of it, if and only if the subject has de dicto knowledge that F (a) where a is the standard term for a. Admittedly, such a stipulation may be more a matter of giving a sense to a kind of locution (ascriptions of de re knowledge) that previously had none, than of ﬁnding out what sense this kind of locution had all along. Pretty clearly, it would be a case of giving rather

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than ﬁnding if one took as canonical terms for heavenly bodies the ofﬁcial names adopted by international scientiﬁc bodies, preferring ‘‘Alpha Tauri’’ over ‘‘Aldebaran.’’ For times, the obvious candidates for standard terms are those provided by chronometry. If one is content with (4.1), there is no need to enter into the problem of de re knowledge about times at all, and so no need to ﬁx on any standard terms for times. If one adopts (7.2), reliance on chronometry is the only obvious way to impose a solution on the problem of aliases. But in that case the one advantage (7.2) appeared to have over (4.1), that of not depending on chronometry, must be recognized to have been illusory. This consideration argues, I claim, in favor of the J-free formalization (4.1) and against the J-laden formalization (7.2). A third general problem with de re belief is the problem of demonstratives (and with them indexicals). When the star Alpha Tauri, alias Aldebaran, is visible in the night sky, one can point to it and say ‘‘that star,’’ and so achieve reference to it. Now it seems someone looking at the star may well know (5) That star is orangish. And yet not knowing the name of the star, she may well not know either (1) or (2). This is, so far, just a special case of the problem of aliases. But demonstratives are especially troublesome because, on the one hand, when available, they seem to provide so direct a way of referring that it is hard to insist that nonetheless it is some other way of referring that provides the canonical terms for reduction of de re to de dicto; but on the other hand, demonstratives themselves are not viable candidates for canonical terms, simply because they are usually not available: if we took demonstratives as canonical terms, most objects would suffer from anonymity most of the time. Demonstratives act, so to speak, as spoilers, making any other candidates for the ofﬁce of canonical term look unworthy, while themselves not being eligible for that ofﬁce. But this problem has been encountered in the case of times in §3, and given the restriction on the discovery principle imposed there, it may be set aside here, as the problem of anonymity was set aside. The problem of aliases, I claim, is enough to make the admission of J undesirable.

9 I have done with the topic of the discovery principle. But what of the knowability principle, and the original, modal version of Fitch’s paradox? I began this essay by recalling that there is a close parallel between temporal and modal. I should now note that while there are many analogies, in connection with Fitch’s paradox there is also one glaring disanalogy, that makes the original, modal problem more

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refractory than its temporal analogue. Perhaps the best way of proceeding would be to begin by simply listing pairs of analogous notions in parallel columns, as shown in Table 10.1.

(1.3) (4.1) (5.2) (7.2)

Temporal

Modal

Discovery Principle present tense, future tense G, F Llinear p → FKp Gp → FKp p → FKPp now J p → FKJp times, instants, moments chronometry

Knowability Principle indicative mood, subjunctive mood , ♦ S5 p → ♦Kp p → ♦Kp p → ♦K♦p actually @ p → ♦[email protected] possibilities, worlds, situations ???

(1) (2) (3) (4)

But, returning to what is formally representable, I have recalled in the left margin in the table the numbers of temporal formulas we have met earlier, and assigned in the right margin numbers to the analogous modal formulas. Fitch’s (1) is dismissable for reason analogous to those that led to the dismissable of its analogue (1.3).⁵ The difﬁculty comes when one seeks a replacement. The absence of any obvious analogue for possibilities of standard chronometric speciﬁcations for times makes (2) and (3) much less satisfactory than (4.1) and (5.2)—and (4) correspondingly much more tempting than (7.2). But the same absence makes the problem of de re knowledge of possible situations connected with (4) at once more critical and more difﬁcult to solve or evade than was the problem of de re knowledge of temporal moments connected with (7.2). I will not enlarge further here, partly because it would be a good exercise for readers to work out the analogy for themselves, but mainly because I would be largely repeating points that have been made by Dorothy Edgington in her proposed solution to the paradox, and by Timothy Williamson in his criticisms thereof.⁶ A further disanalogy emerges in discussion of Edgington and Williamson that is not formally representable, and is therefore not indicated in the above table. It is just this: that, generally speaking, the fact that something is only actually true and not necessarily true tends to matter less to us than the fact ⁵ Though there is a lot more than can be said. For a full exposition of the essentially grammatical fallacy in the paradox, see Rückert (2004). Rückert draws on Wehmeir (2005). ⁶ See Edgington (1985); Williamson (1987a). For further relevant publications, see Brogaard and Salerno (2004).

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that something is only at the present moment true and not permanently true. Or, to put the matter another way, what will be true when the world is older matters more to us than what could have been true if the world had been otherwise, since we hope to live on into ‘‘future worlds’’ but do not expect to transmigrate into ‘‘possible worlds.’’ So far as the present investigation is concerned, it seems that the analogy between mood and tense takes us only so far, and in the end provides us not with a solution, but only with a better understanding of just what makes the problem difﬁcult.

10 Before giving up, however, perhaps we should try the combination of temporal and modal. That is to say, perhaps instead of considering the knowability principle as the principle that anything true could have been known, we should consider it as the principle that anything true could become known. The natural setting for such a principle would be a system like Prior’s logic of ‘‘historical necessity’’ (see Prior 1967: ch. VIII). In the most elaborate version, which he calls ‘‘Ockhamist,’’ Prior uses both tense operators G, H, F, P, subject to the axioms for Llinear , and modal operators , ♦, subject to the axioms of S5. But the modal operators are themselves understood in a tensed way, as meaning necessity and possibility given the course of history up to the present. Prior uses special letters a, b, c, . . . for sentences with the special property that their truth is independent of the future course of history in addition to the usual letters p, q, r , . . . for arbitrary sentences. Not all formulas, but only certain special ones, with the same special property as the special letters, may be substituted for those special letters. These include the special letters themselves, any formula beginning with a modal operator or ♦, and any formula obtainable from formulas of these two kinds using the truth-functions and past-tense operators H and P. A single axiom links the temporal and modal operators: (1) a → a One can obtain by substitution (2) Pa → Pa One cannot derive (3) Fa → Fa Taking for a in (1)–(3) ‘‘A sea ﬁght is occurring,’’ in Prior’s system one can conclude that if a sea ﬁght is occurring or has occurred, then the occurrence of a sea ﬁght is (historically) necessary; but even if a sea ﬁght is only going to occur,

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then its occurrence is (historically) contingent, though once it does occur, it will become (historically) necessary. A version of the knowability principle can be expressed in this context by the formula (4) Ga → ♦FKa And from (4) one can derive, using various tense-logical and modal theorems, the following rough analogues of the corollaries in the proposition of §3: (5) (Pa ∨ a ∨ Fa) → ♦FKPa (6) Ga → ♦FK♦Ga The details will not be given here, because the system is ultimately unsatisfactory. Let me explain how. From (4), by way of its corollaries, one can conclude the following, wherein I contract ‘‘possibly will’’ to ‘‘may’’: (7) If Smith is murdering Jones, then it may become known that Smith has murdered Jones. (8) If the memory of Smith’s victim will always be honored, then it may become known that the memory of Smith’s victim may always be honored. (9) If the universe is always going to be expanding, then it may become known that the universe may be always going to be expanding. What one cannot conclude is: (10) If the memory of Smith’s victim will always be honored, then it may become known that the memory of Smith’s victim will always be honored. (11) If the universe is always going to be expanding, then it may become known that the universe is always going to be expanding. So (4) seems too weak. The strengthening of (4) to (12) Ga → ♦FKGa would provide assurance of (10) and (11), but unfortunately (12) is too strong. For it would also provide assurance of the absurd: (13) If Smith murdered Jones but will forever escape detection, then it may become known that Smith murdered Jones but will forever escape detection. This is Fitch’s paradox, adapted to the present context.

11 In sum, Prior’s Ockhamist framework fails to provide a formula that is not too weak and not too strong, but just right. A glance back at the examples above will

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help us localize the difﬁculty: it is with truths about the actual future (and more particularly about what will always hold throughout that actual future). There are philosophers, however, who question the very meaningfulness of assertions about the actual future (for a recent expression of this view see Belnap and Green 1994, 365–88). And Prior (1967: ch. VII) has developed a logic he calls ‘‘Peircean’’ for them. In this logic one has only the special letters a, b, c, . . ., and only the formulas built up from them using truth-functions, past-tense operators, and four operators amounting to the combinations G, ♦G, F, ♦F. Substitution for the letters a, b, c, . . . is allowed for all formulas so built up. The operators , ♦, G, F do not appear separately, apart from the four combinations just mentioned. The pertinent feature of this logic in the present context is that it bans as meaningless the examples that caused trouble in the preceding section, and (10.4) seems adequate as an expression of the knowability principle for all such sentences as are still accepted as meaningful. Banning statements about the actual future is a radical step. Presumably the friends and relations of Jones know that his memory possibly will always be honored, and possibly will not always be honored. They know ♦Ga and ∼ Ga, where a is (1) The memory of Smith’s victim is honored. The Peircean, however, rejects as meaningless (2) The memory of Smith’s victim will always be honored. unless ‘‘will’’ is either strengthened to ‘‘necessarily will’’ or weakened to ‘‘possibly will.’’ The Peircean it seems, can’t allow the friends and relations to hope that (2) is true, or to fear that it isn’t (this observation is repeated from Burgess 1978). Likewise, cosmologists presumably already know that it is possible the universe will expand forever, and possible that it won’t. The Peircean can’t allow them to wonder if it in actual fact will. So Peirceanism is a radical doctrine. But then, so is the knowability principle. The question is, do the two forms of radicalism cohere? If an adherent of the knowability principle were to embrace Peirceanism, would the resulting position have any coherent motivation? Or would embracing Peirceanism be mere ad hoc epicycling, avoiding counterexamples by declaring them meaningless? This is too large, and too non-logical, an issue to go into here, but at least a word may be said about the historical sources of epistemological views like the discovery and knowability principles on the one hand, and of Peirceanism on the other. Belief in the discovery principle, I said at the outset, has traditionally rested on theological grounds. Belief in the knowability principle has, by contrast, been mainly an expression of a commitment to a certain philosophical theory of meaning, veriﬁcationism. The radical epistemological view that there are no unknowable truths has usually been a consequence of the even more radical

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semantical view that understanding a sentence consists in grasping under what conditions it would be known to be true. Belief in Peirceanism has had several sources. Prior cites late-mediæval logicians who have held a similar view on theological grounds, but the more recent proponents of the view seem to base their adherence on grounds that ultimately are veriﬁcationist. Thus combining the knowability principle with Peirceanism could be viewed as combining two manifestations of an underlying veriﬁcationism. Of course, there are many varieties of veriﬁcationism, and it remains to be seen whether a single variety can cogently motivate both these manifestations simultaneously. A key issue will be the veriﬁcationist’s attitude towards the reality of the past.

11 Logical Types in Some Arguments about Knowability and Belief Bernard Linsky

Of course the foregoing refutation of Fitch’s deﬁnition of value is strongly suggestive of the paradox of the liar and other epistemological paradoxes. It may therefore be that Fitch can meet this particular objection by incorporating into the system of his paper one of the standard devices for avoiding the epistemological paradoxes. If this is possible it will involve a drastic rewriting of the paper, not just a footnote here and there.¹

Over the years a number of arguments have been formulated in elementary modal logic purporting to show that there are limits to what can be known or believed. These include the ‘‘Fitch’’ style arguments that will be the main interest of this paper, versions of the paradoxes of the ‘‘Surprise examination’’ and the ‘‘Preface,’’ and several arguments against analyses of truth in terms of veriﬁability under ideal conditions. A use of iteration of operators and even apparent self-reference seems to reappear in various of these arguments and so one might wonder what exactly is common to these arguments and if that common element reveals something about their validity.² In recent years it has also been claimed that veriﬁcationism is subject to logical difﬁculties revealed by these arguments. It would be a challenge to veriﬁcationism to have a proof that some true sentences simply cannot be known, or believed, even by an idealized agent. The understanding of these arguments is thus a pressing issue for the veriﬁcationist program. Proposals for analyses of truth in terms of veriﬁcation in ideal conditions also confront difﬁculties when ¹ From Alonzo Church’s anonymous referee reports on Frederic Fitch’s ‘‘A deﬁnition of Value,’’ in Chapter 1 of this volume. Church here considers the liar paradox as ‘‘epistemological’’ in its formulation involving propositions that are asserted, following Whitehead and Russell ((1910): p. 62), rather than in the later formulation in terms of linguistic expressions and truth, and so as ‘‘semantic’’. See Church (1976) for his classic treatment of these semantic paradoxes using the theory of types. I am grateful to Joseph Salerno for sharing the reports, which he discovered as this volume was in preparation. ² As I wondered at the end of my (1986).

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one worries about the truth conditions for statements asserting that such ideal conditions do or do not obtain. There appears to be at least self-application of the theory of truth to its own preconditions. In accord with the suggestion from Alonzo Church in the epigram that precedes this paper, I propose below to identify the elements of ‘‘self-reference’’ in these various arguments, distinguish self-reference proper from the use of iteration of operators expressing epistemic conditions, and then provide a uniform account of them making use of the idea of logical types of propositions. The argument with which I begin is due to Frederic Fitch (1963). Fitch gives credit for this argument to an anonymous referee of an earlier version of the paper from 1945.³ As described in the introduction to this volume that referee has recently been identiﬁed as Alonzo Church himself, and two referee reports have been found, although both drafts of the paper are missing. The quotation that begins this paper is from the second report, responding to a revised version from Fitch. Fitch’s original argument is put in terms of ‘‘truth classes’’ of propositions, classes all of whose members are true, but current discussions consider its application to the case of the factive operator ‘‘Knows that’’ (K) which only holds of true propositions. (Hence the ‘‘T’’ axiom in standard logics of knowledge: K φ ⊃ φ): The Fitch Argument 1 1) K (p & ∼Kp) Assumption 2) K (p & ∼Kp) ⊃ Kp & K ∼ Kp Distribution Axiom 1 3) Kp & K ∼ Kp 1,2, Prop Logic 4) K ∼ Kp ⊃ ∼Kp T Axiom 1 5) Kp & ∼Kp 3,4, Prop Logic 6) ∼K (p & ∼Kp) 1–5 reductio But since p & ∼Kp can surely be true in some situations, we would have then something true that couldn’t be known. The argument is brought to point as follows: ∼ K (p & ∼Kp) follows from the last line by necessitation. Suppose that all truths are knowable, including that a given truth isn’t known, that is to say, p & ∼Kp ⊃ ♦K (p & ∼Kp). Then we derive that the antecedent is false, so ∼(p & ∼Kp), in other words it is provable that p ⊃ Kp, a manifest absurdity.⁴ In his ﬁrst referee report, Church offers the following suggestion to Fitch: In spite of the preceding argument I think Fitch has a good defense (but only one). This defense is that there is no law of psychology according to which one who believes ³ See p. 138 n. 5 in Fitch (1963). ⁴ Some discussion of the argument has hinged on the fact that in intuitionistic logic only p ⊃ ∼ ∼Kp can be proved, something that ought not to bother the veriﬁcationist who would use such a logic. This raises another approach to the Fitch argument that I won’t consider.

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a proposition must believe all its logical consequences; on the contrary, historical counter-examples are well known. To be sure, one who believes a proposition without believing its more obvious consequences is a fool; but it is an empirical fact that there are fools. It is even possible that there might be so great a fool as to believe the conjunction of two propositions without believing either of the two propositions; at least, an empirical law to the contrary would seem to be open to doubt. (pp. 2–3)

A ‘‘fool’’ can believe a conjunction but not one of the conjuncts, and so the Distribution Axiom is not in general true. Church’s suggestion is that the above version ‘‘Fitch Argument,’’ in standard epistemic logic that treats ‘K ’ as a sentential operator, can be blocked as invalid at step 2. In what follows this objection will be considered again, but after reformulating the argument within the theory of types, one of the ‘‘standard devices for avoiding the epistemological paradoxes’’ to which Church refers above. Even a version of the Distribution Axiom limited to conjuncts of the same logical type will not make the argument valid. Consider next a proof of a related result, involving in this case a proposition that cannot be believed. ‘‘Moore’s Problem’’ (also known as ‘‘Moore’s Paradox’’) is the seeming oddity in the expression ‘‘p but I don’t believe it.’’⁵ Jaakko Hintikka proposed to analyze the oddity by proving that the sentence might well be true but cannot be believed consistently with the principles of epistemic logic. This might be seen as a challenge to veriﬁcationism in so far as it shows that some truths can’t even be believed, much less veriﬁed or known. Here is a reformulation that uses proposed axioms of the logic of belief to replace Hintikka’s ‘‘reductive,’’ semantic, argument:⁶ The Hintikka Argument 1 1) B(p & ∼Bp) Assumption 2) B(p & ∼Bp) ⊃ Bp & B ∼ Bp Distribution Axiom 1 3) Bp & B ∼ Bp 1,2, Prop Logic 4) Bp ⊃ BBp ‘‘KK’’ principle for B 1 5) BBp & B ∼ Bp 3,4, Prop Logic 1 6) B(Bp & ∼Bp) 5, Conjunction 7) ∼ B(Bp & ∼Bp) Logical Omniscience 8) ∼ B(p & ∼Bp) 1–7 reductio In this argument Hintikka invokes a variant of his notorious KK-thesis (Kp ⊃ KKp) only in this case for belief (Bp ⊃ BBp), as well as requiring a certain amount of consistency on the part of believers, here described as the stronger requirement of ‘‘Logical Omniscience’’. Both assumptions have been challenged.

⁵ See the discussion in Sorensen (1988): chapter 1. ⁶ Presented at p. 69 of his (1962).

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Now consider a third argument in this family, the result of adapting the Fitch argument to belief, so that ‘K’ becomes a ‘B’. Call what we get ‘Fitch-B.’ The Fitch-B Argument 1 1) B(p & ∼Bp) Assumption 2) B(p & ∼Bp) ⊃ Bp & B ∼ Bp Distribution Axiom 1 3) Bp & B ∼ Bp 1,2, Prop Logic 4) B ∼ Bp ⊃ ∼Bp (Disbelief Principle) 1 5) Bp & ∼Bp 3,4, Prop Logic 6) ∼ B(p & ∼Bp) 1–5 reductio Step (4) in Fitch-B cannot be justiﬁed as in the original Fitch argument, as an instance of the T schema, as belief surely does not obey that principle in general. Not everything believed is true. However, it is important to note that while this is a particular instance of a T schema, it is also an instance of a weaker principle, that might be true in general, namely B ∼ Bφ ⊃ ∼Bφ. That the argument can be run for B in this way has been noted, for example by Binkley (1968). Binkley sees it as a version of the KK thesis for belief like the one that Hintikka uses, namely that if one believes φ then one doesn’t believe that one doesn’t believe φ. It is better to view this as an instance of a principle that expresses the ‘‘transparency’’ of (dis-)belief, namely that when one believes that one doesn’t believe φ, one can’t be wrong. While explicitly contrary to Freud’s fundamental insights in psychology, such a principle is in keeping with the generally Cartesian and rationalist tone of much work in the epistemic logic of belief. To indicate that it is a principle on its own. Above I call it the ‘‘Disbelief Principle.’’ This is indeed a principle that one may well want to deny if one wants to block the Fitch argument.⁷ Surely some of our beliefs are opaque to us. However, in addition to indicating another point at which one might object, I hope to present a better logical picture of the nature of such error about our own beliefs. The arguments presented so far all make use of iterated operators, believing that one believes or knowing that one doesn’t know. Some of these principles make strong claims, even about some sort of idealized belief, and so the arguments might be faulted simply for using false premises or ‘‘axioms’’ of epistemic logic. The use of iteration as such does not seem to be at fault. These arguments do not involve self-referential beliefs, or some similar vicious circularity. Still, it may be argued, each involves some strong principle relating beliefs of different logical types, principles that may also easily be denied. The next argument to be considered does not show that a truth cannot be believed, but rather that certain beliefs will guarantee a truth. The correct ⁷ Frederik Stjernberg (1997) shows how a hierarchy of knowledge operators would block the Fitch argument, in a paper that may be one of the targets of Williamson’s (2000a) brief dismissal of this proposal.

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response to this argument will introduce the use of logical types in propositions about belief and knowledge, and so lead us back to a more careful examination of our three arguments so far. The ‘‘Paradox of the Preface,’’ while originally presented to different effect, can be formulated so as to involve a proposition that asserts something about the truth value of a totality of propositions including itself.⁸ Consider a modest author who is convinced, from study of his own earlier work, and the work of others, that his new book is likely to contain a mistake, a false statement. The author then announces this in the preface: ‘‘There is a false assertion in this book.’’ The mere act of making this assertion in the preface has remarkable logical impact. First, it becomes thereby true that there is a false assertion in the book, for if the author had so far avoided making one, the statement in the preface is guaranteed to be false. But in fact we guarantee that it is some other statement that must be false. For if only that one is false, then it must be true, so on pain of contradiction, there must be some other statement in the book that is false. What a powerful way to introduce falsehoods into the body of potentially error free texts! Rather than a case of a truth that cannot be believed we have a case of a belief that cannot but be true. We can symbolize the argument presenting the paradox as about belief, and having the belief that one has a false belief, using an abbreviation ‘F φ’ for ‘one falsely believes that φ’, i.e. ∼φ & Bφ: The Paradox of the Preface (for Belief) 1 1) B ∃p Fp Assumption 2 2) ∼∃p Fp Assumption 1,2 3) B ∃p Fp & ∼ ∃p Fp 1,2, Prop Logic 1,2 4) ∃q (Bq & ∼q) 3, E.G. 1,2 5) ∃q Fq Def F 1 6) ∃p Fp 2–5 reductio? 7) B ∃p Fp ⊃ ∃p Fp 1–6 ⊃ Intro 1 8) ∃p (Fp & ∼(p = ∃p Fp) ) 1,7, Various steps The connection with self-reference is introduced when one considers whether the very belief in the preface is the false one. If one observes the types of propositions, as suggested in a ramiﬁed theory of types, one will say that steps (2) and (5) involve beliefs of different types and so are not in contradiction. A mistake occurs then at step (6); there is no reductio ad absurdum because the p in step (2) is of a different logical type from the q of step (5). Now there is no explicit self-reference here, simply a violation of restrictions on types produced by allowing a proposition to include a quantiﬁer over propositions that ranges over itself. This is, however, how a system of type theory will analyze paradoxes of self-reference. ⁸ Originally from Makinson (1965), I follow the formulation of A. N. Prior (1971): p. 87.

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A simpliﬁed version of Alonzo Church’s (1976) r-types will present enough of the ramiﬁed theory of types to represent the points to be made here.⁹ If premise (1) in the Paradox of the Preface has these type assignments: B (2) (∃p 1 F (1) p 1 ) then when we get to line (4) the assignment of r-type to the generalized variable q will yield: ∃q 2 (B (2) q 2 & ∼q 2 ) hence the q in line (5) will have the r-type 2, while the p in (2) has r-type 1: ∃p 1 (B (1) p 1 & ∼p 1 ) so there is no contradiction and no reductio argument. The Fitch argument and its relatives do not involve illegitimate self-reference, but rather this distinct sort of failure to observe principles of logical types. There is more than only one sort of paradox or fallacy involved in these sorts of ‘‘paradoxes,’’ and they do not all call for the same sort of resolution. Given the range of ‘‘type free’’ solutions to the liar paradox, it has become common wisdom in philosophical logic that type oriented solutions to paradoxes are unintuitive, or violate principles of ordinary language. That doesn’t seem right for this propositional version of the Preface Paradox. My suggestion is that the same is the case for Fitch’s argument. Put another way, one can proﬁtably investigate whether the Fitch argument is valid in ‘‘Russellian’’ intensional logic, i.e. the ramiﬁed theory of types with propositions, independently of whether this is the right way to approach all such paradoxes. Provably self-referential sentences will be banned by such a logic, but that is not the only source of potential violations of principles of types. The Fitch argument does not rely on any sort of self-reference. Determining whether certain arguments are valid in intensional logics with logical types is distinct from the general issue of the appropriateness of using type theory to resolve paradoxes in philosophical logic. Something may be learned from applying different logical tools to different paradoxes. Indeed it is an important step to be able to separate this larger family of results into those that do and those that do not involve self-reference. For example, Roy Sorensen (1988) has gone to great pains to identify those ‘‘assimilators’’ who identify some of what he calls ‘‘blindspot’’ arguments with self-referential paradoxes. An important example is the Surprise Examination Paradox: the case ⁹ The notation includes expressions of the form (ι)/1 which would be the r-type of a predicative, or lowest level, functions of individuals, which are of r-type ι. A proposition of the lowest level will be (a 0-place function) of r-type ()/1. Propositions deﬁned in terms of the totality of propositions of r-type ()/1, i.e. those whose deﬁnitions involve quantiﬁers over that totality, will be of r-type ()/2. When dealing with propositions it is convenient to abbreviate the r-type ()/n simply with the integer n. Thus propositions will be of levels 1,2,3, etc. Finally, when dealing with only predicative functions one can ignore the (potentially crucial) level index, thus, for example, writing (n)/1 as simply (n). The upshot of these abbreviations is that numbers indicate the order of propositions, as Church deﬁnes that term.

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of the teacher who informs the students that there will be an exam during the next week, but that the students won’t know until the day of the exam when it is to be given. Sorensen identiﬁes yet another of his blindspots in the Surprise Exam, namely a proposition that can be true but not known, in this case the truth that there will be an exam, say, on Thursday, but that it will not be known that there will be an exam on Thursday (until Thursday). This is then a case of an assertion p & ∼Kp of the sort we have already encountered. Following a long line of commentators, including Quine, this ‘‘solution’’ to this paradox is simply to point out that the students can’t know that there will be a surprise exam, even though the teacher may announce it in class. A distinct and almost equally long line of commentators has tried to assimilate this puzzle to various paradoxes of self-reference by analyzing the teacher’s assertion that the test will be a surprise as saying something like this: ‘‘There will be a test, but you won’t be able to deduce that there will be from this assertion.’’ While this does produce a puzzle, it seems to miss the straightforward analysis of this problem as one involving blindspots, more propositions that can be true but not known.¹⁰ Indeed some formulations of the ‘‘one day’’ version of the paradox simply involve the teacher saying that there will be an exam tomorrow but it will be a surprise, and so you don’t know it that there will be an exam tomorrow. This is a straightforward use of a Fitch style ‘‘blindspot.’’ The Fitch argument doesn’t involve self-reference, any more than the Surprise exam does on my preferred formulation, even though some paradoxes in this vicinity may well do so. Thus, when formulated in Russellian intensional logic, the ﬁrst part of the Paradox of the Preface involves a violation of the theory of types distinct from an explicit self-reference.¹¹ I now turn to an investigation of whether the Fitch argument survives the imposition of types on assertions about belief. This is a radical proposal in terms of logical form by contemporary standards, if not when Church wrote, but it has a less radical result, namely that of challenging ¹⁰ Even commentators such as Williamson (2000a) who argue that there is something special about the number of days in which the exam could be given, so that at the beginning one can know that there will be an exam, agree that by the last day that is impossible. ¹¹ It appears that in his revision to the ﬁrst draft of his paper in response to Church’s ﬁrst comments, Fitch proposes introducing a principle that if one desires something then one knows that one desires it. In the ‘‘Second Referee Report,’’ page 5, Church responds: ‘‘To further my objection—that there is no law of psychology according to which it can be inferred from the fact that a knows something that therefore a desires something—Fitch replies by pointing out that a might know that a desires p. If, however, Fitch consents to adopt one of the standard devices for avoiding the epistemological paradoxes, this reply will no longer be open to him. For example, on the basis of Russell’s original theory of types, ‘a desires p’ is of higher order than p, whereas the two ‘something’ ’s in my assertion must of course be understood as of the same order.’’ Church here considers the analysis of Fitch’s suggestion in terms of type theory and points out that ‘‘knows that’’ applied to a proposition ‘‘a desires p’’ will be of a different type from ‘‘knows that’’ applied to p. The same would hold for an iterated proposition ‘‘a knows that a knows that p’’ and the proposition ‘‘a knows that p.’’

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certain principles that involve iteration of epistemic operators.¹² Proposing the type theoretic analysis challenges the standard practice, in epistemic logic, and indeed all modal logic, of innocently allowing iteration of operators with regard to syntax. From the point of view of type theory an operator such as B becomes a propositional function applied to propositions. If p is of order 1 then the proposition Bp might have type assignments as following: B (1) p 1 . The whole proposition Bp will then be of order 2. As a result the iteration BBp will involve the following types: B (2) (B (1) p 1 ) Changing the analysis of ‘B’ from an operator that can be indeﬁnitely iterated to a function that differs according to its arguments does not mean that there is any error in the use of modal logic for such notions. Rather what is presented is a different analysis of the syntax of the language used. This allows one to reconsider the validity of various principles, but does not automatically rule any of them out as ill-formed. It should be noted that analyzing some apparent operators, such as the epistemic operators ‘K’ and ‘B’ as second order functions does not require that all operators, including sentential connectives, be so treated. It is possible to analyze a negation ∼p as being the result of applying a one place function to a propositional variable: N (1) p 1 , and so see it as literally a function of propositions as I have proposed for knowledge and belief. In a formulation of type theory, there indeed will be such a function. But it need not be required as the only way of formulating a negation. The language which introduces B and K can keep ∼ whether or not it introduces N . If it does there will be provable an equivalence: N (1) p 1 ≡ ∼p, but ∼ itself can continue to be read as a connective, rather than an expression of N (1) p 1. (Indeed one might consider leaving modal operators as operators in a language where epistemic operators are read as ‘‘propositional attitudes’’ or functions.) The proposal to replace operators with functions can thus be taken selectively, on the basis of considerations that suggest that it does not innocently iterate without raising the type of the proposition to which it applies.¹³ On the proposed account, what appears as one schema, for example the principle B ∼ Bφ ⊃ ∼Bφ, will in fact have to be read as a ‘‘typically ambiguous’’ ¹² Thus Williamson (2000a) argues that the Surprise examination, and several other arguments in epistemic logic incorrectly assume either the KK principle, or some other principle that involves iteration of knowledge claims. But this he does treating K and B as an operator that can be iterated freely, at least syntactically. ¹³ C. A. Anderson has developed a Russellian intensional logic in his (1989) which would allow for negation, for example, to be treated as a connective, while propositional functions can also be represented and the equivalences such as that between N and ∼ above also proved. Despite some unclarity about just how ﬁne the classiﬁcation of types is to be, Whitehead and Russell say at ∗9 · 1 3 1 of Principia Mathematica that a negation is of the same type as the negated proposition. Other passages, however, go against this, suggesting that connectives are functions differing from ‘believes that’ only in being extensional (pp. 6, 8) and that they are ‘‘typically ambiguous,’’ because they apply to different types (p. 43).

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schema ranging over various types for the proposition φ and itself involving functions B of two different types. One instance of this will be for the case where φ is a proposition of the lowest order 1, and so the instance will be what might be called the ‘Disbelief ’ principle: B (2) ∼ B (1) p 1 ⊃ ∼ B (1) p 1 What reason might one have to reject such a principle? If the antecedent B (2) ∼ B (1) p 1 is true, but the consequent ∼ B (1) p 1 is false, then we have a false second level belief about a ﬁrst level belief. How is this coherent? The picture of belief that is suggested by using the theory of types for propositions is of a hierarchy. At the bottom are certain basic propositions. A logic of belief, like a logic of knowledge, is intended to represent not just the particular, potentially arbitrarily collected psychological states of an agent that may be identiﬁed as beliefs. Rather the logic is based on some notion of epistemic consistency (and so of a consequence relation) which is constrained by the logical properties of propositions believed. If propositions believed are distinguished by logical type, then the sort of belief appropriate to each will be distinguished by type. Belief in a basic proposition p 1 of the ﬁrst type will be reported with a second level proposition B (1) p 1 . Belief in that second order proposition, and any other such propositions about ﬁrst level propositions will be reported with a proposition of the type of B (2) p 2 . Thus above the ﬁrst order propositions are propositions that depend on that ﬁrst totality, including propositions that some of those ﬁrst order propositons are believed, then yet another third sort of propositions depending on that second totality, and so on. The failure of the ‘‘BB’’ principle is easy to understand on this model. The proposition asserting belief in a proposition of the ﬁrst level, say p 1, will be a second level proposition, B (1) p 1, and the assertion of belief in that B (2) B (1) p 1 a distinct third level proposition. No logical connection requires that one imply the others. The same holds with the ‘‘Disbelief ’’ principle. The higher level belief may be in error, but it is in no way logically suspect. Given that beliefs must be stratiﬁed into types, there will be no beliefs about all beliefs whatsoever, and any principle such as ‘‘BB’’ or the ‘‘Disbelief’’ principle, which apparently ranges over all beliefs must be seen as ‘‘typically ambiguous.’’ Each speciﬁcation for a distinct level will be a distinct proposition, any or all of which might fail. Admittedly, as an account of natural language expressions about knowledge and belief, and of the semantic paradoxes concerning the predicate ‘true,’ the theory of types may not be the most natural account. I propose that since the

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ramiﬁed theory of types is a natural way to handle the Paradox of the Preface, it may also be a promising way of getting a uniﬁed account of the more selfreferential seeming epistemic paradoxes and the ‘‘blindspot’’ or Moorean cases which involve the recurrent appearance of p & ∼Bp and p & ∼Kp. Denying that operators iterate easily won’t block the original Fitch argument that was given for knowledge. ‘‘Knows that’’ is certainly a factive operator so both: K (2) K (1) p 1 ⊃ K (1) p 1 and: K (2) ∼ K (1) p 1 ⊃ ∼K (1) p 1 surely hold. There is a different aspect of the original Fitch argument about knowledge and the two variants about belief, the Hintikka argument, and my ‘‘Fitch-B,’’ to which one might object after paying attention to types. This objection is motivated by thinking about what an agent might come to know or believe in ideal conditions. I repeat the original Fitch argument, only now with type indices: The Fitch Argument with Types 1 1) K (2) (p 1 & ∼K (1) p 1 ) Assumption (2) 1 (1) 1 (2) 1 (2) (1) 1 ∼ K p Axiom 2) K (p & ∼K p ) ⊃ K p & K 1,2, Prop Logic 1 3) K (2) p 1 & K (2) ∼ K (1) p 1 T Axiom 4) K (2) ∼ K (1) p 1 ⊃ ∼K (1) p 1 3,4, Prop Logic 1 5) K (2) p 1 & ∼K (1) p 1 1–5 reductio? 6) ∼K (2) (p 1 & ∼K (1) p 1 ) My suggestion is that we can also reject step (6) on the grounds that there is no literal contradiction at step (5) to produce a reductio argument. We need not in general accept the principle K (2) p 1 ⊃ K (1) p 1 . The antecedent should certainly be considered well formed, for we do want to express sentences like (1) and also use the distribution principle, but it is not clear that the antecedent always implies the consequent. Think of an idealized agent developing beliefs in order, in some sort of ideal epistemic conditions. There is no reason to believe that what is known at one level is also known at the next higher, or that what is known at a higher level must be known at lower levels. These inferences between knowledge claims fail as a matter of logic, and hence are open to objection on the grounds that they misrepresent the idealized notion of knowledge or belief being formalized. There is no incoherence in not knowing p at the lower level and knowing it at the next higher level. What is known is frequently a function of other beliefs and knowledge. If knowledge is distinguished by types, what is known at one type will be a function of other propositions of the same type which may be evidence or defeators for knowledge claims. Thus a blindspot at one level may be resolved when considered at a higher level, at which one

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knows p after all. Again we don’t even have to accept the validity of the Fitch argument. A similar problem arises with the Hintikka argument. The instance of the ‘‘Distribution Axiom’’ that is relevant will have type assignments as follows: B (2) (p 1 & ∼B (1) p 1 ) ⊃ B (2) p 1 & B (2) ∼ B (1)p

1

The problem is that the ﬁrst conjunct in the consequent is not a simple step of iteration away from producing a believed proposition that will contradict the second. Rather, one will need a version of the ‘‘KK’’ thesis for belief of the following sort: B (2) p 1 ⊃ B (2) B (1) p 1 While well formed, this is certainly even more contentious than its untyped version. The Fitch argument for belief, my ‘‘Fitch B,’’ faces exactly the same problem with types as the original Fitch argument. Thus, while the two arguments for belief require strong principles of iteration and ‘‘disbelief ’’ that may be seen as especially contentious when formulated with types, all three run into further problems involving types in the distribution step. The results of the foregoing discussion can illuminate the question of the logical status of veriﬁcationism, and in particular the relevance of Fitch style arguments to veriﬁcationism. In his paper ‘‘Truth as Sort of Epistemic’’ (Wright (2000)), Crispin Wright has proposed a new formulation of the veriﬁcationist view that to be a truth is to be knowable. After considering some counterfactual formulations of the view that what is true is what would be veriﬁed if tested under ideal epistemic conditions, one of which will be discussed below, Wright proposes that an internal realist adopt what he calls ‘‘Provisional Biconditionals.’’ These require a notion of ideal epistemic conditions Q, perhaps relativized to a proposition p under consideration, Qp , and a notion of what would be believed under those conditions Z (p). The biconditional proposed then is: (o∗ )Q → (p is true ↔ Z (p) ) This is modeled on Carnap’s notion of a ‘‘reduction sentence,’’ such as that which would analyze dispositional predicates such as ‘x is soluble’ with a conditional ‘If x is placed in water then x is soluble if and only if x dissolves.’¹⁴ The reduction sentence replaces the simpler conditional ‘If x is placed in water and dissolves then x is soluble’ for it makes all objects not in water be soluble.¹⁵ To leave the connection ¹⁴ Carnap (1936–7). Carnap sees reduction sentences as giving ‘‘partial deﬁnitions’’ for in cases where the antecedent test condition does not obtain, in this case x is not put in water, the sentences are silent about the term ‘‘soluble.’’ He envisages test conditions being built up to progressively approach a genuine deﬁnition. ¹⁵ The motivation for (o∗ ) in fact treats the ﬁrst conditional as counterfactual. In keeping with the spirit of the account being designed to handle the solubility of substances never put into water, we also want to know about the truth of propositions that are in fact never submitted to

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between truth and veriﬁcation open in so many cases is to retreat from the general veriﬁcationist position. Wright is forced into this by the failure of a number of stronger conditional connections, some of which will be considered below. Wright suggests that using his preferred (o∗ ) veriﬁcationism will have a response to the ﬁrst Fitch argument, as follows. An agent in an ideal epistemic situation faced with the apparent combination p & ∼Zp will have to either stick with the evidence for p and give up ∼Zp, or else stick with ∼Zp and thus drop p. An agent in an ideal situation would resolve the blindspot, or to put the point differently, for agents in ideal epistemic situations there are no blindspots. There may be some things that such an agent doesn’t know, but when the agent is put in an ideal epistemic situation with regard to that fact, the blindspot will be resolved. There is no truth that couldn’t be known by the agent in an ideal epistemic condition for that truth. The response to the Fitch argument, then, is to accept it as sound, but deny that it is an objection to veriﬁcationism. It may thus be true that I cannot know both that p and that I don’t know that p, but for the good reason that in an ideal epistemic situation I would abandon one or the other conjunct. Granted then, this is something that could be true but not known by me (at least), but that is so only because in ideal circumstances it would not be true. One use of the account of iterated belief that I have discussed is to make Wright’s proposal ﬁt easily with a logic of belief. We can treat the operator B in Fitch-B as expressing what would be believed by an agent in an ideal epistemic situation, and show then how the argument is blocked. Beliefs of the ﬁrst level, propositions of type p 1 that are believed, as expressed by propositions B (1) p 1 , will be beliefs about ordinary, non-epistemic, states of affairs. There will also be beliefs about such beliefs, expressed by B (2) (B (1) p 1 ). One interpretation of the operator B is to describe an idealized belief, what a believer after sorting out logical consequences and unhampered by limitations of memory and concentration might be said to at least tacitly believe. Might one not also build into this notion of an ideal believer some feature of being a believer in an ideal epistemic situation? Then we can easily represent the general idea of Wright’s response to Fitch arguments as holding that they are cases where we have more than the usual reason to reject both the ‘‘BB’’ principle and the Disbelief principle. On this account the second level belief operator will describe those beliefs that an ideal agent, or agent in an ideal epistemic situation, can have in the logical position of also holding certain ﬁrst level propositions and beliefs about those propositions. That may differ from what beliefs the agent might be able to have when considering simply the ﬁrst order propositions. There is no reason to hold, as a matter of logic, that what is believed as expressed by ‘B (1) ’ is therefore believed in the way expressed by ‘B (2) ’, or vice versa. veriﬁcation procedures, but which might have been. Using a counterfactual conditional does not make a difference to the sort of objections being considered here.

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A counterexample to Hintikka’s BB thesis might consist of the following situation; B (1) p 1 at the ﬁrst ‘‘stage’’, but then ∼B (2) B (1) p 1 at the second due to some different surrounding beliefs of the same type, indeed, at this stage p 1 is not believed, i.e., ∼B (2) p 1 . The second level beliefs will be either a reﬁnement and improvement or simply an extension of the ﬁrst level beliefs. Similarly for the ‘‘Disbelief ’’ principle. We may well have a situation of revised beliefs where B (2) ∼ B (1) p 1 at the second level, while B (1) p 1 yet, for some reason again, ∼B (2) p 1 . The ﬁrst belief is indeed false, but understandable, especially given the last. Thus not only can one argue that the conclusion of the Fitch-B argument or the Hintikka version is not a problem for veriﬁcationism, one might actually object that these arguments are invalid. On my account a belief might be missed or dropped when moving up a type, precisely because it is part of a blindspot. A belief might be added, as it is reconciled with a larger group of beliefs, including some beliefs about beliefs. At each type there may well be logical connections between beliefs, as expressed with the distribution and conjunction axioms, but all principles involving iteration of belief are up for grabs, and most likely false in certain instances. This notion of resolving blindspots in ideal conditions helps to answer a challenge from T. Williamson ((2000a): 281) presented in his account of the Fitch arguments. Williamson asks how it could be that K (2) p yet not K (1) p. He suggests that this could only be through a convoluted path. ‘‘Perhaps a claim could be known at level i + 1 but not at level i if the route to knowing it involved claims about knowledgei , even though the target claim did not, but it would be bizarre if such contrived cases were crucial to a defence of weak veriﬁcationism.’’ I think that Wright’s notion of resolving blindspots in ideal circumstances presents just such a justiﬁcation. Williamson points out that the ‘‘canonical’’ veriﬁcation of some p would result in ﬁrst level knowledge, K (1) p, while the veriﬁcation of a conjunction K (1) q & p would yield K (2) (K (1) q & p). Deriving that latter proposition by conjoining knowledge claims would require ﬁrst establishing K (2) p rather than K (1) p. It is true then, as he observes, that the canonical way of verifying a certain conjunction would not be by means of canonical veriﬁcations of the conjuncts. A sentence conjoined with a statement about knowledge or belief would have to be veriﬁed in a different way than if it were confronted in isolation. I leave it to veriﬁcationists to determine amongst themselves whether this is out of keeping with the spirit of their program. Williamson has presented another objection to this ‘‘type’’ response to the Fitch argument.¹⁶ The type of an operator like K , he points out, is determined by the content of the propositions to which it applies, i.e. their logical type. ¹⁶ In discussion at the conference on ‘‘The Limits of Warrant’’ held at the University of Waterloo in May of 2001 from which this paper derives.

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The justiﬁcation for the reply to the Fitch argument, however, seems to rely on claims about the procedure employed by an epistemic agent in resolving potential ‘‘blindspots.’’ The objection seems to be that one can’t make claims about the logical types of propositions which rely on a speculative account of how some agent might come to have certain attitudes towards those propositions. My response is that the use of epistemic logic requires a certain amount of idealization in its application. We must decide what idealized epistemic agents are to be represented with a given formalism. In the application I propose, we consider agents that observe type distinctions in their beliefs, i.e. epistemic states are determined in ‘‘stages,’’ whether temporal or not, in which beliefs about beliefs depend on beliefs of lower order. If one application of epistemic logic is to model certain sorts of epistemic agents, this seems to be a legitimate sort of such modeling. Propositions are indeed assigned to types by their contents, but attitudes towards them will depend on each other in ways that do reﬂect a procedure for determining epistemic states. Not every general argument of this sort against every form of veriﬁcationism makes use of iterated operators, however. There are other forms of veriﬁcationism that fall prey to other objections and which just look as though they involve self-reference or at least some sort of variation of type, but in fact do not. Consider a version of the notion that what is true is what would be conﬁrmed under ideal conditions Q, which Wright also discusses: (o) p is true ↔ (Q → Z (p) ) Some time ago Alvin Plantinga (1982) gave an argument against this proposal. Plantinga’s argument is that if one assumes (o) then it will be impossible for some p to obtain without Z (p), in other words, the notion of truth and actual veriﬁcation collapse, provided that one identiﬁes truth with veriﬁcation under ideal conditions. Here is yet another formulation of this argument: The Plantinga Argument 1 1) p is true ↔ (Q → Z (p) ) Assumed account of truth 1 2) Q is true ↔ (Q → Z (Q ) ) 1, Instantiation 1 3) (Q is true ↔ (Q → Z (Q ) )) 2, Necessitation 4 4) ♦(Q & ∼Z (Q ) ) Assumption 4 5) ♦(Q is true & Q & ∼Z (Q ) ) 4, (A is true ↔ A) 4 6) ♦(Q is true & ∼(Q→ Z (Q ) ) 5, counterfactual logic 1 7) (Q → Z (Q ) ) 4–6, reductio 1 8) (Q→ Z (Q ) ) 7, counterfactual logic 1 9) (Q is true) 8, 3, (S5) modal logic 1 10) Q 9, RE, (A is true ↔ A) 1 11) p is true ↔ Z (p) 10, 1, counterfactual logic The course of the argument is as follows: It is ﬁrst shown that it is necessary that if Q obtains that it is veriﬁed, Z(Q), for if at some world the ﬁrst were true

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but not the second, Q would be true and the counterfactual connecting its truth and its veriﬁability (step 2) would be false. But if Q necessitates its veriﬁcation the assumed account of truth yields the conclusion that Q is necessarily true. But then Q occurs vacuously in the ﬁrst conditional (1) and so the account collapses into a straightforward identiﬁcation of truth with veriﬁcation. The earlier tactic of reformulating operators as properties of propositions and thus expressing the counterfactual conditional as a two place relation between propositions does not reveal any of the type differences that occur in the Fitch argument. If one treats the Lewis or Stalnaker possible world semantics for counterfactuals as an analysis of a relation between propositions, and thinks of propositions as sets of worlds, that relation between the antecedent of a counterfactual and its consequent will be of a high order, involving considerable quantiﬁcation over worlds and classes of worlds, etc. As well, possibility may either be thought of in terms of quantiﬁcation over worlds and those as classes of propositions, or just directly as a property of propositions. Fixing on precise type indices for and → will not be easy. What is crucial for the analysis I have proposed for the Fitch argument is the issue of whether those predicates and relations are applied to propositions of different type levels in the course of the argument and whether, when seen that way, the inferences remain justiﬁed. There are such changes of level, as in the move from (7) to (8), where necessity is ﬁrst applied to a relatively low level material conditional and then to a much higher level counterfactual conditional. But these moves do not threaten the validity of the argument as one might argue the shift of levels of belief does affect the validity of the Fitch argument. There might seem to be a whiff of self-reference to the Plantinga argument (although neither Plantinga nor Wright suggest there is any real self-reference here). The issue is rather that if one thinks that truth coincides with what would be veriﬁed in ideal conditions, then turning that analysis on the ideal conditions themselves and wondering whether one can verify that they obtain (in ideal conditions) seems to require some reﬂexive veriﬁcation. But it doesn’t actually do so. All that is required is universal instantiation to apply the condition in (1) to those ideal conditions Q. (1) strictly speaking ought not to be called a deﬁnition. Although p appears on both sides of the biconditional, the notion of truth appears only on the left. However the argument does assume that the Tarski style equivalence (A is true ↔ A) is not only true, but necessarily true, so that ‘A’ and ‘A is true’ are readily interchangable, so (1) comes very close to being circular. Rather, (1) should be called simply an axiom or analytic truth. If it is in order logically then so is the instance that applies to Q. That is as close as this argument comes to involving self-reference. A version of the above argument will still work when operators are treated as higher order predicates and type restrictions are observed. This is not all that one might say about the Plantinga argument from the perspective of type theory, however. Part of the motivating conception of

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type theory is the notion that certain totalities are ‘‘illegitimate.’’¹⁷ The totality of propositions must be seen as only a ‘‘potential’’ inﬁnite, approached by a series of increasingly more inclusive classes of propositions, but never completed. It is not possible to have propositions about all types. But then the notion of a settled state of affairs in which all propositions can be evaluated, whether seen as being ideal conditions for verifying any statement, or as the ‘‘end of scientiﬁc enquiry,’’ does not ﬁt with this perspective. In speciﬁc, then, the proposition Q above will be limited to a speciﬁc type. What type will that be? While formally the argument allows it to be of a low type, it might seem that a proposition describing a situation in which any proposition can be veriﬁed would have to be of a higher type, and hence even impossible because it would have to have an arbitrarily high type. However this may be, the argument, at least for a speciﬁc order of Q, is clear of the sort of type violations considered earlier. As another potential formulation of a knowability thesis for the veriﬁcationist, Wright (2000) considers that a defender of (o) might say that the ideal conditions need not be the same for every proposition, especially when that proposition can be the description of those very ideal conditions. Rather the ideal conditions may vary according to the proposition under consideration, thus for p the ideal conditions will be Qp : (o) p is true ↔ (Qp → Z (p) ) This proposal runs afoul of one of the sort of ‘‘conditional fallacies’’ that Robert Shope (1978) describes. What if it is possible that the obtaining of Qp itself inﬂuences the obtaining of p counterfactually, say Qp →∼ p? Then we could have Qp , Z (p) and ∼ p true in the same possible world if Qp → Z (p) obtains as well (and p is true in the actual world). The ‘‘fallacy’’ is in asserting an ‘‘analysis’’ like (o) when a counterfactual like Qp →∼ p might be true. Again all of this does not require any use of iterated operators or predicates, and so is not subject to the same response as was the Fitch argument and its relatives. This survey of our family of related arguments suggests the following conclusions. The ‘‘Fitch style’’ arguments, both for knowledge and belief, which make use of principles about iterated operators such as BB and Disbelief, may be faulted on the truth of those principles, but not for some sort of illicit self reference in the very iteration. They are not arguments that ‘‘this very truth’’ cannot be known. The Fitch argument for knowledge, can, however, be faulted for a violation of the principles of logical types, though not an explicit self reference. Arguments against accounts that use the notion of ideal epistemic conditions which question how we might know whether such conditions obtain, such as Plantinga’s, also do not make illicit use of self reference or other violation of the theory of types. The same seems to hold about qualms about counterfactual analyses of truth ¹⁷ See Whitehead and Russell, Principia Mathematica, p. 37, and the discussion of the vicious circle principle.

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in terms of veriﬁcation where the truth of a counterfactual hypothesis interferes with veriﬁcation conditions. Church was right to notice that his ‘‘Fitch Argument’’ was ‘‘suggestive of the liar and other epistemological paradoxes.’’ An overview of various epistemological arguments does reveal similarities, but also differences. Some arguments against veriﬁcationism can be met by denying their premises, others rely on dubious principles of epistemic logic, while another seems to violate principles of type theory. Other arguments, however, cannot be so easily dismissed by the veriﬁcationist.

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Pa r t V C a r t e s i a n Re s t r i c t e d Tr u t h

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12 Tennant’s Troubles Timothy Williamson

First, some reminiscences. In the years 1973–80, when I was an undergraduate and then graduate student at Oxford, Michael Dummett’s formidable and creative philosophical presence made his arguments impossible to ignore. In consequence, one pole of discussion was always a form of anti-realism. It endorsed something like the replacement of truth-conditional semantics by veriﬁcation-conditional semantics and of classical logic by intuitionistic logic, and the principle that all truths are knowable. It did not endorse the principle that all truths are known. Nor did it mention the now celebrated argument, ﬁrst published by Frederic Fitch (1963), that if all truths are knowable then all truths are known. Even in 1970s Oxford, intuitionistic anti-realism was a strictly minority view, but many others regarded it as a live theoretical option in a way that now seems very distant. As the extreme veriﬁcationist commitments of the view have combined with accumulating decades of failure to reply convincingly to criticisms of the arguments in its favour or to carry out the programme of generalizing intuitionistic semantics for mathematics to empirical discourse, even in toy examples, the impression has been conﬁrmed of one more clever, implausible philosophical idea that did not work out, although here and there old believers still keep the ﬂame alight. A diffuse philosophical tendency cannot be refuted once and for all by a single rigorous argument. Nevertheless, such an argument can severely constrain the forms in which the tendency is expressed. The tendency labelled ‘anti-realism’ and Fitch’s argument together constitute a case in point. My ﬁrst publication (1982) was a response to the Fitch argument. I argued that it was intuitionistically invalid, and therefore did not show intuitionistic anti-realism to be committed to the absurd claim that all truths are known. Naturally, my aim was not to endorse intuitionistic anti-realism; I found it as deeply implausible then as I do now. But that does not distinguish it from other forms of anti-realism, and such Thanks to Peter Milne and Peter Pagin for comments on a draft of Williamson (2000b), which have in turn beneﬁted the present paper.

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dispositions are not invariant across persons. My aim was rather to assess what forms of anti-realism must be argued against in some way beyond Fitch’s. The advantage in ﬁnal plausibility of those other forms of anti-realism over the brazen assertion that all truths are known is tenuous at best: but it is still worth getting clear about the logical situation. Some of my later work on the Fitch argument (1988b, 1992, 1994a) reﬁned the envisaged response to the Fitch argument and established its formal stability. In The Taming of the True (1997), Neil Tennant objects to the speciﬁc intutionistic anti-realist response to Fitch that I had envisaged, and proposes his own alternative responses, still of a broadly intuitionistic anti-realist kind. In response (2000b), I argued that both Tennant’s objections and his alternatives fail, and that the result illustrates a more general point: that moderate forms of anti-realism tend to be the least stable. Tennant replied at length (2001a). For some time I thought that the problems with his 2001 reply were sufﬁciently evident to make any further response from me unnecessary. Later experience has taught me otherwise. The purpose of this paper is to show that Tennant’s reply fails completely to meet the difﬁculties that I raised in 2000. Since his reply engages with many details of that paper (2000b), while missing the relevance of some of the most crucial ones, the most efﬁcient course is to rehearse the arguments of that paper, interspersing them with discussion of Tennant’s objections as they arise. Thus the present paper constitutes a self-standing critique of Tennant’s treatment of the Fitch paradox that properly includes its predecessor.¹

I The ﬁrst task is to expound the Fitch argument in a form suitable for the subsequent discussion. Anti-realists argue that truth is epistemically constrained. Their arguments are too complex, elusive and at least vaguely familiar to formulate here. We can gesture at them thus: a sentence s as uttered in some context expresses the content that P only if the link between s and the condition that P is made by the way speakers of the language use s; their use must be sensitive to whether the condition that P obtains; that requires of them the capacity in principle to recognize that it obtains, when it does so; thus P only if speakers of the language can in principle recognize that P. In brief: all truths are knowable. We can formalize the anti-realist conclusion in a schema: (1) ϕ → ♦Kϕ ¹ Where appropriate, sentences or paragraphs from Williamson (2000b) have been absorbed into the present text.

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Here ♦ and K abbreviate ‘it is possible that’ and ‘someone sometime knows that’ respectively; ϕ is to be replaced by declarative sentences.² Presumably, the relevant sense of ‘possible’ is not merely epistemic, because the anti-realist takes sensitivity to whether a condition obtains to require a genuine recognitional capacity (a metaphysical possibility of knowing), not a mere incapacity to recognize one’s ignorance (an epistemic possibility of knowing). According to (1), the truth really could have been known. Much in the anti-realist arguments deserves to be questioned. Fitch (1963) introduced a direct objection to their conclusion with an apparent reductio ad absurdum of (1). It requires two highly plausible principles about knowledge: only truths are known, and known conjunctions have known conjuncts. More formally: (2) Kϕ → ϕ (3) K(ϕ ∧ ψ) → (Kϕ ∧ Kψ)³ Principles (2) and (3) jointly entail that nothing is ever known to be an always unknown truth. For (2) yields K(ϕ ∧ ¬Kϕ ) → (ϕ ∧ ¬Kϕ ) and therefore K(ϕ ∧ ¬Kϕ ) → ¬Kϕ, while (3) yields K(ϕ ∧ ¬Kϕ ) → (Kϕ ∧ K¬Kϕ ) and therefore K(ϕ ∧ ¬Kϕ ) → Kϕ, so together they give: (4) ¬K(ϕ ∧ ¬Kϕ ) Principles (2) and (3) are intended as necessary constraints on knowledge, and the propositional logic used to derive (4) from (2) and (3) is necessarily truthpreserving, so by a variant of the rule of necessitation in modal logic we can conclude that what (4) says is not could not have been: (5) ¬♦K(ϕ ∧ ¬Kϕ ) Now consider the special case of (1) with ϕ ∧ ¬Kϕ in place of ϕ: (6) (ϕ ∧ ¬Kϕ ) → ♦K(ϕ ∧ ¬Kϕ ) By (5) and (6): (7) ¬(ϕ ∧ ¬Kϕ ) In classical logic, (7) is equivalent to: (8) ϕ → Kϕ ² Tennant brieﬂy considers other possible readings of K. He complains (1997: 270) that ‘Williamson [ . . . ] appears not to have anticipated the possibility’ of interpreting K in Fitch’s arguments as ‘it is known at t that’ for a particular time t. He has overlooked the discussions of such readings of the argument at Williamson 1982: 204, 1988b: 425–8 and 1994a: 141, 144. It is unnecessary to add to them here. ³ For difﬁculties facing the attempt to evade Fitch’s argument by rejecting (3) see Williamson 1993 and 2000a: 275–85.

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Of course, (8) is deeply implausible. It says in effect that any truth is known. As one instance, it says that there is a fragment of Roman pottery at a certain spot only if someone sometime knows that there is a fragment of Roman pottery there. The corresponding instance of (1) for the same value of ϕ is quite plausible: there is a fragment of Roman pottery there only if it could have been known by someone sometime that there was a fragment of Roman pottery there. But, according to (8), there is a fragment of Roman pottery there only if the possibility of knowing is actualized; that claim is quite unwarranted. Although, by (4), the attempt knowingly to identify a particular example of an unknown truth would be self-defeating, we surely have ample evidence of a less direct sort that not every truth is ever known. Although some believe that an omniscient god makes (8) true, that issue is not very pertinent here. For if we restrict the substitutions for ϕ to sentences of our language, the anti-realist motivation for (1) allows us to read ‘someone’ in the deﬁnition of K as ‘some member of our speech community’; the links between sentences of English and their contents are made by human speakers of English without divine intervention. ‘There is a fragment of Roman pottery at that spot’ is a sentence of our language; surely not every truth expressible in our language will ever be known by some member of our speech community. If you think it matters, give K that restricted reading. We should thus reject schema (8). On the assumption that (8) was derived from (1) using uncontentious principles, we should therefore reject schema (1) too. The seminal presentation of the case for (1) is Michael Dummett’s (1959b, 1975 and elsewhere). Notoriously, he integrates it with a case for a comprehensive anti-realist reconception of meaning in terms of veriﬁcation-conditions rather than truth-conditions, which, he argues, will invalidate classical logic, in particular the law of excluded middle, and justify its replacement by something like intuitionistic logic. The latter was originally proposed as the logic of intuitionistic mathematics, and its intended semantics reﬂect that role, being formulated in terms of the notion of proof. Since the mathematical notion of proof is inappropriate to empirical statements, Dummett envisages a generalized intuitionistic semantics in which a broader notion of veriﬁcation plays the key role.⁴ Even granted that his arguments are not compelling, it is pertinent to ⁴ It is sometimes claimed that one can meet Dummett’s demand simply by treating the notion of truth in a truth-conditional compositional semantics for empirical discourse as veriﬁability. That is a mistake. The key notion in the intuitionistic compositional semantics for mathematical language is ‘ is a proof of ϕ’, not ‘ϕ is provable’ (‘Something is a proof of ϕ’); for example, the semantic clause for → concerns the transformability of proofs of the antecedent into proofs of the consequent, which makes no sense in terms of an undifferentiated notion of provability. Thus the key notion in an analogous veriﬁcation-conditional compositional semantics for empirical discourse is ‘ is a veriﬁcation of ϕ’, not ‘ϕ is veriﬁable’. The truth-conditional clause for negation, ‘¬ϕ is true if and only if ϕ is not true’ (or something similar), cannot be interpreted as a constraint on veriﬁcations, for if something is not a veriﬁcation of ϕ, it does not follow that it is a veriﬁcation of ¬ϕ. The idea of Dummett’s original argument is that the key notion in a compositional semantics should be one

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ask how the attempted reductio ad absurdum of (1) fares under intuitionistic logic, which we may provisionally suppose the anti-realist defender of (1) to have adopted.⁵ The argument from (1) to (7) is intuitionistically acceptable, but the step from (7) to (8) is not. Intuitionistically, (7) is equivalent to: (9) ¬Kϕ → ¬ϕ Intuitionistically, we cannot reach (8) from (9), deleting the negations.⁶ Intuitionists can consistently accept (9) while denying that all truths are known. Since (7) generalizes, they must deny that there is an unknown truth: on their constructivist understanding of the existential quantiﬁer, one could in any case verify that one could never verify that existential claim, because (by (4)) one could never verify a particular instance of it. Intuitionistically, to verify that one could never verify ψ is to verify ¬ψ. Denying that there is an unknown truth does not commit one intuitionistically to asserting that all truths are known. Given (9), the intuitionist can consistently deny the universal generalization of (8) but cannot consistently deny any particular instance of (8), for (9) is intuitionistically equivalent to the double negation of (8). In this respect, the intuitionistic status of ϕ → Kϕ given (9) is exactly like that of the law of excluded middle. For the intuitionist can consistently deny the universal generalization of ϕ ∨ ¬ϕ but cannot consistently deny any particular instance of it, because ¬¬(ϕ ∨ ¬ϕ ) is to which speakers’ use is sensitive, and that their use is sensitive to a given condition in a given context only if they can decide in that context whether it obtains. On such a view, they can decide in a given context whether they have a proof or veriﬁcation of ϕ in that context (for an argument against this decidability claim, see Williamson 2000a: 110–13), but no notion of provability or veriﬁability that would constitute a not wildly subjectivist notion of truth is decidable within the limitations of every given speech context. Thus no notion of provability or veriﬁability that would constitute a not wildly subjectivist notion of truth meets Dummett’s constraints on the key notion in a compositional semantics, even if it is the existential generalization of a notion of proof or veriﬁcation that does meet Dummett’s constraints. ⁵ Dummett’s own response to Fitch (2001) does not appeal to intuitionistic logic; rather, it restricts the knowability principle (1) to atomic sentences. This restriction is hard to reconcile with Dummett’s original motivation for the knowability principle, a motivation that applies to complex sentences just as much as to atomic ones. It will not do to say that the use of complex sentences is indirectly epistemically grounded because their atomic constituents are. For connectives such as conjunction and negation are used as constituents of complex sentences, not by themselves. Thus any epistemic grounding of the use of connectives must derive from an epistemic grounding of complex sentences in which they occur, not vice versa: yet Dummett’s strategy against Fitch is just to avoid any such direct epistemic grounding of the use of complex sentences. Thus his anti-realism unravels. Note also that his original (1959b) examples of sentences that the realist contentiously treated as veriﬁcation-transcendent involved complex constructions such as universal quantiﬁcation and the counterfactual conditional: ‘A city will never be built on this spot’ and ‘If Jones had encountered danger, he would have acted bravely’ are not atomic sentences. See Brogaard and Salerno (2002) (some points of which are anticipated at Williamson 1990: 300) and Tennant (2002) for criticism of Dummett’s response to Fitch and Rosenkranz (2004) for more discussion. ⁶ Williamson (1992) proves model-theoretically that schema (8) is not derivable from schemas (1)–(7) and (9) even in a very strong system of intuitionistic modal epistemic logic.

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intuitionistically valid.⁷ Call an anti-realist position on which (7) is valid and (8) invalid moderately hard anti-realism. It is opposed to very hard anti-realism, on which both (7) and (8) are valid, and soft anti-realism, on which both (7) and (8) are invalid.⁸ A consistent moderately hard anti-realist view can be worked out in some detail within the framework of intuitionistic logic (Williamson 1982, 1988b, 1992). Moderately hard anti-realists may regard the Fitch argument as a further reason for anti-realists in general to adopt intuitionistic rather than classical logic, since their response to the challenge depends on a distinction available within intuitionistic but not classical logic—although of course this was not Dummett’s reason for proposing intuitionistic logic as an appropriate logic for anti-realism. Obviously, (9) itself is a deeply problematic consequence of (1). According to (9), what is never known is not true: thus if no one ever knows that there is a fragment of Roman pottery at a certain spot, there is no fragment of Roman pottery there. That sounds as bad as (8). But there are differences. Schema (8) eliminates the logical distinction between ϕ and Kϕ, for (8) and its uncontentious converse (2) jointly yield ϕ ↔ Kϕ, and therefore (ϕ ) ↔ (Kϕ ) for any sentential context ( ) deﬁned with the standard intuitionistic connectives (in particular, of course, ¬ϕ ↔ ¬Kϕ ). Although (9) eliminates the logical distinction between ¬ϕ and ¬Kϕ, since (9) and (2) jointly yield ¬ϕ ↔ ¬Kϕ, and therefore (¬ϕ ) ↔ (¬Kϕ ), it does not eliminate the logical distinction between ϕ and Kϕ, as the underivability of (8) from (9) shows. The moderately hard anti-realist loses fewer distinctions than does the very hard anti-realist. Indeed, the former can consistently deny the universal generalization of (8), while the latter must assert it. Thus the moderately hard anti-realist, unlike the very hard anti-realist, can assert a gap between what is true and what is ever known. For reasons not peculiar to anti-realism, the acknowledgement of the gap is essentially general; it cannot be made at the level of an individual sentence, because one cannot knowingly present a speciﬁc instance of a never known truth. But how can the moderately hard anti-realist mitigate the implausibility of particular instances of (9)? If one knew that human life was about to be eliminated by a huge meteorite, might one not be entitled to assert that no one will ever know that there is a fragment of Roman pottery at that spot without being entitled to ⁷ See Williamson (1982: 206). Tennant (1997: 267–8) objected that ¬(ϕ → Kϕ) is not intuitionistically inconsistent given (1) (and the other principles used to derive (9)) unless it intuitionistically implies both ϕ and ¬Kϕ, and that, intuitionistically, although it implies ¬Kϕ and ¬¬ϕ it does not in general imply ϕ (it does in the special case when ϕ is decidable, but then ϕ ∨ ¬ϕ holds and the analogy is not useful). The objection rests on an error. Intuitionistically, ¬Kϕ and ¬¬ϕ imply ¬(¬Kϕ → ¬ϕ), the negation of (9); thus ¬(ϕ → Kϕ) is intuitionistically inconsistent given (1) (and the other principles used to derive (9)), although (without those principles) it does not intuitionistically imply ϕ. Tennant (2001a: 277–9) concedes and ampliﬁes this criticism of his objection. ⁸ This adapts Tennant’s terminology (1997: 261).

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assert that there is no fragment of Roman pottery there?⁹ Such examples suggest that the intuitionistic operator ¬ does not correctly formalize the negation that we apply to empirical sentences, such as those of the form Kϕ. Dummett himself distinguishes the negation of intuitionistic mathematics from empirical negation (1977: 337). That is not to say that intuitionistic negation cannot meaningfully be applied to empirical sentences; rather, another negation operator may be needed as well to interpret ‘not’ in empirical discourse. Empirical negation must itself behave non-classically if moderately hard anti-realism is not to collapse into very hard anti-realism, for otherwise empirical negation could be substituted for ¬ throughout the derivation of (8), but it must not behave non-classically in exactly the same ways as ¬, otherwise it would offer no advantage. Major difﬁculties face the attempt to add such an operator (Williamson 1994a). In what follows, I continue to assume that the anti-realist employs intuitionistic negation; I do not seek to minimize the associated problems. Moderately hard anti-realism remains a deeply problematic position. Nevertheless, it avoids the most drastic consequences of very hard anti-realism, and a full critique of it will not simply cite Fitch. Of those engaged in the reﬁnement of Dummett’s programme and the attempted generalization of intuitionistic semantics to empirical discourse, one of the most active has been Neil Tennant (1987, 1997). In his 1997 analysis of the Fitch problem, he argues that the envisaged moderately hard anti-realist line is unstable in the crucial test cases for the Fitch argument, and suggests an alternative soft anti-realist strategy based on restricting the knowability principle (1) (1997: 245–79).¹⁰ Section II below shows that his objection to the envisaged moderately hard anti-realist line is fallacious, and that his later attempt to defend his objection merely changes the subject. Section III shows that under rather general conditions his restricted version of (1) still implies (7) and (9), so that his would-be soft anti-realism collapses into the moderately or very hard view; his later attempt to defend his restricted version of (1) results in its trivialization.

II Tennant objects to the moderately hard anti-realist treatment of the Fitch argument that I described that it is unstable, because in what I presented as ⁹ For related points see Percival (1990). Other relevant discussions of Fitch’s argument in the context of intuitionistic logic include Wright (1993: 427–30), Cozzo (1994), Pagin (1994), Usberti (1995: 65–6, 121–8) and DeVidi and Solomon (2001). ¹⁰ Tennant (1997: 276–8) also rejects the attempt in Edgington (1985) to reconstrue the knowability principle by means of something like an ‘actually’ operator, for reasons given in Williamson (1987, 2000a: 290–301) and Wright (1993: 426–32). See also Percival (1991). Edgington’s idea is developed rigorously by Rabinowicz and Segerberg (1994), Lindström (1997) and Rückert (2004), but none of these papers fully answers the philosophical objections.

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crucial test cases the distinction between (7) and (8) collapses (1997: 268). He correctly points out that in the examples I use to illustrate the argument, I instantiate ‘ϕ’ in (7) and (8) with a decidable sentence: we have a decision procedure whose application would result in a veriﬁcation or falsiﬁcation of ϕ (‘There is a fragment of Roman pottery at that spot’). This is the simplest and most vivid form of the problem that Fitch raises, and a crucial test for any adequate treatment. Tennant also correctly points out that (7) is intuitionistically equivalent to (8) when Kϕ is decidable.¹¹ He then attempts to argue that Kϕ is decidable if and only if ϕ is decidable. If he is right, the envisaged moderately hard anti-realist line fails the crucial test. Before proceeding, I note an obvious condition of adequacy on Tennant’s critique. On pain of irrelevance, it must not depend on a reading of the operator ‘K’ other than that intended by the object of his criticism. It is to be read as ‘someone sometime knows that’, where ‘knows’ itself is understood in its predominant ordinary sense. In this sense, I may fail to know whether a given very large natural number (as presented by a corresponding numeral) is prime, even though I have a decision procedure for ﬁnding out. Of course, I know many things that I am not currently thinking about; my knowledge is stored. But I do not know something merely in virtue of its being routine for me to ﬁnd out, if I do not in fact ﬁnd out, or merely in virtue of its being a logical consequence of other things that I know. This is the understanding of ‘know’ that is evidently in play throughout my presentations of the moderately hard anti-realist line (1982, 1988b, 2000) and in most other discussion of Fitch’s argument. It is a very natural understanding in the context of that argument, which concerns the relation between potential and actual knowledge, a contrast obscured if ‘K’ is itself understood as meaning something potential. In order to preserve the relevance of Tennant’s critique, I will therefore take it for the time being in terms of the usual reading of ‘K’. Another reading will be considered later. Tennant’s objection is sustained if he can demonstrate that Kϕ is decidable whenever ϕ is; the converse does not concern us here. I will argue that his supposed demonstration that the decidability of ϕ implies that of Kϕ is fallacious. Here it is: Suppose that ϕ is decidable. Then here is a decision method for Kϕ: apply the given decision method for ϕ. If you thereby determine that ϕ is true, then you know that ϕ. So you have determined that Kϕ is true. If, on the other hand, you determine that ϕ is false, then you have determined that Kϕ is false, because no one could ever know a falsehood. So if ϕ is decidable, then so is Kϕ. (1997: 262) ¹¹ Tennant claims that ‘the validity of ¬(ϕ ∧ ¬Kϕ) guarantees the validity of ϕ → Kϕ if, but only if, Kϕ is decidable’ (2001a: 265). The ‘only if’ direction, which is inessential to his argument, is an error. The validity of ¬¬Kϕ → Kϕ is easily seen to be sufﬁcient for the validity of ¬(ϕ ∧ ¬Kϕ) to guarantee the validity of ϕ → Kϕ but is intuitionistically a weaker condition than the decidability of Kϕ.

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Our possession of a decision procedure for Kϕ entitles us to assert:¹² (10) Kϕ ∨ ¬Kϕ Intuitionistically, (7) and (10) jointly entail (8). If Tennant is right, the difference between moderately and very hard anti-realism disappears in paradigms of the cases where it was supposed to help. Why does Tennant suppose that our possession of a decision procedure for Kϕ entitles us to assert (10)? He is relying on a principle about the assertibility of disjunctions: (DIS) Our possession of a method whose application will either verify ϕ or verify ψ entitles us to assert ϕ ∨ ψ. (DIS) allows us to assert the disjunction in advance of actually applying the method.¹³ Since the application of Tennant’s decision procedure will supposedly either verify Kϕ or verify ¬Kϕ, by (DIS) our mere possession of the decision procedure, in advance of actually applying it, entitles us to assert Kϕ ∨ ¬Kϕ. At ﬁrst sight, (DIS) looks very plausible. It is surely correct when ϕ and ψ are mathematical. Nevertheless, we can easily see that Tennant’s argument must be unsound on the present reading of ‘K’. For if it were sound, there would also be a sound argument for a much stronger conclusion, namely, that our possession of a decision procedure for ϕ entitles us to assert this: (11) Kϕ ∨ K¬ϕ We can argue for this conclusion in Tennant’s style: Suppose that ϕ is decidable. Then apply the given decision method for ϕ. If you thereby determine that ϕ is true, then you know that ϕ. So you have determined that Kϕ is true. If, on the other hand, you determine that ϕ is false, then you know that ¬ϕ. So you have determined that K¬ϕ is true.

The reasoning for the case where ϕ is true is in Tennant’s own words; the reasoning for the case where ϕ is false in effect merely substitutes ¬ϕ for ϕ throughout that reasoning. By (DIS), we can conclude that our mere possession of the decision procedure for ϕ entitles us to assert (11). But it is utterly implausible to claim that whenever ϕ is decidable, someone sometime will know whether ϕ holds, in the usual sense of ‘know’. Mere possession of the decision procedure does not entitle us to assert that anyone will ever have that knowledge. For in advance of applying the procedure, we may have no reason to think that it will ever be applied; indeed, we may have reason to think that it will never be applied. ¹² Tennant (1997: 268) is explicit that if ϕ is decidable then ϕ ∨ ¬ϕ is intuitionistically acceptable. ¹³ The phrase ‘possession of a method whose application will . . . ’ is to be read as implying ‘recognition that application of the method will . . . ’.

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Perhaps its application is costly in time and other scarce cognitive resources and ϕ is a proposition whose truth-value is unlikely ever to be of interest to anyone. Moreover, it may be unlikely that anyone will ever come to know whether ϕ holds without applying such a decision procedure. Alternatively, we may know that the meteorite is about to strike. Let us provisionally accept the Tennant-style argument for the proposition that if ϕ is decidable then we have a method whose application will either verify Kϕ or verify K¬ϕ. Nevertheless, our mere possession of that method does not entitle us to assert Kϕ ∨ K¬ϕ. Thus the problem lies with (DIS). What has gone wrong is that the application of the decision procedure for ϕ brings about the state of affairs expressed by (11). That is why our mere possession of the method is not enough. To assert (11), we need some reason to think that someone sometime will apply the method. Exactly the same problem affects Tennant’s assumption that we are entitled to assert (10) whenever ϕ is decidable. For, if ϕ is true, the application of the decision procedure for ϕ brings about the state of affairs expressed by Kϕ. Our mere possession of the method, in advance of actually applying it, does not entitle us to assert (10). If, as Tennant assumes, our possession of a decision method for ψ always entitles us to assert ψ ∨ ¬ψ, then it has not been shown that our possession of a decision method for ϕ puts us in possession of a decision method for Kϕ. Alternatively, if we did count the Tennant-style procedure as a decision method for Kϕ, then it has not been shown that mere possession of a decision method in that weak sense for Kϕ would entitle us to assert Kϕ ∨ ¬Kϕ, and Tennant’s argument would still fail because we could not bridge the gap from (7) to (8).¹⁴ For a more dramatic example of the fallacy, consider a paradigm of a potentially undecidable sentence, ‘A city was, is or will be built on this spot’ (compare Dummett 1959b). Assume that no city has ever been built on the spot, and there is no present plan to build one, but equally no special reason why one should never be built there. For the Dummettian anti-realist, we are not entitled to assert ‘Either a city was, is or will be built on this spot or no city was, is or will be built on this spot’, for we have no procedure for determining which disjunct holds. Now imagine someone claiming: We do have a decision method for the sentence. For you can in principle build a city on this spot. Having done so, you will have determined that a city was, is or will be built on this spot.

Although we have the capacity in principle to build a city on the spot, it does not put us in possession of a decision procedure in the relevant sense, for by intuitionistic standards it does not entitle us to assert ‘Either a city was, is or will ¹⁴ As observed in n. 11, we do not need (10) to get from (7) to (8); the weaker ¬¬Kϕ → Kϕ would sufﬁce. But the same considerations apply; the purported decision procedure would entitle us to assert ¬¬Kϕ → Kϕ only by entitling us to assert (10).

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be built on this spot or no city was, is or will be built on this spot’ in advance of exercising the capacity. Indeed, the problem for (DIS) is even worse, for we can take both ϕ and ψ in it to be ‘A city was, is or will be built on this spot’. We possess a method (building a city) whose application will verify ‘A city was, is or will be built on this spot’. Therefore, by (DIS) we are now entitled to ‘Either a city was, is or will be built on this spot or a city was, is or will be built on this spot’. Since even in intuitionistic logic ϕ ∨ ϕ is trivially equivalent to ϕ, we are now entitled to assert ‘A city was, is or will be built on this spot’, in advance of building one or even planning to do so. That is absurd. More generally, such an argument would conclude that whenever one has the power in principle to make ϕ true, one is entitled to assert ϕ in advance of exercising that power or even planning to exercise it. That conclusion involves one in contradictions, since one often has both the power to make ϕ true and the power to make ¬ϕ true, for example when ϕ is ‘I shall count to a thousand by midnight’.¹⁵ By (DIS), one is now both entitled to assert ϕ and entitled to assert ¬ϕ, irrespective of one’s intentions. Evidently, (DIS) can fail for sentences whose truth-values depend on our will; more speciﬁcally, it fails when the the truth-value of ϕ or of ψ depends on whether the method is actually applied. Of course, this problem does not arise when ϕ and ψ are mathematical. If (DIS) does not govern the assertibility of disjunctions, what does? Intuitionistic semantics relies on some notion of canonical veriﬁcation (Dummett 1977: 389–403). One is entitled to make an assertion for which one lacks a canonical veriﬁcation, if one knows that such a veriﬁcation exists. The existence of the veriﬁcation does not consist in its being possessed by anyone; nevertheless, since the intuitionist conceives it as essentially capable of being possessed by someone, it does not import any sort of platonism inconsistent with the intuitionistic view.¹⁶ The natural suggestion is then that a canonical veriﬁcation of a disjunction consists of a canonical veriﬁcation of a disjunct. We may be entitled to assert a disjunction without being entitled to assert any disjunct, because we know that a canonical veriﬁcation exists for some disjunct without knowing which. We are in that position with respect to ϕ ∨ ¬ϕ when we have a genuine decision procedure for ϕ but have not yet applied it. But if we possess the purported decision procedure for Kϕ without having applied it, we do not thereby know that a canonical veriﬁcation for Kϕ or a canonical veriﬁcation for ¬Kϕ exists, even though we know how to bring such a canonical veriﬁcation ¹⁵ Any incompatibility between free will and determinism is irrelevant here. That I am causally determined not to apply a method is compatible with my possession of it in the relevant sense. ¹⁶ For more detail see Williamson (1982: 206–7, 1988: 429–32), where the idea is used to show the invalidity of an argument for ϕ → Kϕ from the intuitionistic semantics of → and the premise that Kϕ is veriﬁable whenever ϕ is veriﬁable (see Hart 1979: 165; Wright 1993: 430); the existence of a canonical veriﬁcation of ϕ does not imply the existence of a canonical veriﬁcation of Kϕ. Tennant describes this objection as ‘compelling’ (1997: 264).

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into existence. Consequently, we do not know that a canonical veriﬁcation for Kϕ ∨ ¬Kϕ exists; so by intuitionistic standards we are not entitled to assert the disjunction. Tennant’s response to this critique is to insist on a different reading of ‘K’ (2001a: 273–7). In his terminology, he understands ‘know’ to mean virtually or implicitly know rather than occurrently know, with a corresponding difference in the interpretation of ‘K’ (2001b: 109). In this sense, in merely possessing a decision method for primeness we already know whether any given natural number is prime. Unfortunately, he never squarely faces the obvious problem that this makes his discussion prima facie quite irrelevant to the point at issue, namely, the stability of the moderately hard anti-realist response to Fitch as I proposed it, which involves a logical distinction between (7) and (8) for decidable ϕ on the usual reading of ‘know’, not Tennant’s. We have just seen that his arguments do not work on the usual reading. To make his arguments relevant, Tennant would have to show that the usual reading of ‘know’ was somehow unavailable to the moderately hard anti-realist. The boldest, least plausible strategy would be to argue that such a reading makes no sense. But it is hopeless for the anti-realist to pretend that actually applying a decision procedure makes no cognitive difference at all. Indeed, Tennant describes in his own terms what cognitive difference it makes: The whole point of having a decision procedure is to discover the canonical form of expression of a proposition that, at the outset, can be identiﬁed only by description: as the result of applying the decision procedure. (2001a: 276)

We are supposed to discover the canonical form by applying the decision procedure. We thereby discover the canonical form only if we did not already know what it was. But in Tennant’s special sense we did already know what it was, since we had a procedure for ﬁnding out. Thus the point of applying the procedure is to gain knowledge in the usual sense of the canonical form. Consequently, Tennant himself relies on the coherence of something like the usual reading.¹⁷ Indeed, he speaks of it as of a genuine sense of ‘know’ (2001a: 276, 2001b: 109). Although he writes favourably of a conceptual reform that would impose his special reading on ‘know’, not all moderately hard antirealists need feel bound by such a reform; moreover, eliminating all unreformed epistemological terminology would leave us unable to articulate the point of applying a decision procedure. ¹⁷ Tennant says of a passage in which I emphasize the cognitive difference that applying a decision procedure makes that it ‘displays a vestige of realist thinking’ (2001a: 274); if so, Tennant has not explained how he himself can do without that vestige of realist thinking. Tennant’s further complaint concerning the conditional ¬Kϕ → ¬ϕ and ‘a curious asymmetry (between truth and falsity)’ (2001a: 275) raises an issue that is discussed in a more acute version in Williamson (1994), to which the reader is referred. That issue is separate from Tennant’s main argument.

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Tennant points out that his special reading of ‘know’ is better suited to epistemic logics that assume logical omniscience. He claims that ‘if this idealizing assumption were disallowed, the original Fitch argument would not go through’ (2001a: 274). But that is to ignore crucial differences. The only aspect of logical omniscience used in the argument is principle (3), that knowledge of a conjunction implies knowledge of its conjuncts. But one can accept that very weak closure principle even for knowledge in the usual sense without accepting logical omniscience in general. The modest idea that in knowing a conjunction one knows its conjuncts does not commit one to the extravagant idea that in knowing anything one knows anything that it entails. Moreover, there are revisions of the Fitch argument that do not even rely on (3).¹⁸ Nothing that Tennant says compels the moderately hard anti-realist to formulate their knowability principle (1) in terms of a reading of ‘know’ that satisﬁes logical omniscience. Thus the effect of Tennant’s insistence on his special reading of ‘know’ is that his arguments completely fail to engage with the version of moderately hard anti-realism that he was supposed to be attacking. Indeed, they fail to engage with just about any form of anti-realism that endorses the knowability principle (1) on the usual reading of ‘know’, since the most pertinent version of the Fitch argument will then involve that reading throughout. To refrain from endorsing the principle that every truth is capable of being known in the usual sense is to take a signiﬁcant step back from full-blooded anti-realism. Do Tennant’s arguments establish something of interest on his preferred reading of ‘know’, even though they do not establish what they were supposed to? On such a reading, (11) is no longer obviously absurd when ϕ is decidable but not occurrently decided. His purported decision procedure for Kϕ is at much less risk of violating the constraint, which he accepts, that ‘[p]roper decision procedures do not interfere with the states of affairs’ that they are supposed to determine (2001a: 276–7). Nevertheless, crucial unclarities remain. First, Tennant’s argument that the decidability of ϕ implies the decidability of Kϕ appears to commit him to a version of the highly controversial principle that when one knows, one knows that one knows. For the key passage is this: If you [ . . . ] determine that ϕ is true, then you know that ϕ. So you have determined that Kϕ is true. (1997: 262)

The principles invoked in the two sentences can be formalized by schemas (D1) and (D2) respectively, with Dy for determination by you as true and Ky for your knowledge: (D1) Dyϕ → Kyϕ (D2) Kyϕ → DyKϕ ¹⁸ See n. 3.

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By substitution of ‘Kϕ’ for ‘ϕ’ in (D1): (D3) DyKϕ → KyKϕ By transitivity from (D2) and (D3): (D4) Kyϕ → KyKϕ In other words, if you know something, you know that someone sometime knows it. But that principle is extremely doubtful, even on a special reading of ‘know’ that satisﬁes logical omniscience (Williamson 2000a: 114–34). In terms of epistemic logic: a knowledge operator can satisfy deductive closure without satisfying the S4 principle. An argument that relies on assumptions that are jointly as strong as (D4) is some way from establishing its conclusion. Second, it is not altogether clear what Tennant means by ‘decidable’. He is attracted by a version of moderately hard anti-realism different from the one I envisaged. On his version, one accepts schema (7) and rejects schema (8), but accepts this restricted version of (8) (2001a: 266–7): (8a) [ϕ∧ (ϕ is decidable)] → Kϕ If one accepts the Tennant-style argument from the decidability of ϕ to (11) on his special reading, (8a) is a simple consequence given principle (2) (the factivity of knowledge), since K¬ϕ implies ¬ϕ. Clearly, if (8a) is not to collapse into (8), one must reject: (8b) ϕ → (ϕ is decidable) For of course (8a) and (8b) jointly entail (8). Thus ‘ϕ is decidable’ cannot be regarded as a notational variant of ϕ ∨ ¬ϕ, as it often is in intuitionistic writings, for that would immediately validate (8b). Moreover, that reading disqualiﬁes undecidability as a genuine option, as Tennant takes it to be (2001a: 266), since ¬(ϕ ∨ ¬ϕ ) is intuitionistically inconsistent. At ﬁrst sight, it is unclear how an intuitionist can reject (8b). For suppose that we have a proof of ϕ. Then that proof decides ϕ, and thereby constitutes a proof of its decidability. Thus (8b) seems to be intuitionistically provable, because we can transform any proof of its antecedent into a proof of its consequent. Presumably, the way to block (8b) is to insist that its consequent concerns our actual possession of a decision method, not the mere possibility in principle of possessing one. Attempts to validate (8b) by means of the intuitionistic semantics for the conditional can then be blocked like similar attempts to validate (8).¹⁹ But then a more than virtual notion of possessing a decision procedure is doing the crucial work. To clarify his envisaged version of moderately hard anti-realism, Tennant needs to explain how it is supposed to reconcile virtual and non-virtual aspects of cognition. ¹⁹ See n. 16.

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Tennant’s alternative version of moderately hard anti-realism remains undeveloped. On an understanding of the language relevant to the version that I mooted, the decidability of ϕ does not imply the decidability of Kϕ in any sense conducive to (10). Tennant fails in his attempt to collapse (7) into (8) for decidable ϕ. He has located no instability in the envisaged version of moderately hard anti-realism.

III Tennant also proposes another response to Fitch’s argument, this time a soft anti-realist one, by constructing and defending a modiﬁed knowability principle. Having deﬁned a sentence ϕ to be Cartesian if and only if the contradiction ⊥ does not follow from Kϕ, he endorses this restricted variant of (1), formulated as a rule of inference:²⁰ (♦KC)

ϕ; ergo ♦Kϕ, where ϕ is Cartesian

Informally: (♦KC) says that truth entails knowability except when Fitch’s problem occurs. Tennant admits that it may be undecidable whether a given step is an instance of a rule like (♦KC), because it is undecidable whether ⊥ follows from Kϕ. He claims that this does not matter, on the grounds that we will apply it only when we do know that the condition is met. Since (♦KC) is intended for use only in a few philosophical arguments, not for systematic application in mathematics or science, the undecidability is supposed not to defeat its purpose. Tennant’s rule looks desperately ad hoc. He replied in detail (2001b) to a similar charge from Michael Hand and Jonathan Kvanvig (1999). However, his reply depends on a misunderstanding of the nature of the charge. He writes as though what is wrong with ad hoc principles is that they are restricted to the point of total or partial triviality. For instance, he argues that the restriction in (♦KC) is analogous to restrictions in other principles that are nevertheless ‘substantive, informative and important’ (2001b: 110, 111, 113).²¹ He seems to assume that if a principle P∗ is more restricted than a principle P, then P is ad hoc only if P∗ is also ad hoc.²² He assumes that ‘[a]d hoc emendations to general ²⁰ For (♦KC) to be a well-deﬁned rule, the notion of consequence used to deﬁne ‘Cartesian’ should be given independently of (♦KC) and cannot be assumed to be closed under it (see Tennant 1997: 275). This does not affect the argument in the text. ²¹ The examples are unconvincing. They too look ad hoc. Moreover, his restricted thesis about truth treats ‘This sentence is false’ as a premise of the Liar paradox (2001b: 110), whereas the mere well-formedness of the sentence is what makes trouble. His restricted thesis about wondering (2001b: 113) results from a derivation that makes the radically idealizing assumption (SI) that ‘a rational thinker is one whose attitudes are self-intimating to the thinker himself’ (1997: 248) so that I rationally wonder whether something is the case only if I believe that I so wonder (ibid.: 255). ²² Some such assumption is required for the relevance of his claim that ‘The restricted thesis about truth [Tarski’s] to which almost every philosopher subscribes is in fact even more restricted

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laws in natural science’ detract from their applicability (2001b: 112). However, imagine a scientist who has always maintained that emeralds are always green, but at time t is confronted with a blue emerald, and adopts the revised theory that emeralds are always grue, where something is grue at a time if and only if either it is green and the time is before t or it is blue and the time is not before t. The gruesome, gerrymandered revised theory is clearly ad hoc, even though it is just as general, substantive, informative and important (if correct) as the old theory. The problem is not triviality but ill-motivated complexity. Although the new theory predicts the same data before t as the old theory, and improves on the old theory with respect to the datum at t, the previous evidential support for the old theory does not transfer to the new theory, even before counterexamples to the new theory emerge. Analogous problems face (♦KC). If Fitch’s argument forces anti-realists to restrict the original knowability principle (1), then something is wrong with their original meaning-theoretic arguments for (1). Until we have an adequate diagnosis of the fallacies in those arguments, we cannot assume that such considerations confer any support whatsoever on (♦KC) or any other attempted approximation to (1) that does not immediately succumb to the Fitch argument. A subtle fallacy in an argument can easily mean that it establishes nothing of interest whatsoever. (♦KC) is a gruesome principle. We need not dwell on the charge that (♦KC) is ad hoc, for that is not the worst of its problems. The point of restricting (♦KC) to Cartesian cases is to enable the soft anti-realist to avoid asserting (8), that something is true only if known, even for decidable sentences. Since Tennant holds that (7) collapses into (8) when ϕ is decidable, his soft anti-realist rejects (7) too. The restriction on (♦KC) blocks the original derivation of (7) from (1). But Tennant overlooked a more complex derivation of (7) from (♦KC) and some plausible assumptions. The argument will be conducted in Tennant’s preferred background logic, which is not the standard intuitionistic one but his weaker system IR of intuitionistic relevance logic (Tennant 1997: 343–4). The differences do not affect the arguments below. For deﬁniteness, let ϕ be the decidable sentence ‘There is a fragment of Roman pottery at that spot’ (we assume a suitable context). Introduce a proper name ‘n’ by the stipulation that it is to designate (rigidly) the number of books actually now on my table. Thus ‘n’ is not a numeral such as ‘9’ but rather a name whose reference is ﬁxed by an empirical description. Let ‘E’ be the predicate ‘is even’. We ﬁrst argue that the conjunction ϕ ∧ (Kϕ → En) is Cartesian. For suppose that K(ϕ ∧ (Kϕ → En) ) is inconsistent, in the sense that ⊥ follows from than’ a thesis about truth that Tennant wants to show not to be ad hoc, given that ‘Tarski can hardly be accused of making an ad hoc restriction to his disquotational Thesis about truth’ (2001b: 111).

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K(ϕ ∧ (Kϕ → En) ) according to the logic adverted to in the deﬁnition of ‘Cartesian’. Then this story contains an inconsistency: STORY: I ﬁnd a fragment of Roman pottery at this spot and identify it correctly; I thereby come to know that there is a fragment of Roman pottery there. I also count the books actually now on my table and discover that the number is even; I deduce that if someone sometime knows that there is a fragment of Roman pottery at that spot then n is even.²³ By putting the two pieces of knowledge together, I acquire the knowledge expressed by the conjunction ϕ ∧ (Kϕ → En). Thus K(ϕ ∧ (Kϕ → En) ) holds.

But STORY is obviously consistent; we cannot exclude its truth on purely logical grounds (just try!). Of course, n may in fact be odd, in which case, since that number could not have been even, STORY expresses an impossible state of affairs. Nevertheless, STORY itself is still consistent; we cannot discern by reason alone that the description which ﬁxes the reference of ‘n’ picks out an odd number. Someone who asserts En because he failed to see one of the books is not guilty of an inconsistency. Although we might produce an inconsistent story by substituting for ‘n’ throughout STORY a numeral with the same reference as ‘n’, it does not follows that STORY itself is inconsistent. More precisely, the result of substituting a coreferential numeral for ‘n’ in the sentence K(ϕ ∧ (Kϕ → En) ) is a different sentence; the inconsistency of the latter does not imply the inconsistency of the former. Thus ⊥ does not follow from K(ϕ ∧ (Kϕ → En) ), so ϕ ∧ (Kϕ → En) is Cartesian. For the rest of the argument, let be the consequence relation of a system of modal epistemic logic based on IR with the additional rule (♦KC) and the axiom schemas (2) and (3).²⁴ Since its condition is met in this case, (♦KC) gives: (12) ϕ ∧ (Kϕ → En) ♦K(ϕ ∧ (Kϕ → En) ) Moreover: (13) ϕ ∧ ¬Kϕ ϕ ∧ (Kϕ → En) For ¬α α → β holds even in IR (Tennant 1997: 344); from ¬Kϕ Kϕ → En we can derive (13) by the rules for ∧. Now (12) and (13) yield: (14) ϕ ∧ ¬Kϕ ♦K(ϕ ∧ (Kϕ → En) ) ²³ β α → β in IR (Tennant 1997: 342). ²⁴ may differ from the consequence relation used to deﬁne ‘Cartesian’. Tennant uses the weaker set of inference rules K(ϕ ∧ ψ) Kϕ and from , ϕ ⊥ to , Kϕ ⊥ in place of (3) and (2) respectively (1997: 259–60). Nevertheless, his discussion makes clear that on his view we can reasonably treat instances of (2) and (3) as theorems of epistemic logic.

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The next step is a Fitch-like argument for: (15) K(ϕ ∧ (Kϕ → En) ) En For (3) and ∧-elimination yield K(ϕ ∧ (Kϕ → En) ) Kϕ, while (2) and ∧elimination yield K(ϕ ∧ (Kϕ → En) ) Kϕ → En, and even in IR we can then move to (15).²⁵ Since the rules used to derive (15) are truth-preserving in all possible situations, not just the actual one, if the premise of (15) expresses a possibility, so does its conclusion (if α → β is a theorem of a normal modal logic, so is ♦α → ♦β): (16) ♦K(ϕ ∧ (Kϕ → En) ) ♦En Now (14) and (16) yield: (17) ϕ ∧ ¬Kϕ ♦En Uncontentiously, it is not contingent whether n is even. Since I can count the books on my table, it is decidable whether n is even; hence n is either odd or even. But if n is odd, it could not have been even, for the mathematical properties of numbers are not contingent. Thus n could have been even only if it is even. We can symbolize that as the argument En ∨¬En, ¬En → ¬♦En, ♦En En, since ‘n’ is a rigid designator.²⁶ Thus, treating the uncontentious auxiliary assumptions concerning En as part of the background logic, we can strengthen (17) to: (18) φ ∧ ¬Kφ En Since the two cases are symmetric, we can now repeat the argument for (18) with ‘odd’ in place of ‘even’ to derive: (19) φ ∧ ¬Kφ ¬En But (18) and (19) together yield: (20) ¬(φ ∧ ¬Kφ ) That is to make (7) a theorem. But the point of Tennant’s restricted knowability principle (♦KC) was precisely to enable the soft anti-realist not to assert (7) for decidable φ (as in the present case), since on Tennant’s view (7) collapses into ²⁵ The cut rule used to chain inferences together does not hold unrestrictedly in IR, but fails only in case of a redundant premise or conclusion, which (15) does not contain. ²⁶ For the reason sketched in section I, the argument assumes a nonepistemic reading of ♦. The mere epistemic possibility that n is even does not entail that n is even. (♦KC) is in any case quite unpromising on an epistemic reading of ♦. If ♦ is read as ‘for all we know’, the principle will be unacceptable to soft anti-realists of the sort for whom Tennant seems to intend (♦KC), since, according to them, we sometimes know empirically that a given decidable proposition will never in fact be decided. If ♦ is read as ‘for all we know a priori’, (♦KC) is more or less trivialized because ♦Kφ then says little more than that φ is Cartesian.

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(8), the principle deﬁnitive of hard anti-realism, for decidable φ. Thus Tennant’s restriction is futile. Evidently, the foregoing critique of Tennant’s principle (♦KC) depends on careful respect for the distinction between the logical notion of inconsistency and the metaphysically modal notion of impossibility. One or other of En and ¬En expresses a metaphysical impossibility, but each of them is logically consistent, since the reference of ‘n’ is ﬁxed empirically. Similarly, the sentence ‘George W. Bush = Tony Blair’ is logically consistent, even though it expresses an impossibility. Unfortunately, Tennant’s reply to the critique displays a startling insensitivity to the distinction. In expounding (♦KC), he writes: It should be clear to anyone with a sympathetic understanding of the spirit of the proposed restriction that for a proposition to be Cartesian one ought to be unable to derive absurdity from it modulo any necessarily true propositions. It is a logical convention of long standing that mention of theorems as premises can be suppressed. (2001b: 264)

This passage conﬂates necessary truth and theoremhood. Of course, given the cut rule, the use of theorems of a given logic as premises of derivations in that logic does not enable one to reach any conclusions that could not be reached without those premises. But it is certainly not ‘a logical convention of long standing’ that mention of necessary truths as premises can be suppressed. The undecidable G¨odel sentence for ﬁrst-order arithmetic is a necessary truth, but that does not mean that mention of it as a premise can be suppressed in the proof theory of ﬁrst-order arithmetic. Similarly, one cannot declare ‘George W. Bush = Tony Blair’ logically inconsistent just on the grounds that its negation expresses a necessary truth. Nevertheless, Tennant is quite explicit in his notion of a Cartesian proposition that: To say that absurdity is not derivable from Kφ is equivalent to saying that absurdity is not derivable from Kφ in conjunction with any set X of necessarily true propositions. (2001a: 269–70)

We had best understand Tennant as deﬁning ‘Cartesian’ in terms of a special consequence relation for which, by stipulation, all necessities are theorems. For the time being let us read his term ‘proposition’ as equivalent to ‘sentence’, since elsewhere he treats the constituents of arguments in the logically standard way as linguistic (1997: 313–15); in his reply (2001a) he does not object to my treatment of premises and conclusions as sentences. We will reconsider the talk of propositions later. According to Tennant, ‘We are dealing primarily with logico-mathematical possibility and necessity here’ (2001a: 269). On Tennant’s proposal, if n is odd then ¬En is a logico-mathematical necessity because ‘n’ is a rigid designator; since En follows from K(φ ∧ (Kφ → En) ) by (15), absurdity is derivable from K(φ ∧ (Kφ → En) ) in conjunction with the necessity ¬En; thus φ ∧ (Kφ → En) is not Cartesian after all. By parallel reasoning, if n is even

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then φ ∧ (Kφ → ¬En) is not Cartesian. Either way, one half of my argument is supposed to break down. Tennant seems to assume that logico-mathematically necessary truths are knowable a priori, for he glosses the account quoted above of what it is for absurdity not to be derivable from Kφ thus: Whether this deﬁnition calls for the consideration only of sets X all of whose members are knowable a priori, or calls for the consideration also of sets X some of whose members might be knowable only a posteriori, is an issue of principle on which we are not at present forced to take a stand. (2001a: 270)

Tennant is not forced to take a stand on the issue of principle only if he is entitled to assume that if n is odd then ¬En is knowable a priori and if n is even then En is knowable a priori. But recall that the reference of ‘n’ was ﬁxed by the description ‘the number of books actually now on my table’. Thus we cannot know a priori whether n, so presented, is even! That was the crux of my argument. En and ¬En are not sentences of the language of mathematics, because ‘n’ is not a term of that language: although it refers to a number, it does so in a non-mathematical way. Tennant describes ‘n is even’ as ‘a mathematical proposition’ on the grounds that ‘n’ is a rigid designator (2001a: 270), but it is unclear what he means by ‘mathematical proposition’. At any rate, his discussion ignores the signiﬁcance of the empirical way in which the reference of ‘n’ was ﬁxed.²⁷ In order to make Tennant’s discussion relevant to the argument that I presented, we should understand him as deﬁning ‘Cartesian’ in terms of a special consequence relation for which, by stipulation, all necessary truths may occur as premises in the derivation of absurdity, whether or not they are knowable a priori. The result does not constitute a formal system, but never mind. Consider the following variant of (♦KC): (♦KC∗ )

ϕ ; ergo ♦Kϕ , where ¬¬Kϕ holds

We can derive (♦KC) from (♦KC∗ ) by showing that if ϕ is Cartesian, ¬¬Kϕ holds. Suppose that ¬Kϕ holds. Then ¬Kϕ is permitted to occur as a premise in the derivation of absurdity, in the sense used in the deﬁnition of ‘Cartesian’. So absurdity is derivable from Kϕ in that sense. Therefore ϕ is not Cartesian. Thus if ¬Kϕ holds, ϕ is not Cartesian. By an intuitionistically valid form of contraposition, if ϕ is Cartesian, ¬¬Kϕ holds. Thus (♦KC) is a simple consequence of (♦KC∗ ); one can also argue for the converse, although that is not our present concern. But (♦KC∗ ) is not a distinctively anti-realist principle. For a realist who accepts classical modal logic, (♦KC∗ ) is trivially truthpreserving, since ♦ is equivalent to ¬¬ simply by the duality of the two modal operators. Of course, it is odd to present ♦Kϕ as derived from the ostensible premise ϕ rather than from the condition for the applicability of (♦KC∗ ), that ²⁷ In his examples, Tennant replaces ‘n’ by a numeral (2001a: 264, 271), thereby obscuring the vital point.

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¬¬Kϕ holds, which is what really guarantees the truth of where ♦Kϕ. But that oddity is harmless once one appreciates that the connection signalled by ‘ergo’ outside the scope of the ‘where’ clause need not be knowable a priori. For exactly the same reason, Tennant’s principle (♦KC) as now interpreted, a corollary of (♦KC∗ ), is also not distinctively anti-realist; for a realist who accepts classical modal logic, it is trivially truth-preserving. The two principles may not be quite so innocent from an intuitionistic perspective, since the inference ¬¬α to ♦α is structurally analogous to the intuitionistically invalid inference from ¬∀x¬α to ∃xα. However, that extra piece of logical content from an intuitionistic perspective is, if anything, a slight concession to classicism, not the articulation of a distinctively anti-realist claim about the possibility of knowledge. Thus Tennant’s stipulations about the notion of derivability in the deﬁnition of ‘Cartesian’ are relevant to my original objection to (♦KC) only by voiding (♦KC) of all interest as a formulation of an anti-realist principle of knowability. I had in mind considerations of the kind above when I wrote in my original critique ‘A more liberal interpretation of inconsistency might trivialize ♦KC; it is not what Tennant intends’ (2000b: 110). Perhaps it was what Tennant intended, and he has indeed fallen into the trap that I warned against. Does it make a difference if we suppose that by ‘proposition’ Tennant means something signiﬁcantly more coarse-grained than a sentence, even though he does not mention the distinction between sentences and propositions in this connection? Let ‘q’ be a numeral with the same reference as ‘n’. If proper names are directly referential, then the two sentences Eq and En express the same proposition, even though the sentences do not have the same cognitive signiﬁcance for us. If knowledge is a relation to propositions, it follows that knowing Eq a priori is knowing En a priori, although one can be in a position to express one’s knowledge by one sentence without being in a position to express it by the other. Such a view might ﬁt what Tennant says in his discussion of decidability about different forms of expression of a given proposition (2001a: 276–7). Would this approach enable Tennant to restrict the use of necessarily true propositions in the derivation of absurdity to those knowable a priori under some mode of presentation or other (such as Eq)? The appeal to a directly referential semantics for proper names deals with only one class of examples. An analogue of my argument can be developed for any decidable sentence ϕ whatsoever, by substituting for En the sentence Aϕ, where A is the rigidifying ‘actually’ operator, so that Aϕ is a priori equivalent to ϕ, but ♦Aϕ entails Aϕ and ♦¬Aϕ entails ¬Aϕ. To extend the approach just sketched to such examples, Tennant would need to argue that Aϕ expresses the same proposition as some sentence that one can use to express a priori knowledge. It is hard to see what principled justiﬁcation there could be for such a claim, short of the identiﬁcation of all necessarily equivalent propositions. But on that approach there is just one necessary truth, which is known a priori under the

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mode of presentation ‘0 = 0’. We can then reproduce the derivation of (♦KC) from (♦KC∗ ), and regain just the trivialization of Tennant’s principle that the appeal to coarse-grained propositions was supposed to avoid. Thus Tennant’s response to the critique of (♦KC) serves only to emphasize his difﬁculties; his attempt to locate a fallacy in it merely trivializes his own view. Should Tennant’s soft anti-realist seek, instead of (♦KC), a knowability principle with a stronger restriction to avoid (7)? The natural suspicion is that such a restriction would have to be very draconian indeed, and thereby risk trivialization again. But the search is in any case ill-motivated, for reasons already indicated in the discussion of Tennant’s unsuccessful response to the charge that (♦KC) is ad hoc. The original knowability principle (1) was the outcome of an anti-realist argument (albeit a very dubious argument). If (1) has false consequences, then something must be wrong with the argument. If the argument is irreparably fallacious, one has lost one’s reason for postulating any knowability principle at all, restricted or unrestricted. If the argument can be repaired, the nature of the repairs should dictate the nature of the restrictions on the resultant knowability principle.²⁸ Tennant does not indicate any form of argument for anti-realism that would motivate a principle restricted in the manner of (♦KC) on a non-trivializing interpretation. To say that ϕ is non-Cartesian is not to explain on anti-realist terms how ϕ could be unknowably true, how speakers’ use of ϕ could be sensitive to a condition they could not in principle recognize to obtain or how ϕ could express its content without such sensitivity; it is merely to say that broadly logical considerations do (not not) rule out knowledge of its truth. Without such an explanation, from the perspective of principled anti-realism it is quite premature to endorse anything like (♦KC). If Fitch’s argument does not by itself refute all forms of anti-realism, it certainly shows how much would have to be done before there was a working anti-realist semantics for empirical language, even in the toy examples that we have been considering. The attempts on behalf of anti-realism to deal with the Fitch problem give every sign of a degenerating research programme. ²⁸ A failure to ﬁt the philosophical arguments for anti-realism may also affect the revisions of knowability principle proposed in Edgington (1985), Melia (1991), Rabinowicz and Segerberg (1994), Kvanvig (1995), Lindström (1997) and Rückert (2004), although the point cannot be argued here; see also the comments on Dummett (2001) in n. 5. The upshot of the treatment of Fitch’s argument in Usberti 1995 is a much more drastic restriction of ϕ in (1) to mathematical sentences, which excludes those containing K. The proposal is grounded in Usberti’s anti-Dummettian analysis of the arguments for an intuitionistic approach (see also Williamson 1998). For a discussion of Fitch’s argument in the context of classical logic see Williamson (2000a: 270–301, 318–19).

13 Restriction Strategies for Knowability: Some Lessons in False Hope Jonathan L. Kvanvig

The knowability paradox derives from a proof by Frederic Fitch in 1963. The proof purportedly shows that if all truths are knowable, it follows that all truths are known. Antirealists, wed as they are to the idea that truth is epistemic, feel threatened by the proof. For what better way to express the epistemic character of truth than to insist that all truths are knowable? Yet, if that insistence logically compels similar assent to some omniscience-like claim, antirealism is in jeopardy. Response to the paradox has drifted toward a common theme, a theme I will argue is a non-starter in resolving the paradox. Seeing this point will also make clear the philosophical inadequacy of simply viewing the paradox as a refutation of a wide range of antirealisms. Re s p o n s e s t o t h e Pa r a d o x One way to respond to this problem for antirealism is to question the proof itself, and there have been a number of questions raised about the proof. Such questioning seems to lead nowhere, however. The simplest form of the proof goes as follows. Where we understand the operator K as ‘it is known by someone at some time that’, we begin by assuming (1) K(p & ∼Kp). If we distribute the K operator across the conjunction, we get (2) Kp & K∼Kp. In thinking about the issues discussed here, I have been helped immensely in a number of conversations and blog interchanges with the following, whom I would like to thank: Mike Beaty, Bryan Frances, Michael Hand, Stephen Hetherington, Carrie Jenkins, Robert Johnson, Matt McGrath, Julien Murzi, Joe Salerno, Fritz Warﬁeld, Jonathan Weinberg, and Tim Williamson.

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Since knowledge implies truth, K∼Kp implies ∼Kp; hence (2) implies (3) Kp & ∼Kp, allowing us to prove by reductio (4) ∼ K(p & ∼ Kp). Since (4) is a theorem, we can derive by the Rule of Necessitation (5) ∼ K(p & ∼ Kp), which is equivalent to (6) ∼ ♦K(p & ∼ Kp). This claim, however, is pretty obviously inconsistent with the claim that all truths are knowable. All that is needed is for the value for p in (6) to be a truth that nobody knows, in which case p & ∼Kp is a truth. By the knowability principle, it must be knowable, i.e., (7) ♦K(p & ∼ Kp). Since (7) contradicts (6), we learn that not all truths can be known (or that all truths are known, for those who enjoy the bizarre, half-baked, inane, and philosophically barmy¹). The options for ﬁnding a logical ﬂaw in this proof are quite limited. The only rules of inference it employs beyond those of propositional logic are these: (K-Dist) K(p & q) Kp & Kq (KIT) Kp p and the metalinguistic rule (RN) (p) ⇒ ( p). Of these rules, the ﬁrst is the most likely candidate to be challenged, but Timothy Williamson (1993) has shown how to generate paradox without relying on (K-Dist) at all. Given this result, the only hope for avoiding the paradox is to deny that knowledge implies truth or to deny the rule of necessitation.² In light of the well-entrenched character of these rules, it is not surprising to ﬁnd the literature on the paradox turning in a different direction in attempting to save antirealism from the dark force of the knowability paradox. The dominant ¹ Do not say here: if you understand what we mean by the claim that all truths are known, it is not a bizarre or barmy claim. It’s a sentence of English; we all speak the language; you don’t get to reinterpret into your favored alternative idiolect or dialect. ² This is not to say that other strategies have not been tried. One common attempt is to reinterpret the disappointing results in intuitionistic language, i.e., to hold that, in that language, it is not so bad a thing to have to deny that there are unknown truths. I shall not comment here on this strategy beyond pointing out how philosophically strained it is.

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strategy has become to deny that the antirealist commitment to the epistemic character of truth involves any commitment at all to the claim that all truths are knowable. Instead, the heart of antirealism is to be found in some weaker claim. Dorothy Edgington (1985) proposes that antirealists need only hold that all actual truths are knowable. Michael Dummett (2001) insists that knowability is required only for basic statements, and Michael Hand (2003) provides a sophisticated defense of a similar point of view. Cesare Cozzo (1994) develops an alternative to the knowability claim in terms of idealized arguments, and Neil Tennant (1997: ch. 8) argues that the only truths that must be knowable are those for which the assumption that they are known is logically consistent, leading to a cottage industry regarding whether his approach is a complete non-starter.³ The body of literature pursuing such restriction strategies—strategies for coping with the paradox that deny that the idea that truth is epistemic commits the antirealist to the claim that all truths are knowable—comprises an enormous percentage of the writing on this subject, and has become the favored approach among antirealists for disarming the paradox. When one takes into account the additional literature aimed at undermining such strategies, the dominant issue displayed by recent literature on the knowability paradox is whether any such restriction strategy can successfully disarm the paradox. Nowhere in this body of literature is the strategy itself questioned. Instead, the questions are two: is the strategy faithful to, and theoretically sustained by, the antirealist commitment to the epistemic character of truth, and does the restriction yield a claim that avoids Fitch’s result? We do not need answers to these questions, however, if such approaches to the paradox are red herrings, and that is what I claim here. To defend this point, I will ﬁrst explain a different approach to the paradox that is clearly a red herring. In the process, we will learn more about the heart of the knowability paradox, enough to see clearly that restriction strategies are simply lessons in false hope.

T h e i s m a n d K n ow a b i l i t y The knowability paradox is typically thought of as deriving from two assumptions. The ﬁrst is that all truths are knowable and the second is that some truths are unknown. Upon generating a contradiction from these two assumptions, we are required to discharge, leaving us to conclude that the knowability claim is false (and antirealism thereby threatened). In the presentation given above, however, a different characterization of the paradox might be given. The presentation above could be characterized as a proof ³ See, e.g., Hand and Kvanvig (1999); Williamson (2000a); DeVidi and Kenyon (2003). Replies and further discussion can be found in Tennant (2001a), (2001b), and (2002); and Brogaard and Salerno (2002).

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that there have to be unknown truths, for it begins by assuming that a particular truth is known (the truth that p is an unknown truth), and derives from that claim the impossibility of knowing this particular truth. For philosophers of a theistic bent, this characterization of the paradox may disturb, since it threatens the idea that there is an omniscient being. Given such a disturbance, theistic philosophers may see themselves as having a strong reason to ﬁnd some ﬂaw in the proof, hoping thereby to prevent the knowability paradox from refuting their theistic perspective. Such a response, however, is confused. After more sober reﬂection, the theistic philosopher may see the ﬂaw in this reaction to the paradox. The theistic philosopher may come to see that the above proof is no threat to theism unless it is true that there is some unknown truth, for, if there is no unknown truth, then omniscience does not require that it be known that there is some claim that is both true and unknown. Yet, if there is an omniscient being, then there aren’t any unknown truths! Hence, if one thinks there is an omniscient being, the above proof can be dismissed as a challenge to that viewpoint. It is no more interesting a challenge to theism than any argument that presumes an omniscient being must know what is false. So the theistic philosopher can move on to other interesting areas of philosophy, knowing that the knowability paradox is of no concern. So characterized, the supposed reaction by the theistic philosopher is both right and wrong. It is right in that the proof above does not threaten the claim of omniscience, but it is wrong in supposing that nothing paradoxical remains about which the theistic philosopher need be concerned. One way to see this point is to notice that the paradox does not depend simply on whether one accepts the two assumptions in question, the knowability assumption and the non-omniscience assumption. The central perplexity involved in the paradox does not depend on the idiosyncrasies of one’s favored philosophy, but rather on a perplexing lost logical distinction between what is actually the case and what might be case. It is obvious that knowledge implies the possibility of such, but what is not obvious is what the Fitch proof attempts to demonstrate: that, to put it carelessly, possible knowledge implies actual knowledge. Should that distinction disappear, it would be ﬁtting to ﬁnd ourselves in a state of perplexing philosophical stupor. How could it be that there is no logical distinction between actuality and possibility in this way? We might try for equilibrium by reminding ourselves that there are philosophical domains in which the distinction between actuality and possibility is lost. For example, modal logicians have long been comfortable with the idea that what is actually necessary is not logically distinct from what is possibly necessary. The comfort experienced by this thought will not last long, however. We are comfortable with the lost distinction in this domain because we have a semantical theory to which to appeal to explain why there is no logical distinction here, and we became comfortable with denying the distinction here only after the development of the semantical theory that makes intelligible the loss of such a

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distinction.⁴ Merely formulating S5 systems with the distinctive axiom central to the proof that eliminates the logical distinction between actual and possible necessity is not enough. As the history of the philosophy of modal logic shows, it took a semantical explanation to motivate the contemporary orthodoxy in favor of S5. Proof rules for the modal operators do not, by themselves, yield the degree of understanding necessary to rid us of the philosophical puzzle; only something more, such as is provided by Kripke semantics for quantiﬁed modal logic will help.⁵ Nothing similar can be said when we return to the context of the knowability paradox, however: we have no semantical basis whatsoever for being sanguine about a lost distinction between actual and possible knowledge. So, even if our theistic philosopher should dissent from the assumption that there are unknown truths, said philosopher has as much reason as anyone to view Fitch’s proof as establishing a very troubling conclusion. There is nothing about theism that yields an explanation as to why actual and possible knowledge are not logically distinct. It is for this reason that the theistic response to the paradox is a red herring, even though such a philosopher can take refuge in holding that theism itself is not at stake. The imagined theistic philosopher denies that there are unknown truths, and thereby achieves serenity in the face of the paradox. Such serenity is warranted, however, only if the theistic perspective does more. It will need to explain why, to speak again in the loose and popular vernacular, there is no logical distinction between actual and possible knowledge.

A n t i re a l i s m a n d K n ow a b i l i t y Antirealists, I maintain, do something similar to what the imagined theist has done. The imagined theist denies the assumption of non-omniscience, thereby claiming to avoid any perturbation from the paradox. The now-dominant antirealist strategy is to deny the knowability claim, substituting for it some careful emendation with weaker implications, also thereby claiming to avoid the reach of the paradox. Yet, if the theistic response to the paradox is a red herring, one should wonder why the antirealist restriction strategy isn’t as well. What reason can an antirealist give on behalf of a restriction strategy that will render respectable such a response to the paradox in contrast to the theistic response? Here is what antirealists will need to say in response to the claim above that the heart of the paradox concerns a lost logical distinction between actuality ⁴ Note here how rare it is to ﬁnd a defender of S5 prior to a development of the semantics in question by Kripke, and the corresponding paucity of deniers of S5 after this development. ⁵ There is a way that the remarks in the text are a bit misleading, for it is not the mere fact of having a formal semantics that does the trick here. What is important is that the pure semantics connect up with ordinary meaning, which then gives us the explanation we seek. For more on the distinction between these two kinds of semantics, see Alvin Plantinga’s (1979) distinction between pure and depraved semantics.

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and possibility. They can insist on more precision; they can say that we need to speak with the sophisticates rather than the vulgar. The lost distinction is not one between actual and possible knowledge, for even false (contingent) claims are objects of possible knowledge (in worlds where they are true). Fair enough; so let’s try to be more careful. The antirealist is likely to put the careful point this way: the lost distinction is a lost distinction between actual known truths and possible known truths. That is, a careful presentation of the lost distinction is: (LD) ∀p( (p & ♦Kp) ⇔ (p & Kp) ). The proof from p & Kp to p & ♦Kp is trivial, depending only on the modal principle that what is actual is possible. So, the antirealist can claim, the heart of the paradox is found in demonstrating that p & Kp follows from p & ♦Kp. That proof, however, requires assuming that all truths are knowable.⁶ So (LD), the careful expression of the heart of the knowability paradox in terms of a lost logical distinction between actuality and possibility, is derivable only on the assumption that all truths are knowable. Hence, a perfectly respectable strategy in responding to the paradox is to weaken the knowability assumption in such a way that (LD) can no longer be derived. So long as the resulting restriction still expresses the idea that truth is epistemic, the antirealist has bested the theist above, for not only can the antirealist claim that their position is not undermined by the paradox but also that the paradox has been disarmed since the lost distinction at the heart of the paradox follows only by assuming a claim that is false and necessarily so. In case we needed reminding, we might be reminded as well that strange consequences often follow from necessarily false assumptions. It is time for a philosophical lament here, however. This response to the theist analogy is valuable because of its demand for precision regarding the lost logical distinction at the heart of the paradox. It is mistaken, however, in claiming that (LD) is the proper formulation of that distinction. The paradox is generated from two assumptions, the assumption that all truths are knowable and the assumption that some truths are not known. The proof from the latter assumption to the former is trivial; the proof from the former to the latter is just Fitch’s proof. Given these two proofs, the obvious formalization of the lost distinction is not (LD) but (LD∗ ) ∀p(p → ♦Kp) ⇔ ∀p(p → Kp). Whereas (LD) is not a theorem, but instead can be proven only by assuming that all truth are knowable, (LD∗ ) is a theorem so long as Fitch’s proof is valid. I will express (LD∗ ) in ordinary English by saying that there is no logical distinction between universally knowable truth and universally known truth. This more ⁶ Once we get to ∼ ♦K(p & ∼ Kp), as above, we can only get a contradiction by noting that p & ∼Kp is true and hence knowable by the knowability principle.

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careful articulation still codiﬁes a lost logical distinction between actuality and possibility with respect to what is known. With this more careful formulation, the analogy with the theist above is restored. The theist is both right and wrong in being undisturbed by the paradox, and the antirealist is in the same boat. If the idea that truth is epistemic doesn’t require that all truths are knowable, then the paradox does not threaten to undermine antirealism any more than it threatens to undermine theism. Neither theism nor antirealism of this restricted variety has anything to say that is relevant to the paradoxicality engendered by Fitch’s proof. Each view denies a different assumption in Fitch’s proof, but the paradoxicality involved in (LD∗ ) depends in no way whatsoever on the truth of the assumptions used to generate the paradox. Antirealists may still ﬁnd comfort in undermining (LD) by pursuing a restriction strategy, but they should not pretend that undermining (LD) solves the paradox. Im p l i c a t i o n s Prior to encountering the literature on the topic, the taking of (LD) as the proper careful articulation of the threat of the paradox should strike one as surprising. The obvious careful articulation is (LD∗ )—after all, Fitch’s proof is a derivation of the left side of (LD∗ ) from its right side. There is a larger point to note as well. Critics of antirealism, such as Williamson, view the paradox as a refutation of (most versions of ) antirealism, with Fitch’s proof simply a display of a surprising logical result to this effect.⁷ Such approaches to the paradox, however, leave the paradoxicality in question unresolved. What is paradoxical here is not that Fitch has discovered a proof that threatens antirealism, but rather that Fitch has discovered a proof that threatens a logical distinction between actuality and possibility. One way to put this point is to notice that the omniscience-like claim, though not likely to be thought true (especially when we envision the quantiﬁers restricted to ﬁnite minds), is not obviously impossible. Contrast this point with the fact that the knowability claim, if true, is supposed to be necessarily true: it is a purported implication of a proper understanding of the nature of truth. Yet (LD∗ ) claims that the two are logically equivalent, which they cannot be without having the same modal status. A satisfactory response to the paradox cannot simply swallow this result without explanation. We have already seen one example of a satisfying response to a similar situation, where we have a semantic explanation of the lost logical distinction between actual and possible necessity. If we could have the same here, we’d have a solution to the paradox. It may be there is some alternative ⁷ A point he made most recently to me at the Modalism and Mentalism conference in Copenhagen at the end of January 2004, and most clearly made in print in Williamson (1987) and (1993).

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explanation as well that is not semantic in character, but it is hard to see at this point what such a weaker explanation might look like. I want to consider the issue of whether there might be a non-semantical explanation of the lost distinction in a moment, but I ﬁrst want to emphasize the paradoxicality of the lost distinction by contrasting it with results that are merely surprising but not paradoxical to prevent some deﬂationary approach to the paradox that claims that the result is merely surprising. Consider, for example, Gödel’s incompleteness results. These results are surprising, and threaten important philosophical perspectives, such as Hilbert’s formalism. These results themselves are not paradoxical, however. They present no challenge to anything like the edicts of common sense or the viewpoint of received opinion. That makes these results quite different from a lost logical distinction between actuality and possibility. One might disagree here with my characterization of the Gödel results, arguing that they involve real paradox, but that point can be granted without implying that there is nothing paradoxical resulting from Fitch’s proof. If I’m wrong that the Gödel results do more that threaten important philosophical perspectives, then they may be paradoxical; but if so, they join the class of things already including the results of Fitch’s proof rather than showing that Fitch’s proof is merely surprising. Consider for another example Vann McGee’s (1985) apparent counterexample to modus ponens. The result of McGee’s arguments is not merely surprising, but paradoxical. Modus ponens is so well-entrenched a part of our ordinary view of things that our reaction to his arguments is that they must contain a mistake. Suppose, however, that we are wrong. If we are wrong, and McGee is right, some explanation is in order. We need to know how it could be that our ordinary view of things could be so mistaken. It is worth noting that McGee attempts just such an explanation: logical rules should be thought of as more akin to generalizations and lawlike statements in science which can be useful and instructive even if not always completely accurate. My intention in citing McGee’s explanation is not to endorse it, nor to endorse his arguments that modus ponens is not an exceptionless logical rule.⁸ The point is only that when a proof conﬂicts with ordinary understandings, a further explanatory burden must be shouldered. So it is not enough simply to accept the surprising character of Fitch’s result. One must also shoulder the ⁸ My own view of the matter is that it is preferable to abandon importation/exportation in response to his arguments. If his example is put in counterfactual form, this response becomes obvious: to say that if Reagan were to lose, then if Anderson were to lose, Carter would win, is to say something false; whereas to say that if both Reagan and Anderson were to lose, Carter would win, is to say something true. Only the former, however, is of any use to McGee’s argument. McGee’s argument, of course, involves indicative conditionals rather than counterfactual ones. As a result, more argument is needed to get around his claims. The extra arguments needed, I believe, involve refusing to adopt the assertibility condition semantics he employs, but since that is beyond the scope of the present essay, I will leave that topic for another time and place.

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philosophical burden of explaining how the proof could be correct since it implies a lost distinction between actuality and possibility. Even worse would be to respond as follows. ‘‘We’ve carefully considered the rules involved in the paradox, the rules of (K-Dist) and (KIT), plus the metalinguistic rule of (RN) are so inherently plausible that the conﬂict they create with the intuitive logical distinction between possibility and actuality is not paradoxical at all. The results are surprising and unanticipated, but not paradoxical.’’⁹ Someone is living in logical denial. No argument can conclusively show that this approach is mistaken, since the difference between what is paradoxical and what is merely surprising is, perhaps, only a difference in degree and not in kind. Even so, there is a distinction to be drawn here between the unanticipated and the seemingly contradictory, and Fitch’s proof engenders the latter experience and not simply the former. It is not merely surprising when we are told that what looks like a modal truth is logically equivalent to what looks like a non-modal truth (especially when, given ordinary assumptions, the ﬁrst would apparently be necessarily true if true at all and the second would apparently not be). We can’t simply afﬁrm the rules, and say, ‘‘I guess we were wrong; non-omniscience really is impossible.’’ That’s simply not an adequate explanation of what’s gone wrong; more accurately, it is not an explanation at all. As we have seen, the paradigm example of a satisfying explanation in this regard is a semantical one. Moreover, as already noted as well, a purely syntactic explanation in terms of axioms of a system and proof rules for it, is particularly unsatisfying. Suppose, for example, that in the modal domain, we showed multiple contexts in which the introduction of a possibility operator on a formula yielded nothing logically distinct from the original formula. Such is the case, if S5 is to be believed, for logical necessities. It is also true of other modal systems, however. For example, obligation statements and possible obligation statements cannot be distinguished logically in certain deontic systems. We learn to accept such results, if we do, by being told a semantic story. In the deontic case, if we interpret obligation statements in terms of ideal worlds, worlds where everything is done properly, then we can see actual and possible obligations collapse. An actual obligation statement takes us to what is true in an ideal world, and a possible obligation statement takes us to another world where the obligation statement is interpreted with reference to ideal worlds. On the assumption that all worlds are accessible from every world, the class of ideal worlds will be the same, whether accessed from our world or some other possible world, generating a logical equivalence between possible obligation and actual obligation. If we accept this semantical story, we understand the lost distinction here. The important point to note is that understanding is not achieved merely by multiplying contexts ⁹ This response is in the spirit of Williamson’s approach to the paradox both in print and in conversation, as well as that of Carrie Jenkins’s contribution to this volume.

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in which there is a similar lost distinction. All such multiplication would do is to replace a rather speciﬁc paradoxicality with a more general one. What is needed is some understanding of why an apparent logical distinction is lost, and, without such understanding, we cannot say ‘‘paradox lost.’’ This approach to the paradox in terms of trying to accustom us to the loss by generalizing on the syntactic features involved in the paradox has a history going back to J. L. Mackie’s (1980) early paper on the paradox. Seeing the failure of the strategy in other contexts should make us suspicious here, and it is worth taking a look at the details to see why this suspicion is correct. Here’s an attempt along these lines.¹⁰ Consider the operator ‘‘it is written on my blackboard that’’ and the operator ‘‘it is true that,’’ and the idea that anything true might be written correctly on my blackboard. If we call these operators W and T respectively, we won’t be able to get an analogue of the knowability paradox out of them, since only the latter is factive. To get a paradox, we’d have to generate a contradiction from this assumption: WT(p & ∼WTp). From this formula, we can get WTp & WT ∼ WTp, since, we assume for now, both operators distribute over conjunction. We also assume that the operators can be split by the following rule: WTp Wp & Tp. Using this rule, we can get T ∼ WTp from the second conjunct (and &-Elim), and then get ∼WTp from this formula given the factive character of truth: Tp p. We may also wish to put the two operators together into a single operator WT . Intuitively, this operator is supposed to mean something like ‘‘written truthfully on my blackboard,’’ but, formally speaking, the crucial idea is that WT is both distributive and factive—that is, it borrows distributivity from the W operator and factivity from the T operator. Because it is both factive and distributive, we can generate the analogue of the contradiction crucial to the knowability paradox a bit more quickly: WT (p & ∼ WT p) WT p & WT ∼ WT p (by distribution) WT p & ∼ WT p (by factivity) ¹⁰ I owe a great deal to Michael Hand regarding this approach. In fact, I think it fair to say that I simply would not have seen the possibility or signiﬁcance of proof-theoretic insight without the long discussions we have had together.

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So it appears that not everything true can be correctly written on my blackboard, and thus that WT is an analogue of the K operator. Both are distributive and factive, and hence the knowability paradox is but a special case of a more general phenomenon: ﬁnd any distributive and factive operator, and the crucial contradiction in the knowability paradox will follow. There are some niggling problems with this attempt at generalizing so as to provide a syntactic explanation of the lost logical distinction that constitutes the heart of the knowability paradox. These problems are not my fundamental reasons for rejecting this strategy, but they place important limitations on any attempt to pursue this generalization strategy. The ﬁrst problem to note is that neither of the ways above of demonstrating a contradiction is quite adequate. To see the problem, let us generalize here beyond W and T and the combined operator WT and think in terms of p and p, for any operators and . The idea above is to allow the combined operator to inherit the preferred formal features of the individual operators themselves, but there is no guarantee that this result can be achieved. Suppose we construct a new operator that is a combination of knowledge (‘‘it is known by someone at some time that’’) and necessity (‘‘it is necessary that’’). Call this the ‘‘known-to-be-necessary’’ operator. Assume as well that the combined operator inherits formal features from the individual operators of which it is a combination. Since &-I works within the context of necessity and since knowledge implies belief, we can infer from the governing of p and the governing of q by this new operator that the conjunction of the two is believed by someone at some time. This inference is faulty, however, since the ﬁrst could be known by someone and the second known by someone, but the conjunction believed by no one. For another example, combine obligation-for-everyone and knowledge-bysomeone, and you get knowingly obligatory (known-by-someone-to-be-obligatory-for-everyone). Being obligatory preserves &-I, so if both p and q are knowingly obligatory, then someone believes both p and q (because knowledge implies belief ). This inference is obviously absurd, however. The lesson here is that one can’t combine operators and expect to be able to apply the usual rules for either of the individual operators that were put together to form the combined operator. Notice further in the example about writing truly that if we keep the operators separate, we can’t prove the contradiction. Consider how to try. We begin by representing the claim that a speciﬁc truth is not truthfully written on my blackboard as: Tp & ∼(Wp & Tp). Then the reductio assumption will have to be: W(Tp & ∼(Wp & Tp)) & T(Tp & ∼(Wp & Tp)).

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If we then distribute the W and T operators, respectively, we get: WTp & W ∼ (Wp & Tp) & TTp & T ∼ (Wp & Tp). Since truth is factive, we get from the latter two conjuncts: Tp & ∼(Wp & Tp), from which we can get ∼Wp. This point is somehow supposed to contradict the ﬁrst conjunct WTp, but since we can’t apply the factivity claim about truth inside the W operator, we can’t demonstrate the contradiction. We can avoid this problem by eliminating the truth operator in our representation: p & ∼(Wp & p). The assumption for reductio can then be W(p & ∼(Wp & p)) & (p & ∼(Wp & p)). From this claim we derive by the distributivity of W: Wp & W ∼ (Wp & p) & p & ∼(Wp & p). The latter two conjuncts give us ∼Wp, which contradicts the ﬁrst conjunct of this formula. There are two points to note about this derivation. First, there is no factive operator in this proof. Even so, it is reminiscent of Fitch’s proof since we represent the claim that p is an unwritten truth as the claim that ∼Wp & p, just as we represented the idea that p is an unknown truth in Fitch’s proof as the claim that ∼Kp & p. To claim, however, that we have an analogue of Fitch’s proof that mirrors its dependence on an operator that is factive and distributive is mistaken. The only operator in this representation is the W operator, and it is not factive. This point is not important when we are looking for non-syntactic generalizations of the knowability paradox, for the above proof is reminiscent enough of Fitch’s proof that any explanation of the lost distinction engendered by Fitch’s proof will shed light on this proof as well. The syntactic strategy, however, hopes that mere duplication of syntactic form will relieve our perplexity, and there is no such duplication here. The second point to note will take a bit longer to develop, but it too casts doubt on the idea that we have a syntactic analogue of knowability here. Any syntactic explanatory value for the above derivation depends on the interpretation

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assumed of the operator in question, since no one should have an issue with the idea that there are distributive operators ψ for which the formula (p & ∼p) implies a contradiction. Neither should anyone think that the existence of such operators appropriately addresses the knowability paradox. Instead, if any help is found in syntactic mimicry here, it comes from the assumed interpretation of the W operator in terms of what is written. Once we begin thinking carefully about the concept of writing, however, problems appear. To see them, let’s think ﬁrst about asserting. To assert a claim is not just to utter a sequence of phonemes that conventionally expresses the proposition in question. Instead, to assert is to express that very proposition. When speaking only of the sequence of phonemes and the associated noises, I will term the act in question an act of uttering; when the proposition itself is expressed by such uttering, I will term such an act of asserting. What I mean by the term ‘proposition’ here is simply a bearer of truth-value. Thus, I leave open whether propositions are sentences or abstract objects of some sort, and claim that there is a speech act involving the uttering of phonemes by which bearers of truth-value are expressed—namely, the speech act of asserting. We should note that uttering has formal properties that asserting does not. For one thing, I can’t avoid uttering ‘‘it is raining’’ by uttering ‘‘I believe it is raining,’’ but I do not (always) assert it is raining by asserting I believe it is raining. Furthermore, a string of phonemes is not itself a bearer of truth-value, since propositions have that property exclusively. As a result, the string of phonemes that, when uttered, express a bearer of truth-value is not itself a bearer of truthvalue, but only a vehicle by which a bearer of truth-value is expressed. The lesson is that if we wish to consider operators that are both factive and distributive, we will not be able to appeal to the utterance operator, but only to the assertion operator, since only the latter governs items capable of being true or false. Suppose then that we focus on the assertion operator. Once we notice this difference between uttering and asserting, the claim that asserting has the distributive property is in doubt. When I utter ‘‘p and q’’ I clearly utter ‘‘p’’ and I clearly utter ‘‘q’’. But when I assert p and q, do I assert p? Well, when I assert I believe p, I don’t assert p, so if you think that assertion distributes across conjunction, what’s the difference? Once the question is formulated, the answer is obvious: the difference is that I’ve logically committed myself to p by asserting p and q. So, in order to preserve the distributive character of assertion, we have to take the concept of asserting to include not simply what propositions one expresses by uttering a string of phonemes that make up a simple declarative sentence. We’ll also have to count you as asserting at least some claims to which you are logically committed in virtue of the assertions you make by saying declarative sentences. Which ones? Trying to sort among logical consequences leads to enough of a mess that the road most easily traveled takes us to the point of including all of them.

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That’s a mistake. I can assert that Fermat’s last theorem is unprovable even though everything I assert logically commits me to the truth of that theorem (since it has been proven). Antirealists are wont to appeal to idealizations, so it wouldn’t be surprising to ﬁnd some appealing here to the concept of what is assertible by an ideal rational agent, substituting for the concept of assertion the concept of what a logically omniscient being is committed to in virtue of what s/he asserts. Too much idealizing, methinks. The being would have to be quite unlike us, capable of knowing an uncountably inﬁnite number of things and propositions with uncountably inﬁnite components. If we want to speak of God here, theists like myself will have no problem with the discourse, but to think of such a being in terms of some ﬁnite extension of our own abilities and capacities is intolerable. These same points hold for the concept of what is written. Everything written is inscribed, but sometimes only a string of morphemes is inscribed and sometimes the writing expresses a proposition as well. In my terminology to inscribe a sentence is the scribal form of uttering a string of phonemes, and writing relates to a proposition in scribal form in the way asserting relates in a vocal form to a proposition. As before, we’ll have the same reasons to focus on the concept of writing rather than inscribing, since what is inscribed is not itself a bearer of truth-value; but when we consider the writing operator W, we ﬁnd that it is distributive only if the operator includes a reference to the logical consequences of what is written, and then the operator is not that of writing. For it is one thing to write down a claim, and it is another thing for what one has written to commit one logically to some further claim. This problem about the W operator is not likely to detain the proof-theoretician for long. For one thing, there is no reason we can’t interpret the W operator as ‘‘logically implied by what is written.’’ Such an operator would purportedly show the falsity of the intuitive idea that anything true can be logically implied by something written truthfully on my blackboard. Even so, there are costs to the syntactic generalization strategy. The W operator, on this understanding, is now logically complex, requiring reference to some correct logic for its interpretation. By contrast, in the knowability paradox, the rules of inference are intrinsic to the formal shorthand for the ordinary concept of knowledge. The more complex the operator, the more tempting it is to attribute the perplexing result of the proof to the complexity of the operator and the difﬁculty in processing this complexity. That is, the temptation is to treat it like we do barber sentences (‘‘there is a barber who shaves all and only them who do not shave themselves’’): once we see the implications, we relieve our perplexity by simply reminding ourselves of the logical complexity of the sentence, so that the appearance of possibility is misleading. Interpreting the W operator in this complex way suggests the same kind of response, but it is a response that is not appropriate for the knowability paradox. Given these disparate reactions to two proofs that can be formally represented in the same way, it is not clear that

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sameness of formal representation has any power here to relieve our perplexity at the particular lost logical distinction resulting from Fitch’s proof. This last claim raises a more general point about such a syntactic generalization strategy of ﬁnding analogues of knowability by searching for operators that mimic the distributivity and factivity features upon which Fitch’s proof relies. To see the issue, note that we can generalize the form of Fitch’s proof with a number of operators, such as the ‘‘true belief’’ operator, the ‘‘truly wished for’’ operator, the ‘‘truly imagined’’ operator, the ‘‘truly desired’’ operator, etc. In each case, the existence of a thing not truly X-ed will purportedly be incompatible with the idea that any truth can be truly X-ed. Note here that we don’t need to idealize to some logically omniscient X-er nor do we need to talk of the logical implications of what is X-ed in order to ﬁnd an analogue of the knowability results, so no distinction between the arguments is naturally brought to mind by the degree of complexity of one operator over another. Does this variety of operators somehow explain the paradoxicality of a lost logical distinction between possible and actual universally known truth? I can’t see why. A more plausible response to such generalizing is simply to characterize the more general paradoxicality in question. Instead of saying that there’s a knowability paradox, we’d say instead that there is a paradox about any mental state operator that is factive and distributive. Syntax generalized is just paradox generalized, not paradox lost. To generate paradox lost, my preferred explanation would be semantic. Since I’ve given no argument that no other explanation is possible, it is worth considering what other non-semantic approaches might look like besides the syntactic generalization strategy just rejected. A pragmatic explanation might appeal to the notion of structural interference, claiming that the process of X-ing when applied to a conjunction can cause problems since in X-ing the ﬁrst conjunct, one may affect the truth-value of the second.¹¹ Well, not quite, since the claim in question is an eternal truth if a truth at all (it quantiﬁes over all individuals who X and all times), so a more careful claim would be that the X-ing of the ﬁrst conjunct entails the falsity of the second. These claims may be correct, but I have some questions about it. In order to carry the explanation through, one will have to distinguish, in the case of the knowability paradox, between the executability of the basic steps of a procedure for generating knowledge (say, for knowing a conjunction) and the executability of the entire procedure itself, holding that only the basic steps are required to be executable, leaving unanswered why the basic steps are assumed to be executable.¹² Surely more is required here than simply assuming that these steps are executable. ¹¹ For the latter perspective on the paradox, Michael Hand (2003). ¹² See Hand (2003). Hand uses the analogy of recursion theory to show how a formal system can be developed by informal reference to procedures that might be executed, when the formal construction itself should not be thought to require such. Carrying the analogy through would lead to the conclusion that not even the basic steps of a veriﬁcation procedure need to be thought of as executable.

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This point doesn’t by itself cast doubt on this pragmatic approach to the paradox, however, so let’s assume that this problem can be overcome. Deeper problems can be seen, though, if we simply attend to what the account is good at explaining and what it is not. Put pithily, it is good at explaining why antirealists shouldn’t say that all truths are knowable, and it is not good at explaining the lost logical distinction expressed by (LD∗ ). It is clearly designed for the former purpose. Given this approach, one should deny that all truths are knowable by noting that those with antirealist sympathies should only go so far as to say that all truths for which structural interference is not an issue are knowable. To go further is to risk refutation by Fitch’s proof. The notion of structural interference is not designed to explain the lost logical distinction at the heart of the knowability paradox, however, and it is not very successful when turned to that purpose. To see the problem, I want to compare the omniscience-like conclusion of Fitch’s proof with a more famous claim of the same sort, the claim that God exists. One’s intuitive, pre-philosophical attitude toward this claim should be that it is a contingent matter whether there is a God (just as it should be contingent whether all truths are known). There is a plausible path of reasoning to the denial of the contingency claim, however. It begins by claiming that God is the most perfect being, that He exempliﬁes maximal greatness. We thus identify the claim that God exists with the claim that maximal greatness is exempliﬁed. The ﬁnal steps toward a denial of the contingency assumption about God’s existence is to clarify what maximal greatness involves (it is to display the maximal amount of any great-making property that has an intrinsic maximum) and argue that modal stability is itself a great-making property whose intrinsic maximum is existence in all possible worlds. There are two quite natural responses to this threat to the contingency of the theistic claim. The ﬁrst is to question the proof itself, to question the implications of the concept of maximal greatness, especially to doubt whether maximal modal stability is itself a great-making property. In doing so, one may look for analogues of the property, or one may simply construct formal notions that are claimed to have the modal stability property of being necessarily instantiated if instantiated. This strategy has a long history of threatening the ontological argument, from Gaunilo’s perfect island to Arnauld’s existent lion. This approach is like the syntactic generalization strategy rejected earlier. It is, however, a more promising approach here, since the examples used do not simply mimic the problematic proof, but constitute reductios of it. What they show is that the proof contains a mistake, even if we cannot identify exactly where the mistake occurs, thereby reafﬁrming our intuitive sense of the contingency of the theistic hypothesis. This ﬁrst response to the ontological argument thus could be used only to try to resurrect the syntactic generalization strategy, not to make sense of the pragmatic explanation relying on the notion of structural interference. Since we are now only considering the latter issue, we should move past this ﬁrst response

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to look at the second kind of response. This response questions the account of the theistic claim itself. Why should we think that the claim that God exists is logically equivalent to the claim that maximal greatness is exempliﬁed? After all, it’s not as if the meaning (sense) of the term ‘God’ is the same as the meaning (sense) of the term ‘maximally great being.’ Seeing what defenders of the argument do at this point shows why the pragmatic approach to the paradox is unsatisfying. Defenders of the ontological argument sometimes simply stipulate an understanding of ‘God’ in terms of maximal greatness. Such an approach leaves untouched the intuitive sense of the contingency of the theistic hypothesis, and thus provides no useful model for the pragmatic approach to the knowability paradox to emulate. What is needed instead is some way of explaining away some apparent contingency. An alternative to this stipulation approach to the issue tries to argue against the contingency claim. The argument takes the form of a reductio of the denial of contingency, beginning with the supposition that there is a God and also that there is a distinct being greater than God. The argument then proceeds by asking what understanding of God one might have that would call for allegiance, or worship, or religious commitment to the lesser being. This line of argument could be resisted by insisting that a proper conception of God has no religious signiﬁcance whatsoever,¹³ but that escape route will strike most as fairly extreme. My point here, however, is not to defend this approach, but to show it can be used to explain away the apparent contingency of the theistic hypothesis. The apparent contingency is resisted by pointing out that we think in these terms because there is nothing about the meaning or sense of the claim that God exists that yields a denial of contingency, and yet there is an argument for this conclusion. Put in the language of the analytic/synthetic distinction, the claim is not necessary because analytic, but it is necessary nonetheless. The important point to note is that the argument for this claim is not simply the original argument for the necessity of a maximally great being. Applying this point to the context of the knowability claim, we can see that the pragmatic approach to Fitch’s proof in terms of structural interference is not plausibly taken at all as providing an analogue of this way of defending the necessity of the claim that God exists. To function in the same way, the pragmatic approach would need to provide a basis to argue, independently of Fitch’s proof, that the omniscience-like conclusion of that proof is necessarily false. Even the most superﬁcial understanding of this approach shows that it would be complete pretense to assert that it can provide such an argument. Once we appreciate the design plan of the structural interference approach, we can see why that approach seems so irrelevant to the lost logical distinction in question. The reason is that it wasn’t designed to answer the question of how ¹³ See, e.g., Richard Taylor (1982). Taylor endorses arguments for the existence of God, but takes this result to have no religious signiﬁcance at all.

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there could be such a loss. It was designed to answer the question of whether an epistemic conception of truth requires afﬁrming that all truths are knowable. Looked at from this perspective, it is a powerfully promising idea. It holds out the promise of being able to explain why ‘‘I don’t exist’’ can’t be veriﬁed even though it might be true, and how ‘‘no thinkers exist’’ can’t be conﬁrmed even though possible. Such conclusions are essential to an adequate defense of semantic antirealism, and it is a mark in favor of the philosophical fecundity of the idea of structural interference that it blocks these problems for antirealism while at the same time showing why Fitch’s proof fails to refute the view. The lesson to learn here is that it is sometimes best to let tools be used for what they were intended, rather than to try to force them to accomplish a job for which they were not intended. So independently of any use to which antirealists might put the concept of structural interference, we ﬁrst need a solution to the paradox itself.

Conclusion In short, the paradox should disturb us all, antirealists and realists alike. It is true that the difference between a paradox and a merely surprising logical result is often not a difference in kind but only a difference in degree. Even so, there are distinctive marks of each that we look for when assessing what kind of a result we have achieved. If, for example, the result is merely one that we had no reason to think was true, we should classify such a result as a surprise. Or, again, if the result is merely one that threatens a particular philosophical perspective, such as antirealism, we should still classify the result as merely surprising. But when the result threatens some aspect of received opinion, especially received opinion on logical matters themselves, we should not classify the result as merely surprising. In the present case, the lost logical distinction is part of a ﬁrmly entrenched understanding of the nature of the modalities of necessity, possibility, and actuality. It is not a partisan distinction that only certain philosophical perspectives could endorse, and in this way, it is paradoxical to face a derivation that undermines the distinction, in the same way it is paradoxical to be told that two grains of sand constitute a heap or that motion is impossible. The perplexity engendered by Fitch’s proof is paradoxical, and the paradox cannot be addressed either by embracing Fitch’s proof as a refutation of antirealism or by ﬁnding a version of antirealism that involves no commitment to the knowability claim itself. What we need is either an explanation of the failure of Fitch’s proof or an explanation of the lost logical distinction between actuality and possibility that it implies—nothing short of that constitutes a proper philosophical response to the paradox.

14 Revamping the Restriction Strategy Neil Tennant

Ab s t r a c t This study continues the anti-realist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the anti-realist’s knowability principle ‘ϕ , therefore ♦K ϕ ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of ϕ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before applying the proposed restriction. A similar restriction is proposed for the converse inference, the so-called Rule of Factiveness ‘♦K ϕ therefore ϕ ’. The proposed restriction appears to block another Fitch-style derivation that uses the KK -thesis in order to get around the Cartesian restriction on applications of the knowability principle. Korean saying: Joong-i je meo-ri mot kkak-neun-da. Translation: A (buddhist) monk cannot shave his own hair.¹

1 . In t ro d u c t i o n In The Taming of The True a restriction was proposed on the anti-realist’s Knowability Principle, which can be expressed as a rule of inference in natural deduction as follows: ϕ ♦K ϕ This paper would not have been written without the stimulation, encouragement and criticism that I have enjoyed from Joseph Salerno, Salvatore Florio, Christina Moisa, Nicholaos Jones, and Patrick Reeder. ¹ Thanks to Sukjae Lee for the motto.

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It was proposed that this principle be limited, in its applications, to Cartesian propositions ϕ. A proposition ϕ is Cartesian just in case K ϕ ⊥. So the restricted Knowability Principle would be ϕ where K ϕ ⊥ ♦K ϕ This way of restricting the Knowability Principle may well be suspected of being overly ‘local.’ It might be advisable to have a more ‘global’ restriction. In general, a step of inference from ϕ to ♦K ϕ (with or without an extra condition on ϕ) takes place within a proof which will have some set undischarged assumptions: ϕ ♦K ϕ If we limit ourselves to the ‘local’ restriction, we ignore the contribution of the set of assumptions, focusing instead on the fact that, via , we have just reached the conclusion ϕ, whatever our starting point might have been: ϕ ♦K ϕ

where K ϕ ⊥

But if our grounds for ϕ are indeed , then the inferred possibility of knowing that ϕ surely presupposes the possibility of knowing that . Indeed, if it were impossible to know the joint truth of the assumptions in , how could one be conﬁdent in inferring from the intermediate conclusion ϕ to the knowability claim ♦K ϕ? These considerations lead to the thought that the restriction strategy, instead of looking down at ϕ within should rather look up at . The proposal, then, is that the restricted Knowability Principle should take the form of the following rule of inference, with a rather more exigent pre-condition for its applicability: Globally Restricted Knowability Principle where K ⊥ ϕ ♦K ϕ Here K is deﬁned in the usual Frobenian way as {K δ|δ ∈ }. When K ⊥, we shall say that is Cartesian. In logics whose relation of deducibility is not effectively decidable, the correctness of applications of the globally restricted rule is accordingly not effectively decidable. This is a modest price to pay, however, when one is concerned to avoid Fitch’s paradox.

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The main purpose of this study is to explain, investigate and defend this new proposed global restriction on applications of the Knowability Principle. A similar proposal will be made concerning its converse, the Rule of Factiveness for the compound operator ♦K : ♦K ϕ ϕ As will become clear from details that will emerge below, we need to restrict Factiveness too, in order to avert a different proof of Fitch’s paradox, which exploits the KK -thesis but does not fall foul of the global restriction on its application of the Knowability Principle.

1.1. A retraction, for the record The present author’s claim, made in Tennant (2000), at p. 829 and repeated in Tennant (2002), at p. 140, to the effect that ♦K ϕ is factive, was incautious. While it is valid so long as the sentence ϕ in question concerns only non-epistemic facts, the inference is not guaranteed always to preserve truth if we allow ϕ to contain occurrences of K (and adopt the KK -thesis). A related claim, however, still stands: to the extent that ♦K is factive, ♦ is not to be analyzed as the familiar alethic modal operator. Its contribution to truth- or assertability-conditions of sentences in which it is preﬁxed to K will have to be elucidated in terms of possibilities of investigative outcomes, at future times, within the actual world. Those possibilities will be strongly constrained by relevant contingencies in the actual world. It is this feature of the possibilities adverted to within ♦K that make the use of an ordinary alethic ♦ inappropriate.

1.2. Global restrictions on rules of inference A ‘global’ restriction on a rule of natural deduction is one that imposes some pre-condition for applicability by adverting to syntactic features of the proof other than the forms of the sentences standing as the immediate premises, or as the conclusion, of applications of the rule. As soon as any ‘global’ restriction is proposed on a rule of natural deduction, the possibility arises that its strictures can be rendered toothless by applying other rules of inference in a roundabout fashion that creates an artiﬁcial deductive context that meets the pre-condition in the letter, but not in the spirit, of the proposed restriction. The most obvious way to do this is by constructing proofs that are not in normal form.² Thus the most obvious prophylactic ² Such proofs involve inferring a sentence as the conclusion of an application of an introduction rule, and then treating that sentence as the major premise for an application of the corresponding

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against such deductive chicanery is to insist that, when determining whether the pre-condition is met, the proof that is to result from the contemplated application of the restricted rule should be in normal form. With this said by way of foreshadowing, we shall defer detailed illustrative examples to their most natural points of entry below. 2 . T h e Ne w Re s t r i c t i o n o n K n ow a b i l i t y

2.1. How the new restriction works on Fitch’s original proof We recall the proof of the Fitch paradox as given in Tennant (1997), at pp. 260–1.³ (1) K (j ∧ ¬Kj) ¬K j (∧I ) j ( K ) j ∧ ¬K j ⊥ (1) ( ⊥ ) K (j∧ ¬Kj) ⊥ where the embedded proof is

(I ) K (j ∧ ¬Kj)

j ∧¬Kj ¬Kj ⊥

(1)

(K ∧) K (j∧ ¬Kj) Kj ⊥ (1)

The reader will easily verify that the new, global restriction blocks this proof of Fitch’s paradox at its application of the rule (♦K ). For the premise-set for that step is {ϕ , ¬K ϕ}. Hence K = K {ϕ , ¬K ϕ} = {K ϕ , K ¬K ϕ}. And the latter set is not Cartesian: K ¬K ϕ ¬K ϕ Kϕ ⊥ We see, then, that the global restriction can do the old work required of the local restriction. elimination rule. Such sentence-occurrences within a proof are called maximal, and their presence is what prevents the proof from being in normal form. By contrast, a proof in normal form is one that contains no maximal sentence occurrences. ³ Natural deductions will be set out in tree form below. The reader unfamiliar with this format for proofs is advised that with applications of so-called ‘discharge rules’ the parenthetically enclosed numeral ‘(i)’ has an occurrence labeling the step at which the indicated assumption-occurrences higher up at ‘leaf nodes’ of the sub-proof(s) are discharged by applying the rule in question. A discharged assumption no longer counts among the assumptions on which the conclusion of the newly created proof depends.

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2.2. How the new restriction works on a proof of Salerno The strengthened restriction also does important new work that the local restriction cannot do. The following is a short proof of a result brought to my attention by Joe Salerno (albeit using a different proof).⁴ Consider the following proof of ψ from the set of assumptions {p, ¬Kp}. Note that the application of (♦K ) uses the old, ‘local’, restriction. (2) Kp

¬Kp ⊥ (1) K (1) (2) K ( p ∧(Kp →K )) p K p →K p ∧ (K p →K ) K ( p ∧ (K p →K )) Kp →K Kp K K( p ∧( K p →K )) ⊥ p ∧(Kp →K ) K K( p∧ (K p →K )) (1) K :

Note that the step labeled (1) is an application of the following rule of inference in modal logic: (i)

( )

where

is the sole assumption of the subordinate proof

(i)

Note also that the ﬁnal step of the proof : ♦K ψ ψ is an application of the aforementioned Rule of Factiveness of ♦K . It is worth stressing that its application here would be correct even if it were subject to the restriction that ψ should not contain any occurrence of K . (We shall return to this rule later.) Given the pattern of occurrences of ψ within , we could evidently take any contingent atomic proposition q (not involving K ) and complete a new proof of Fitch’s paradox as follows: {p, ¬Kp} {p, ¬Kp}

[ψ/q] [ψ/¬q] q ¬q ⊥ ⁴ Personal communication. See also Brogaard and Salerno (2006).

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Remember that these two substitution instances of involve the old, ‘local,’ restriction on (♦K ). But those steps of (♦K ) will not go through when we impose the new, ‘global,’ restriction. For at the point where (♦K ) is applied, the subordinate proof of p ∧ (Kp → Kq) (resp., p ∧ (Kp → K ¬q)) has as its set of assumptions the non-Cartesian set {p, ¬Kp}.

2.3. A possible objection to the new restriction A possible objection to the new global restriction is that it is all too easy to comply with. The thought might be that one could still construct a Fitchian reductio by sneaking around the restriction to Cartesian in the Globally Restricted Knowability Principle. The trick would be to successively discharge all the members of a non-empty, non-Cartesian by means of terminal applications of →-introduction within the subordinate proof in the proof-schema below. One would thereby reduce to the empty set the set of assumptions of the resulting subordinate proof for globally restricted (♦K ); in which case the restriction in question—that the set of assumptions be Cartesian—would be trivially met: (1) (2) 1, 2,

(n−1)(n ) , n−1, n

j (1) →j 1 (2) 2 → ( 1 →j)) (n−1) ( 2 →( 1 →j)) ) (n ) )) n → ( n−1 →( ( 2 →( 1→j)) K ( n → ( n−1 → ( ( 2 →( 1→j)) ))) n−1 → (

The objector who takes this line this far will not, however, be able to press it much further. For now we see that any use of ♦K (δn → (δn−1 → (. . . (δ 2 → (δ 1 → ϕ ) ) . . .) ) ) as a premise for a rule that involves stripping away the preﬁx ♦K (or, ﬁrst stripping away ♦, and thereafter stripping away K ) will presumably result in the multiply nested conditional eventually being exposed. Therewith arises a need to assume δ 1 , . . . , δn in order to winkle out ϕ for whatever Fitchian mischief is up the objector’s sleeve—mischief which would have been blocked by the new global restriction before the currently contemplated stratagem of multiple →-introductions. Yet there is no guarantee that these forced extra assumptions δ 1 , . . . , δn will be of the required form (enjoying a dominant occurrence of

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or of K , say) that might be called for in the application of whatever rule(s) might have been used to strip away both ♦ and K .

2.4. The importance of normal forms for the new restriction As foreshadowed earlier, in order for the new global restriction to be effective, we must insist that it be applied only within the context of a proof in normal form. The reason for this is that by resorting to proofs that are not in normal form, one can ‘hide from view’ the full set that provides the genuine grounds for the possible knowledge-claim ϕ. Only in the context of a proof in normal form will all those grounds be displayed as undischarged assumptions on which the premise ϕ (for an application of the Knowability Principle) depends. An example illustrating this point would be the following, which I owe to Salvatore Florio (Florio, unpublished). It exploits the rule called (λ) in Tennant (2000), at p. 837:

K, Kj ( )

Kj

⊥

⊥

(i)

(i)

The proof using (λ) is as follows. (2) p∧¬Kp Kp ¬Kp (3) K (p∧(Kp→¬Kq)) ⊥ (3) (2) ¬Kq p∧¬Kp p∧(Kp →¬Kq) K (p∧(Kp→¬Kq)) (1) p Kp→¬Kq (4) Kp→¬Kq Kp Kq ¬Kq p∧(Kp →¬Kq) p∧(Kp→¬Kq) ∃j(j∧¬Kj) ∃j (j∧(Kj →¬Kq)) K(p∧(Kp →¬Kq)) ⊥ (2) (3) ( ) ∃j(j∧(Kj →¬Kq)) ⊥ (4) ⊥ (1)

This proof, absurdly, reduces an arbitrary proposition of the form Kq to absurdity, on the basis of there being an unknown truth. The proof, however, is not in normal form, for the major premise of the ﬁnal step of ∃-elimination had been derived two steps earlier as the conclusion of a step of ∃-introduction. By means of the resulting abnormality, this proof is able to harbor an application of the Knowability Principle without, apparently, violating the condition that the ultimate premises for that application should be Cartesian. This can be seen by normalizing the foregoing proof. We do so by applying the appropriate reduction procedure in order to get rid of the maximal occurrence of the sentence ∃ϕ (ϕ ∧ (K ϕ → ¬Kq) ):

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(3) p ∧ ¬Kp Kp ¬Kp (2) ⊥ K (p ∧(Kp → ¬Kq)) (2) (3) ¬Kp K (p ∧(Kp→¬Kq)) p∧(Kp →¬Kq) p ∧¬Kp (1) Kp→¬Kq Kp p Kp→¬Kq Kq ¬Kq (†) p∧(Kp →¬Kq) K(p∧(Kp →¬Kq)) ⊥ (2) ( ) ∃j(j ∧ ¬Kj) ⊥ (3) ⊥ (1)

We ﬁnd that in the normalized proof the Knowability Principle is, after all, being incorrectly applied at the step marked ( † ); for its ultimate premise (at this application) is p ∧ ¬Kp. And this, as we already know, is not Cartesian. It took conversion into normal form to detect this violation of the global restriction on the Knowability Principle. 3 . Re s t r i c t i n g t h e Ru l e o f Fa c t i ve n e s s It was pointed out earlier that the Rule of Factiveness ♦K ψ ψ could be restricted in its applications to sentences ψ that do not contain any occurrences of K . Call the Rule of Factiveness of ♦K , restricted in this way, F . The restriction in question is very well motivated for the anti-realist who is also an internalist about knowledge. For if, as some internalists do, one subscribes to the so-called KK -thesis: Kψ KK ψ an unrestricted Rule of Factiveness U F will instate the unwanted Fitch inference, even for Cartesian propositions ψ. For consider the following proof that ψ implies K ψ.⁵ ( K) :

K KK K KK K

(1) (KK ) (1) ( )

⁵ Compare Brogaard and Salerno’s proof of the KK-knowability paradox in Brogaard and Salerno (2002).

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Given this proof, one will be able, disastrously, to inﬂate possibility to actuality (for Cartesian propositions ψ),⁶ by means of the following proof . (2)

(2) Π:

K K

(2) ( )

( K) , i.e.

(1) K (K K ) K K K (1) ( ) KK K (2) ( ) K

The way to avoid this madness, without losing a proper grip on knowability, is to refuse to grant the unrestricted factiveness of ♦K . It should be possible to hold the KK -thesis without making every truth known (via ), and without inﬂating the possible truth of a Cartesian proposition to its actual truth (via ). Suppose one holds that a knower’s knowing is always reﬂectively accessible to the knower, so that if x knows that ϕ, then x knows that x knows that ϕ. It follows that if it is known that ϕ, then it is known that it is known that ϕ. That is, the KK -thesis holds. Suppose now that ψ is true. The anti-realist wants to say that it is possible, in principle, for someone to know that ψ. That envisaged in-principle possibility will, if and when actualized (say at time t), bring with it the knowledge (at t) that ψ is known (at t). So, it is possible also that it be known that ψ is known (i.e., ♦KK ψ). But does this intuitively imply that ψ is actually known, or will ever actually be known (i.e., K ψ)? Of course it does not!—for the envisaged possibility might never be actualized. There is a degree of serendipity in empirical (and even mathematical) inquiry, which even the anti-realist must recognize. So we cannot treat ♦K as reliably factive when applied to propositions, such as K ψ, that have K dominant. Our formal rules of inference must answer to our pre-theoretic intuitions. We cannot issue the carte blanche of the unrestricted rule F .

3.1. Two proposals for restricting Factiveness How, though, might F be restricted? Clearly, it is the rot within the proof that has to be stopped. The pre-theoretic intuitions mulled through above tell us that this much of is in order:

:

( K)

K K KK KK

(1) (KK ) (1) ( )

⁶ This was observed by Brogaard and Salerno in Brogaard and Salerno (2006).

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It is only the ﬁnal step of that should give us pause: ♦KK ψ ( UF ) Kψ We have envisaged circumstances in which one would be warranted in asserting the premise ♦KK ψ, without being warranted in asserting the conclusion K ψ. So: what is wrong with this application of the rule U F ? What might be the general defect exhibited? Could we restrict applications of the rule so as to avoid just those would-be applications that are defective in this way?

3.1.1. The ﬁrst proposal for restricting Factiveness A ﬁrst stab at the problem might be to insist that applications of the rule ♦K ϕ ϕ may be made only when the sentence ϕ is ‘about basic, non-epistemic facts.’ One could give here a myriad examples from the language of physics, mathematics, chemistry, biology etc., of sentences that are about basic, non-epistemic facts. The proposal would be that sentences like these could be substituends for ϕ in applications of the rule of factiveness. For such sentences, surely, the only reason why it might be possible to know that they are true is that they are indeed true. This is not the case, however, with ‘epistemic’ sentences θ. Here, there can be reason to hold it possible to know that the truth of θ is known, without this being a reason for holding that the truth of θ is indeed (or will ever be) known. Our inquiries might, by the heat-death of the universe, never have taken the turns required, even though, had we conducted our investigations otherwise, we could have come to know the truth of θ. Restriction to K -free ϕ would certainly block the ﬁnal step of , since it depends on taking K ψ as a substituend for ϕ in the statement of the rule. But this seems rather drastic as a proposed logical inoculation against the possibility of incurring -type rot.

3.1.2. A second proposal for restricting Factiveness Another way to formulate a restriction that would render the proof ill-formed (at its ﬁnal step) would be to observe that the premise for that would-be application of the Rule of Factiveness stands as the conclusion of the application, labeled (1), of the rule ♦ , in whose subordinate proof there had been an application of the rule KK : (1) K ( K) (KK ) K K (1) ( ) K KK K

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We can perhaps make more vivid the reason why one might wish to formulate the restriction this way. Let us identify the different occurrences of K by means of italics and boldface. The proof would then look like this: (1) K (K K ) ( K) K K K (1) ( ) KK K The idea would be that ♦K is not factive, whereas ♦K is. ♦K is not factive because K was introduced by an application of the rule (KK ).

3.1.3. The importance of normal form, again As with the global restriction on the Knowability Principle, this restriction will work only when we have ensured that the proof is in normal form. In order to illustrate this, let us remind ourselves how proofs not in normal form characteristically arise. They come from joining together two proofs, the conclusion of one of them being an undischarged assumption of the other. The occurrences of the sentence serving as such a ‘point of accumulation’ can thereby be maximal: a point of locally increased, and—given the overall context—unnecessary logical complexity. Such local complexity can be eliminated by applying a suitable reduction procedure. The result of applying the appropriate reduction procedure might well be a new proof in which other sentences now have maximal occurrences. But these will be of lower complexity than the original one. By repeatedly applying the appropriate reduction procedures, the proof will eventually be transformed into one in normal form. In the context of epistemic logic, here is a simple proof—call it —that appears to be entirely in order. It does not even sin against the newly proposed restriction on the Rule of Factiveness of ♦K (the one that is framed in terms of earlier applications of the rule ♦ ). (1) K (j ∧ ) Kj Φ: ( ) K (j ∧ ) (1) Kj j Below is another proof—call it . The reader ought to look ahead at in order to follow the explanatory comments about to be entered. In , the sentence ψ is indicated as having been proved outright, from the empty set of assumptions. Since we are idealizing the logical abilities of our knowers, we may accordingly infer K ψ—for a logical saint knows every logical theorem. In we also employ the rule Kθ Kχ K (θ ∧ χ)

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This rule can be justiﬁed, in its application within , by appeal to the more obvious rule aK θ aK χ aK (θ ∧ χ) which employs a logical form making explicit provision for the knower a. (Claims of the form K ϕ are really short for ∃x(xK ϕ ).) The latter rule, applied to the materials involved in , would mediate the inference aKK ϕ aK ψ aK (K ϕ ∧ ψ) —for, since any logical saint knows the truth of the theorem ψ, she will know also the conjunction θ ∧ ψ, for any truth θ that she knows. (Here, the truth θ takes the form K ϕ.) Here at long last is the proof . ∅ Ω:

(2) Kj j KKj K Kj K (Kj ∧ ) (2) ( ) K (Kj∧ )

Let us substitute K ϕ for ϕ within the proof , so as to obtain the following proof . (1) K(Kj ∧ ) KKj (1) Φ: ( ) K (Kj ∧ ) KKj Kj The time has come now to join together the proofs and . Note that the conclusion of is the undischarged assumption of . When we make that sentence ♦K (K ϕ ∧ ψ) the point of accumulation, we obtain the following proof, which is not in normal form, and which purports to establish the Fitch result ϕ K ϕ: ∅ (2) Kj K j K Kj K (K j ∧ Kj ) ( ) (2) ( ) K (Kj ∧ ) KKj Kj

K (Kj ∧ )

(1)

KKj (1)

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The advocate of restricting the Rule of Factiveness so that it yields only nonepistemic conclusions will already have objected that the substitution-instance of sins against this restriction. Moreover, he might (mistakenly, as it happens) complain that the other bruited restriction, banning applications of the rule KK within subordinate proofs for applications of the rule ♦ , does not work: witness the proof just formed, in which the latter restriction does not appear to be violated. Appearances, however, can be deceptive. Note that the proof in question is not in normal form. It turns out that, if we normalize it, it is transformed into a normal-form proof in which that second proposed restriction is violated. To see this, proceed as follows. First, apply the obvious reduction procedure to the proof just constructed, so as to get rid of the maximal occurrence of ♦K (K ϕ ∧ ψ). The result is ∅ (1)

Kj KKj K j K (K j ∧ ) KKj (1) Kj KKj Kj

in which there is a new maximal sentence occurrence, namely the occurrence of K (K ϕ ∧ ψ). Upon applying the appropriate reduction procedure in order to get rid of this occurrence, we obtain the following proof in normal form: (1) Kj ( K) j (KK ) Kj KKj j: (1) ( ) KKj Kj —which of course we had already faulted earlier, on the basis of our second proposed restriction on applications of the rule of factiveness.

3.1.4. Blocking an S4-like route to Fitch Consider the following purported proof , involving a Cartesian proposition ϕ:⁷ (1) j j Kj (1) -E Kj S4 Kj j ⁷ This proof is due to an anonymous referee.

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The topmost step, an application of the Knowability Principle, complies with the global restriction. The purported result is unacceptable: that every Cartesian proposition, if possible, is true. Indeed, so is the inference from ♦ϕ to the claim ♦K ϕ, the penultimate conclusion of the proof . This ‘proof’ highlights a point made in §1.1. There are two possibility operators at work, and they need to be distinguished. The one that is introduced by applications of the Knowability Principle adverts not to metaphysical possibilities (which may be contrary to actual fact), but rather to the possibility of an agent coming to know, in accordance with the contingent facts governing his own world, that a certain proposition is true. This epistemic possibility operator should accordingly be distinguished from the metaphysical possibility operator. We shall use and ♦ respectively. The ‘proof’ accordingly becomes (1) j j Kj (1) -E Ξ Kj S4 (??) Kj j in which the allegedly S4-like step is now clearly not valid. While the step ♦♦θ ♦θ is formally correct and valid for the metaphysical possibility operator ♦ of S4, and hence also its substitution instance ♦♦K ϕ ♦K ϕ is formally correct and valid, matters are very different with the step ♦

Kϕ Kϕ

By way of counterexample, consider the Cartesian proposition ‘Grass is purple’ as an instance of ϕ. It is metaphysically possible that grass be purple; but it happens, in our world, not to be. In any other possible world in which grass were purple, however, it would be possible (in that world) to know that it was purple. Hence such possible knowledge would also be a metaphysical possibility. That is, the premise ♦ K ϕ of the last displayed rule of inference, for this choice of ϕ, is true. Its conclusion K ϕ, however, is false—for grass’s being purple is not anything we could come to know in this world. Since the rule of inference in question is invalid, we can prohibit any application of it in proofs. The purported ‘proof’ is not a proof.

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3.1.5. Summary of discussion of how to restrict Factiveness We see, then, that there are still two ways of restricting the Rule of Factiveness, so as to avert the Fitch paradox in the presence of the KK -rule. The ﬁrst restriction is easy to apply: simply ban applications of the Rule of Factiveness that involve conclusions containing K . The second restriction, however, seems at this stage to be just as effective. But we need to bear in mind that it stops the rot only when we have converted the proof to be appraised into normal form. In this regard, the second restriction is like the global restriction proposed earlier for applications of the Knowability Principle. These applications, too, can be assessed for correctness only when the proof in question has been transformed into normal form. This is not the ﬁrst philosophical problem for which the technique of converting proofs into normal form has afforded useful insights. Dag Prawitz used normalization techniques to frame a fertile conception of intuitionistic consequence. (See Prawitz 1974 and Prawitz 1977.) Reduction procedures also form the centerpiece of Michael Dummett’s inferential theory of meaning, and his arguments in favor of intuitionistic logic as the right logic. (See Dummett 1977, the two famous essays ‘The Philosophical Basis of Intuitionistic Logic’ and ‘The Justiﬁcation of Deduction’ in Dummett 1978, and Dummett 1991b.) Normalization lies at the heart also of the present author’s characterization of relevance in deduction.⁸

4. Conclusion We have undertaken here only the most preliminary explorations of prooftheoretic measures designed to stave off certain threats of paradox in an anti-realist epistemic logic. We are still a long way, of course, from having a fully adequate proof-theory governing the interaction among ♦, and K (let alone a formal semantics, with respect to which one might be able to establish the soundness and completeness of whatever proof system is devised). The aim here has been to clear the way for an eventual proof (if such can be found) of a metatheorem to the effect that a suitable system of proof (in epistemic modal logic) embodying the globally restricted Knowability Principle is Fitch-free: that is, it affords no proof of ϕ from K ϕ. Establishing such a result, however, is beyond the scope of the present paper. This discussion has suggested a proof-theoretic path for the anti-realist to follow, without being Fitched. The way forward is to formulate the Cartesian restriction on the Knowability Principle by reference to the ultimate grounds that one could know for the truth of a proposition. As our discussion revealed, the ⁸ See Tennant, ‘‘Logic, Mathematics, and the Natural Sciences’’, in Dale Jacquette (ed.), Philosophy of Logic (Amsterdam: North-Holland, 2006), 1149–66.

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technique of normalizing proofs is crucial for detecting incorrect applications of the Knowability Principle. And the use of a natural deduction format affords us structural insights into proofs, and the resources by means of which one can frame some of the delicate but philosophically motivated restrictions that might be called for on applications of certain rules of inference. With an eye to such resources, other logical or epistemic principles, besides the Knowability Principle, can be adopted in more liberal or more exigent formulations, by way of cautious development of an anti-realist, epistemic logic. The guiding requirement will be that every truth be knowable, without implying that it need ever be known.

Pa r t V I Mod a l a n d Ma t h e m a t i c a l Fi c t i o n s

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15 On Keeping Blue Swans and Unknowable Facts at Bay: A Case Study on Fitch’s Paradox Berit Brogaard

In t ro d u c t i o n What has come to be called the knowability paradox was ﬁrst published by Frederic Fitch as Theorem 5 (1963: 139). It is equivalent to the claim that if every truth is knowable, then every truth is known: (T5) ϕ → ♦Kϕ ϕ → Kϕ where ♦ is possibility, and ‘‘Kϕ’’ is to be read as ‘‘ϕ is known by someone at some time’’. Let us call the premise the knowability principle and the conclusion near-omniscience.¹ Here is a way of formulating Fitch’s proof of (T5). Suppose the knowability principle is true. Then the following instance of it is true: (p & ∼Kp) → ♦K(p & ∼Kp). But the consequent is false, it is not possible to know p & ∼Kp. That is because the supposition that it is known is provably inconsistent.² The inconsistency requires us to deny the possibility of the supposition, yielding ∼♦K(p & ∼Kp). This, together with the above instance of the knowability principle, entails ∼(p & ∼Kp), which is (classically) equivalent to p → Kp. Since p occurs in none of our undischarged assumptions, we may generalize to get near-omniscience, ϕ → Kϕ. QED. (T5) is today considered by many to be a paradox for a number of related reasons, among others, that it threatens to show that the very thesis that is thought to discriminate a mature semantic anti-realism from a naïve idealism entails that I am indebted to Joe Salerno for invaluable discussion, and to Joe, John Divers, Michael Hand, David Jehle, Julien Murzi and two anonymous referees for Oxford University Press for helpful comments. ¹ I used to call the conclusion omniscience. But, of course, ϕ → Kϕ does not entail omniscience, i.e., that there is someone who knows all truths (∃∀), but only the weaker claim that all truths are known by someone (∀∃). Thanks to Michael Hand for suggesting that I rename it. ² If p & ∼Kp is known, then it is true, giving p & ∼Kp, and so ∼ Kp. Also, if p & ∼Kp is known, then the left conjunct is known, giving Kp.

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very idealism. A number of strategies have been developed to avert the paradox, and several of them have provoked signiﬁcant and interesting debate.³ What has rarely, if ever, been noted, however, is that Fitch-like paradoxes threaten to undermine not only semantic anti-realism, but also potentially a number of other anti-realisms with superﬁcial resemblance to semantic anti-realism. One form of anti-realism that is troubled by a Fitch-like paradox is what has come to be called strong modal ﬁctionalism.⁴ Strong modal ﬁctionalists hold that possible world talk, like literary ﬁction, is literally false. Nonetheless, they think the ﬁction provides the resources for an analysis of modal claims. In this note I develop a Fitch-like paradox for strong modal ﬁctionalism. I argue that the most promising strategy to avoid paradox is to reject the claim that modal claims are to be analyzed in terms of the contents of the ﬁction of possible worlds. It is hoped that by looking at the parallel case of modal ﬁctionalism light can be shed on the threat posed by Fitch’s paradox to semantic anti-realism. Mo d a l Fi c t i o n a l i s m Since Gideon Rosen’s centerpiece of 1990, modal ﬁctionalism has been taken seriously by many as a way to employ the resources of possible-world semantics without any of the usual ontological commitments. Modal ﬁctionalism holds that possible world talk is to be treated on a par with talk of ﬁctional objects, such as Sherlock Holmes. Like talk of ﬁctional objects, possible world talk is literally false (or untrue).⁵ It is literally false that there is a brilliant detective at 221b Baker Street. Likewise, modal ﬁctionalists say, it is literally false that there are merely possible worlds, and merely possible objects. Thus, while there might have been blue swans, there is no possible world where there are blue swans. What distinguishes ﬁctionalists from eliminativists is that ﬁctionalists hold that modal claims can be explicated in terms of possible worlds, as long as quantiﬁcation over possible worlds occurs within the scope of an implicit story preﬁx (e.g., ‘‘according to the possible worlds ﬁction’’). Quantiﬁcation within the scope of a story preﬁx is, familiarly, not existentially committing. The content of the possible worlds ﬁction is standardly taken to be David Lewis’s theory of possible worlds (Lewis 1968), including an encyclopedia, that is, a list of all literally true non-modal propositions. Following Rosen, p is a non-modal proposition just in case ‘‘it contains no modal vocabulary and entails neither the existence nor the non-existence of things outside our universe’’ (Rosen 1990: 335). ³ For an overview of the literature, see Brogaard and Salerno (2004). ⁴ This position originates in Rosen (1990). Rosen also considers an alternative position which he calls ‘‘timid modal realism.’’ As we will see, this position is salvageable from paradox (or at least, salvageable from this paradox). For a rich overview of the literature, see Nolan (2002). ⁵ More carefully: possible world talk that incurs a commitment to merely possible worlds or merely possible individuals is literally false (or untrue).

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Where p is a sentence of quantiﬁed modal logic, p∗ is a translation of p into the language of possible worlds, and W is a story preﬁx which reads: according to the Lewis story, Rosen’s formulation of modal ﬁctionalism may be given as follows: (Fic) p ↔ Wp∗ (Fic) says that a modal claim will be true iff its translation into the language of possible worlds is true in the Lewis story. So, for example, ‘‘there might have been blue swans’’ is true iff, according to the Lewis story, there is a possible world where there are blue swans. Likewise, ‘‘there are no blue swans’’ is true iff according to the Lewis story, in the actual world, there are no blue swans. A problem for modal ﬁctionalism was noticed independently by Rosen (1993) and Stuart Brock (1993).⁶ In a nutshell, it is that since in the Lewis story it is true at each world that there exists a plurality of worlds, we can derive, by (Fic), that necessarily there is a plurality of worlds. Since necessity entails truth, there in fact exists a plurality of worlds, which is not something the ﬁctionalist should tolerate. However, the objection has been shown to be unproblematic if careful attention is paid to the translation scheme offered by Lewis in 1968 for translating sentences in the language of quantiﬁed modal logic into sentences in the language of counterpart theory.⁷ In the 1968 translation scheme, the sentences of quantiﬁed modal logic translate into sentences that quantify over worlds and their parts. Thus, where ‘‘U’’ means world, and I is the existing-wholly-in (or parthood) relation, a modal sentence of the form: ♦∃xFx translates as: ∃x∃y(Uy & Ixy & Fx) This says that there is a world of which something that is F is part. Assuming the letter of the 1968 translation scheme, ‘‘necessarily, there is a plurality of worlds’’ translates as: ∀x(Ux → ∃y∃z(Iyx & Izx & Uy & Uz & y = z)) This says that all worlds have at least two worlds as parts. But this is false, according to the Lewis story, since no world has any other world as a part. Hence, the objection fails. However, too careful attention to the 1968 scheme leads to Hale’s dilemma (see Hale 1995).⁸ The ﬁctionalist cannot say that modal realism is possible. For ⁶ For a much more substantial treatment of the history of the debate and alternative solutions to problems, see Nolan (2002). ⁷ The 1968 translation scheme is the one found in Lewis (1968) [1983]. For discussion, see also Nolan (2002). ⁸ I am simplifying the ﬁrst horn of the dilemma.

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assuming the letter of the 1968 translation scheme ‘‘it is possible that there is a plurality of worlds’’ translates as: ∃x(Ux &∃y∃z(Iyx & Izx & Uy & Uz & y = z)) This says that some world has at least two worlds as parts. But this is false, according to the Lewis story. Nor can he say that modal realism is impossible. For, Hale argues, the ﬁctionalist must provide some analysis of the story preﬁx ‘‘according to the Lewis story.’’ Most plausibly this explication will involve a non-material conditional of the form: if modal realism is true, then p. But if it is necessarily false that there is a plurality of worlds, then this conditional will be trivial; ‘‘according to the Lewis story, p’’ will be true for any p. Hence, the ﬁctionalist will be committed to the truth of any modal claim.⁹ The same problem arises if the ﬁctionalist holds that the story preﬁx is primitive. The ﬁctionalist ought to accept some closure principle of the form, Wp, p ⇒ q Wq. But where p is impossible (e.g. ‘‘there is a plurality of worlds’’), p entails any claim. So, Wq obtains for any q. If q is ‘‘there is no world where there are blue swans,’’ we can derive (by Fic), the far-fetched claim ‘‘ ∼♦(there are blue swans)’’. John Divers has offered the following resolution of Hale’s dilemma (Divers 1992). When the realist assents to ‘‘there is a plurality of worlds’’ his intention is that the ‘‘conventional world-restriction of quantiﬁcation should not apply’’ (Divers 1999a: 323; see also Divers 1999b). But, Divers argues, if restricted to alethic modality, the T axiom, p ♦p, ‘‘approaches the status of analyticity’(1999b: 218). Hence, the realist must assent to ‘‘it is possible that there is plurality of worlds.’’ In such cases, Divers argues, the operand modality must be read as redundant. That is, where p is unrestricted, the realist must assent to ♦p p.¹⁰ Where the realist translates the unrestricted possibility claim ♦p as p, the ﬁctionalist will thus do well to translate it as Wp. That is, the ﬁctionalist can assent to the following instance of (Fic): If p is read as an unrestricted possibility claim, then (♦p ↔ Wp) Since the ﬁctionalist is thus able to assent to the contingent falsehood of modal realism, Hale’s dilemma is blocked. ⁹ The suggestion can be found in Rosen (1995). Rosen also suggests that the ﬁctionalist could block Hale’s dilemma by, for example, rejecting ‘‘classical’’ semantics for non-material conditionals, or allowing truth-values to the statement involving the ‘‘world’’ predicate only when the predicate occurs within the scope of the preﬁx. Hale (1995) thinks these suggestions are desperate. Disallowing truth-values to the statements involving ‘‘free’’ occurrences of the ‘‘world’’ predicate would leave claims like ‘‘there is exactly one world, namely the actual’’ without a truth-value. Rejecting the claim that counter-possible conditionals are all trivially true is plausible for conditionals with metaphysically impossible but logically possible antecedents. But Rosen’s suggestion is the more radical one that even counter-possible conditionals with logically impossible antecedents may fail to be trivially true. For further problems with these strategies, see Divers and Hagen (2006). ¹⁰ p p trivially follows. S5 and ♦p p yields the trivial system (Tr), in which no signiﬁcant modal distinctions can be drawn.

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There is, however, a different path to disaster even assuming Divers’s counterpart-theoretic translation principles for possibility claims.

A Fi t c h - l i k e Pa r a d o x A collapse ensues owing to a Fitch-like proof of the following theorem (where p is an unrestricted locution):¹¹ (T2) p → Wp∗ (Wp → p) Like the logic of Fitch’s proof, the logic of the Fitch-like proof of T2 is modest: minimal, normal modal logic, and three intuitive story-preﬁx principles: (A) (B) (C) (D)

W(p & q) Wp & Wq WWp Wp Wp ∼W∼p ∼♦p ∼p

(A) is a distributivity principle that says that if in the Lewis story, p and q, then in the Lewis story, p, and in the Lewis story, q.¹² (B) says that if the Lewis story states that according to the Lewis story p, then the Lewis story states that p. (C) states: if in the Lewis story, p, then it is not the case that in the Lewis story, not-p. (D) is the inference of a necessary falsehood from an impossibility. If these resources are not already believable, we can say this. (A) is entailed by the uncontroversial assumption that conjunction in the Lewis story is classical. Denying (B) would yield the implausible result that the Lewis story may deny p yet assert about itself that it holds that p (certainly, Lewis would have disapproved of such a theory).¹³ (C) follows from the uncontroversial assumption that modal realism is classically consistent. (D) follows from the duality of the modal operators. The proof employs a theorem derived from these resources, Theorem T1—viz., for any p, it is not the case that according to the Lewis story, both not-p, and according to the Lewis story, p. (T1) ∼W(∼ p & Wp) ¹¹ The star is not required on the right-hand side of (T2). Starred statements are translations of the statements of quantiﬁed modal logic into statements of counterpart theory. But, given Divers’s translation scheme, unrestricted locutions of the form ‘‘♦p’’ translate as ‘‘p.’’ So, the star drops off. Thanks to an anonymous referee here. ¹² Of course, to avoid obvious counterexamples, we will ultimately need an account of tense operators occurring within the scope of a story preﬁx. Thanks to David Jehle here. ¹³ Notice, further, that (B) is validated on Divers’s (1999a) explication of the story preﬁx: ((Wp) ↔ (the Lewis story → p)), where the box is primitive. For if (the Lewis story → (the Lewis story → p)), then (the Lewis story → p). It is also validated on the following subjunctive explication of the story preﬁx: Wp ↔ if the Lewis story were true, p would be true. For if (if the Lewis story were true, then if the Lewis story were true, then p would be true), then (if the Lewis story were true, then p would be true).

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The proof of (T1) runs as follows. Suppose for reductio that W(∼p & Wp). Then, by (A), W∼p & WWp. The right conjunct, by (B), entails Wp. This, by (C), entails ∼W∼p. So, we derive: W∼p & ∼W∼p. Contradiction. Rejecting our assumption, by reductio, gives us ∼W(∼p & Wp). QED. Where p is an unrestricted locution (e.g. ‘‘there is a plurality of worlds’’), a Fitch-like proof of (T2) may be developed as follows:¹⁴ (1) (2) (3) (4) (5) (6)

∀p(p → Wp∗ ) ♦(∼p & Wp) → W(∼p & Wp) ∼W(∼p & Wp) ∼♦(∼p & Wp) ∼(∼p & Wp) (Wp → p)

from (Fic), left to right from 1, Divers instance of T1 from 2, 3 from 4, D from 5

We suppose at (1) that, for any unrestricted modal or non-modal locution p, if p is true, then according to the Lewis story, the translation of p into the language of counterpart theory obtains. (2) substitutes the unrestricted possibility claim, ♦(∼p & Wp), for p in (1). Following Divers’s realist translation schema for unrestricted claims, ♦p p, the modal realist translates ♦(∼p & Wp) as ∼p & Wp, and the ﬁctionalist translates it as W(∼p & Wp). Line (3) is an instance of the theorem, (T1). Line (4) follows trivially from lines (2) and (3). By (D), we derive line (5) from line (4). In classical logic line (6) is equivalent to line (5). Therefore, if ﬁctionalism is true, then every unrestricted claim that is true according to the Lewis story is true: T2 p → Wp∗ (Wp → p).¹⁵ According to the Lewis story, there is a plurality of worlds. By T2 and (Fic), we can derive that there is a plurality of worlds. This, of course, should not be acceptable to the ﬁctionalist.

K e e p i n g Re a l i s m a t Ba y How will the ﬁctionalist respond? Well, there is, of course, always the option of denying our initial logical resources. I will not rule out that this can be done in a principled manner. Assuming, however, that no such route is available to the ﬁctionalist, how can he avoid paradox? In this section I will look at four of the main strategies that have been employed in order to avoid Fitch’s original ¹⁴ In line 2 the star drops off as the result of applying Divers’s translation scheme for unrestricted locutions to an unrestricted locution. Thanks to an anonymous referee here. ¹⁵ Divers and Hagen (2006) think Divers’s solution to the problem of unrestricted claims is undermined because the T-theorem (i.e., truth entails possibility) holds for unrestricted claims. But this assumption is controversial. Lewis explicitly denies it (see 1968[1983]: 39–40). My argument does not rest on this assumption. Of course, it may be thought that if we deny the T-theorem for unrestricted claims, then the problem of possible unrestricted claims does not arise. But this is not so. For the ﬁctionalist wants to say that unrestricted claims are false in spite of being possible. So, for them, the T-theorem does not play a role in arguing for the possibility of an unrestricted claim.

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result, and try to determine whether the ﬁctionalist can avail himself of similar strategies.

The Intuitionistic Strategy Since Fitch’s proof is classically, but not intuitionistically, valid, the paradox can be avoided by rejecting classical logic.¹⁶ Like Fitch’s proof, the proof of T2 is classically, but not intuitionistically, valid. Without classical logic we cannot derive line (6) from line (5). An intuitionist, however, is committed to Wp →∼ ∼p. This entails: ∼p → ∼Wp. But the latter is evidently absurd from a ﬁctionalist stance. It reads: if it is not the case that p, then it is not the case that according to the Lewis story, p. So, if it fails to be true that there is a plurality of worlds, as the ﬁctionalist claims, then it is not the case that according to the Lewis story, there is a plurality of worlds.¹⁷

The Modal Fallacies Strategy Another important strategy that has provoked signiﬁcant and interesting debate is that offered by Jon Kvanvig (1995, and forthcoming). Kvanvig argues that Fitch’s result is invalid, owing to a fallacious substitution into a modal context. The problem, Kvanvig says, is that Kp is implicitly quantiﬁed. Explicitly it reads ∃x∃t(Kxpt), which says that there is someone x and a time t such that x knows at t that p. But, on Kvanvig’s neo-Russellian account of quantiﬁed expressions, quantiﬁed expressions cannot, in general, be legitimately substituted into modal contexts, hence, the failure of the substitution of the Fitch conjunction, p & ∼Kp, into the knowability principle. Kvanvig’s solution has been criticized on various fronts (see, e.g., Williamson 2000b: Chapter 12; Brogaard and Salerno 2007; Jenkins, Ch. 18 of this volume). However, even if it succeeds, the ﬁctionalist cannot avail himself of Kvanvig’s strategy. For the whole purpose of strong modal ﬁctionalism is to be able to analyze the sentences of quantiﬁed modal logic which, familiarly, allows for substitution into modal contexts. A closely related but equally unsuccessful strategy is this. For the Fitch-like proof to go through it is crucial that ♦(∼p & Wp) is an unrestricted possibility claim. So, might not the ﬁctionalist simply deny that ♦(∼p & Wp) can be read as an unrestricted possibility claim? Unfortunately, this is not an option. For the ﬁctionalist is prepared to say that it is possible that both ∼(there is a plurality of worlds), and according to the Lewis story, there is a plurality of worlds. That is, he is prepared to say: ♦(∼p & Wp). However, on a restricted reading, ¹⁶ For discussion of the intuitionistic strategy, see e.g. Williamson (1982); Wright (1993a [1987]). ¹⁷ Parallel reasons have been given for rejecting the intuitionist solution to Fitch’s paradox. See e.g. Percival (1990).

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♦(∼p & Wp) cashes out to the obviously false claim: ‘‘For some world w, w does not have a plurality of worlds as part, but according to the Lewis story, it does.’’ Thus, the substitutional fallacy move is unsuccessful.

The Rigidiﬁer Strategy A third strategy that has provoked signiﬁcant debate is that of Dorothy Edgington (1985). Edgington’s strategy is to bypass the knowability principle altogether. Instead, she requires of knowability the less general thesis: (AKP) Ap → ♦KAp where ‘‘Ap’’ is to be read ‘‘it is actually the case that p.’’ Since KA(p & ∼Kp) is not provably inconsistent, this strategy avoids paradox. The ﬁctionalist might, similarly, propose to replace ﬁctionalism with the following weaker thesis: (FicA) Ap ↔ WAp∗ There is, however, little reason to think the ﬁctionalist would want to do that. For if he accepts p Ap, and WAp Wp, which we can reasonably expect, then (FicA) entails p → Wp∗ . This is all we need to get the Fitch-like proof going.

The Restriction Strategy A more recent and widely discussed strategy to block Fitch’s original paradox is to restrict the universal quantiﬁer in ‘‘all truths are knowable.’’ Neil Tennant, for instance, favors what he calls the ‘‘Cartesian’’ restriction (Tennant 1997: 274).¹⁸ A proposition ‘‘p’’ is Cartesian just in case ‘‘Kp’’ is not provably inconsistent. Tennant’s Cartesian knowability principle may be stated thus: all Cartesian truths are knowable. (CKP) p → ♦Kp, where p is Cartesian. It should be apparent that the Cartesian restriction blocks Fitch’s paradox, since Fitch’s result requires the substitution ‘‘p & ∼Kp’’ for ‘‘p’’ in ‘‘p → ♦Kp’’. ‘‘p & ∼Kp’’ is not Cartesian, as K(p & ∼Kp) is logically impossible. The ﬁctionalist’s best option may be to follow Tennant’s lead and reformulate the ﬁctionalist principle as follows: (Fic∗ ) p ↔ Wp∗ , where Wp* is not provably inconsistent Provided that Wp∗ is provably inconsistent only when p is an unrestricted claim, we can offer the following more effective formulation of ﬁctionalism. (Fic∗∗ ) p ↔ Wp∗ , where the quantiﬁers in p are restricted to worlds. ¹⁸ A related proposal can be found in Dummett (2001).

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We, furthermore, propose that the ﬁctionalist treat the unrestricted modal operators as primitive S5 operators.¹⁹ On a non-redundancy interpretation of the unrestricted modal operators, the possibility of modal realism does not entail its truth. The ﬁctionalist can thus coherently deny ‘‘there is a plurality of worlds,’’ but assent to its possibility. But there is a potential danger in relying upon restriction to avoid paradox. One major obstacle to Tennant’s restriction strategy, for example, is that there are other Fitch-like paradoxes that are not averted by the restriction.²⁰ Tennant has subsequently promised to develop his restriction strategy to protect against these further paradoxes. Whether the ﬁctionalist who restricts is susceptible to similar criticism remains to be seen. If he is, there is then the option of following Tennant’s lead and developing more suitable restrictions. But there is a further worry about restriction strategies. Against Tennant’s strategy it has been argued that the restriction on knowable truth is unprincipled—that no reason has been given, other than the threat of paradox, to restrict the knowability principle (see for instance, Hand and Kvanvig 1999; DeVidi and Kenyon 2003; Hand 2003). A related charge against Tennant’s restriction strategy is that we must admit that however plausible the knowability principle is for a restricted class of sentences, it is to be rejected as a general principle. This is a stern confession on the part of the semantic anti-realist, who claims to have on offer an epistemic theory of truth. A related charge can be issued against the proposed restriction of ﬁctionalism, i.e. (Fic∗∗ ). Any reinterpretation of modal discourse must be inferentially adequate. The reinterpretation of modal discourse, for example, must inherit the inferential advantages of using discourse about possible worlds (Rosen 1990: 330; and Divers 2004: 665). However, the proposed reinterpretation of modal discourse does not inherit the inferential advantages of possible world semantics. For the modal realist can account for the validity of our standard modal inferences. For example, where p is world-restricted, the modal realist can account for the validity of p ♦p by translating p into an idiom of counterpart theory, inferring the ﬁrst-order consequence that there is a world in which p, and then translating this consequence back into an ordinary modal idiom. Since the unrestricted modal operators are redundant, validity is even easier to account for when p is unrestricted. By contrast, the ﬁctionalist can account only for the validity of p ♦p, where p is world-restricted. Hence, ﬁctionalism does ¹⁹ An unrestricted modal operators is a modal operator that embeds an unrestricted statement, for instance, the possibility operator occurring in ‘‘it is possible that there is a plurality of worlds.’’ The proposed strategy is entirely motivated. First, if Hale is right, then the ﬁctionalist cannot offer an adequate and comprehensive analysis of all possibility claims in non-modal terms. The ﬁctionalist will need to admit primitive modal operators in order to account for the meaning of the story preﬁx. Second, the modal realist, too, must acknowledge two kinds of modal operators. The modal realist analyzes the restricted modal claims of quantiﬁed modal logic in terms of possible worlds, but must, if Divers is right, treat unrestricted modal operators as redundant. ²⁰ See e.g. Williamson (2000b); Brogaard and Salerno (2002, 2006, 2007).

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not inherit the inferential advantages of using discourse about possible worlds without the ontological costs. We have been assuming strong modal ﬁctionalism. Strong modal ﬁctionalism holds that the ﬁction of possible worlds provides the resources for an analysis of modal claims, and so that modal claims depend on the content of the ﬁction of possible worlds. Some of the problems with ﬁctionalism can be avoided if, as Rosen suggests, we take ﬁctionalism to provide ‘‘not a theory of possibility, but merely a theory linking the modal facts with facts about the story PW’’ (1990: 354). The result is timid modal ﬁctionalism.²¹ Just like strong modal ﬁctionalism, timid modal ﬁctionalism licenses only a subset of transitions from modal idioms to idioms of counterpart theory. However, this may be unproblematic insofar as the timid ﬁctionalist does not regard (Fic∗∗ ) as purporting to shed light on the nature of modal truth, but merely takes it to link the sentences of quantiﬁed modal logic with the sentences of counterpart theory.

Ge n e r a l L e s s o n s Of the four solutions to Fitch’s result considered, only restriction strategies can be extrapolated to block the Fitch-like result developed above. This shows that restriction strategies are potentially more effective tools for avoiding Fitch-like paradoxes than are the other strategies we have considered. Unless it can be argued that different anti-realisms call for fundamentally different resolutions of Fitch-like paradoxes,²² the aptitude of restriction strategies to block Fitch-like paradoxes should count in their favor. The downside is that the restriction strategist cannot take the principles he is restricting as providing the resources for an analysis of truth. The result of restriction is thus timid anti-realism. Is timid semantic anti-realism an acceptable position? Well, one of the motivations for semantic anti-realism is Dummett’s manifestation argument (Dummett 1976; 1977: Chapter 4; 1978). The argument is often taken to be that, for reasons having to do with the manifestability of meaning, truth is to be understood epistemically, in terms of what our epistemic capacities allow us to verify in principle. But if it can be shown that manifestability ²¹ For a defense of timid modal ﬁctionalism, see Brogaard (2006). ²² Fitchy paradoxes may be symptomatic of a common malady in (a range of ) anti-realist positions. It would be interesting to ﬁnd out what the malady is. Is there a deep metaphysical point to be made about why these different positions give rise to this distinctive kind of paradox? On the face of things, it is not easy to see what that might be. Anti-realism and modal ﬁctionalism both consist in the non-acceptance of the objectivity of a certain subject matter (e.g. modality), but apart from this superﬁcial similarity, they are obviously very different. In a recent paper (Brogaard and Salerno 2005), it was argued that the malady in question is a kind of conditional fallacy. Perhaps, then, it is not that modal ﬁctionalism is similar in some deeper respect to semantic anti-realism, but rather that strong anti-realisms run into some kind of conditional fallacy. Thanks to John Divers here.

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considerations entail no such result, then it is open to argue for a timid semantic anti-realism. A timid semantic anti-realist could appeal to veriﬁability merely in order to counter the realist claim that even truths that can be consistently known may be beyond our epistemic reach.²³ In conclusion: the present case study indicates that Fitch-like paradoxes present a major obstacle, not only to semantic anti-realism, but also potentially to a number of other anti-realisms. The Fitch-like paradoxes give anti-realists reason to restrict. Restriction leads to timid anti-realism. Fitch’s proof can thus be construed as an argument for timid (as opposed to strong) anti-realism. ²³ This, in fact, may be all that Tennant is committed to. See Tennant (2001b).

16 Fitch’s Paradox and the Philosophy of Mathematics Ot´avio Bueno

In t ro d u c t i o n It is intuitively plausible to suppose that the sheer fact that something true can be known should not be sufﬁcient for it to be actually known. According to Fitch’s paradox, however, under very reasonable assumptions, we basically obtain this result (see Fitch 1963, and Brogaard and Salerno 2004). More precisely, we obtain a reductio of the claim that two apparently acceptable principles are consistent.¹ The ﬁrst principle is the knowability principle, according to which all truths can be known by somebody at some time: (KP) ∀p (p → ♦Kp), where K is the epistemic operator ‘‘it’s known by somebody at some time that,’’ and ♦ is the modal operator ‘‘it’s possible that.’’ The second principle states that we are non-omniscient; that is, there is a truth that nobody ever knows: (NonO) ∃p (p ∧ ¬Kp). It’s easy to see that (KP) and (NonO) entail ♦K(p ∧ ¬Kp).² However, we have independent reasons to believe that ¬♦ K(p ∧ ¬Kp).³ Thus, if we keep the claim My thanks go to Jody Azzouni, Joe Salerno, Amie Thomasson, and Ed Zalta for extremely helpful discussions. Thanks are also due to Joe Salerno and two anonymous referees for very perceptive comments on earlier versions of this work. Their comments led to substantial improvements. ¹ I follow here Berit Brogaard and Joe Salerno’s elegant presentation of the paradox (see Brogaard and Salerno 2004). ² Consider the following instances of, respectively, (NonO) and (KP): (1) (p ∧ ¬Kp) (2) (p ∧ ¬Kp) → ♦K(p ∧ ¬Kp) where (2) is obtained by substituting (p ∧ ¬Kp) for variable p in (KP). ♦K(p ∧ ¬Kp) then follows immediately. ³ In fact, the argument to this effect is based on four reasonably straightforward inferences:

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that all truths are knowable (KP), we have to deny that we are non-omniscient (NonO); that is, we have to assert that all truths are actually known: (O) ∀p (p → Kp). In other words, it looks as though, given the assumptions of the argument above, we are licensed to move from the claim that something true is knowable to the conclusion that it’s actually known.⁴ But, prima facie, this seems paradoxical. In what follows, I’ll take the paradox to be an inference that ultimately licenses us to conclude, given a true sentence p and the possibility of knowing p, that p is actually known; that is: (FP) p → (♦Kp → Kp). In this paper, instead of examining the paradox in a generic context, I’ll consider the impact it has on particular epistemological views about mathematics. I will assume therefore, for the sake of argument, that the reasoning leading to Fitch’s paradox is valid (which is indeed the case given the logical assumptions made). Having this focus provides a speciﬁc context to assess the nature and limitation of the paradox, while also showing the paradox’s signiﬁcance for current debates in the philosophy of mathematics. More speciﬁcally, I’ll examine two versions of Platonism (standard and full-blooded Platonism) and two versions of nominalism (mathematical ﬁctionalism and agnostic ﬁctionalism). And I’ll argue that, given the speciﬁc assumptions about mathematical knowledge that these views make, (FP) brings trouble for some, but not for all, of them. In particular, full-blooded Platonism and—to a certain extent—mathematical ﬁctionalism are in trouble with the paradox, but traditional Platonism and agnostic ﬁctionalism don’t seem to be. I conclude with a dilemma that this situation poses for Platonism, and the prospects it offers for nominalism. (a) (b) (c) (d)

If K(p ∧ q), then Kp ∧ Kq. (A conjunction is known only if the conjuncts are known.) If Kp, then p. (A statement is known only if it’s true.) If p is a theorem, then p. (All theorems are necessarily true.) If ¬p, then ¬♦p. (If it’s necessary that ¬p, then p it’s impossible that p.)

Assuming these inferences, here’s the argument for ¬♦K(p ∧ ¬Kp): (1) (2) (3) (4) (5) (6)

K(p ∧ ¬Kp) Kp ∧ K¬Kp Kp ∧ ¬Kp ¬K(p ∧ ¬Kp) ¬K(p ∧ ¬Kp) ¬♦K(p ∧ ¬Kp)

Assumption (for reductio) from (1), by (a) from (2), by (b) from (1)–(3), by reductio, discharging assumption (1) from (4), by (c) from (5), by (d).

⁴ The reason for this conclusion is that, as we saw, Fitch’s paradox establishes that the conjunction of (KP) and (NonO) leads to a contradiction. This, in turn, establishes the conditional: ∀p(p → ♦Kp) → ∀p(p → Kp), where the antecedent is (KP) and the consequent is the negation of (NonO), that is, (O). Now, if we assume (KP) and p (that is, if we assume that ‘p’ is true), we obtain the conditional: p → (♦Kp → Kp).

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Fi t c h’s Pa r a d o x a n d Fu l l - b l o o d e d P l a t o n i s m According to full-blooded Platonism (FBP), all mathematical objects that logically possibly exist actually do exist (Balaguer 1998: 53).⁵ The notion of possibility is here taken in its broadest sense, namely, as logical possibility (Balaguer 1998: 5–6 and 69–75). In other words, the picture of the mathematical reality that emerges is one of plenitude: all logically possible mathematical objects exist. From non-Cantorian sets to non-separable Hilbert spaces, from unusual metric spaces to yet unknown solutions to weird differential equations, everyone is welcome to this truly over-populous mathematical paradise. But why should anyone be happy with this extravagant ontology? Surprisingly enough, argues Balaguer, because this is the only defensible version of Platonism. Balaguer’s message is clear: if you want to advocate the existence of mathematical entities, don’t be shy, countenance all of them! Why should only Cantorian sets be accepted? What about the non-Cantorian ones? And why should we stop at sets, and not also introduce functions, numbers, and categories? The mathematical realm is a rich and multifarious domain, and that’s how it has to be if we are to solve the most pressing problem faced by Platonism: how can we have any knowledge of this realm if we have no contact with it, no form of access whatsoever? According to the FBP-ist, by extending the ontology of mathematics, we can answer this central question regarding mathematical epistemology. If all logically possible mathematical objects exist, we can explain the possibility of mathematical knowledge and the reliability of mathematical beliefs. To provide an account of the possibility of mathematical knowledge, the FBP-ist needs only to account for the fact that we can know that mathematical theories are consistent. Basically, FBP reduces the problem of knowing the truth of mathematical theories to that of knowing their consistency.⁶ After all, if FBP is true, the consistency of a mathematical theory M leads to its truth, since if M is consistent it will truly describe some part of the mathematical realm. And in this way, for the FBP-ist, knowledge of the consistency of M yields knowledge of M ’s truth. ⁵ There are worries about how to formalize properly the claim that characterizes FBP. Balaguer discusses some of these worries, without completely resolving them (see Balaguer 1998: 5–8). As Balaguer himself acknowledges, the formulation that comes closer to capturing FBP involves secondorder quantiﬁcation. Let ‘Y ’ be a second-order variable and let ‘Mx’ mean ‘x is a mathematical object’; in this case FBP can be (roughly) formulated as:

∃xMx ∧ ∀Y (♦∃x (Mx ∧ Yx) → ∃x (Mx ∧ Yx) ). For the sake of argument, I’m going to grant that there is a workable formulation—and a formalization—of FBP, although this is actually doubtful (see Restall 2003). ⁶ Since in this paper I’m focusing primarily on mathematical theories, following Balaguer, I’ll use the notions of consistency and logical possibility interchangeably. Nothing hinges on this.

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Moreover, to provide an account of the reliability of mathematical beliefs, all that needs to be explained is why it is the case that (as a general rule) if mathematicians accept p, then p.⁷ To indicate how this can be achieved, Balaguer provides the following argument: (i) FBP-ists can account for the fact that human beings can—without coming into contact with the mathematical realm—formulate purely mathematical theories. (ii) FBP-ists can account for the fact that human beings can—without coming into contact with the mathematical realm—know of many of these purely mathematical theories that they are consistent. (iii) If (ii) is true, then FBP-ists can account for the fact that (as a general rule) if mathematicians accept a purely mathematical theory T, then T is consistent. Therefore, (iv) FBP-ists can account for the fact that (as a general rule) if mathematicians accept a purely mathematical theory T, then T is consistent. (v) If FBP is true, then every consistent purely mathematical theory truly describes part of the mathematical realm, that is, truly describes some collection of mathematical objects. Therefore, (vi) FBP-ists can account for the fact that (as a general rule) if mathematicians accept a purely mathematical theory T, then T truly describes part of the mathematical realm. (Balaguer 1998: 51–2) After putting forward this argument, Balaguer indicates why its premises should be accepted, and, as a result, concludes that FBP leads to a new Platonist epistemology. For the purposes of this paper, there is no need to review all of Balaguer’s considerations (for details, see Balaguer 1998: 52–3). I’ll focus on the key premises of his argument—particularly premise (ii). Note that (vi) follows from (iv) and (v) only if FBP is true. Otherwise, we have at best the conditional: If FBP is true, then (vi) is true—which is, of course, substantially weaker than the claim that (vi) is true. In other words, FBP-ists can provide an account of mathematical knowledge and the reliability of mathematical beliefs only by asserting the truth of FBP. Without this assertion, the account doesn’t get off the ground. But does Balaguer’s overall strategy succeed? To support premise (ii), Balaguer develops an anti-Platonist—in fact, a factionalist—epistemology for Platonism. The epistemology is ﬁctionalist in ⁷ As Balaguer acknowledges, in explaining the possibility and reliability of mathematical knowledge, he follows related moves made by Hartry Field’s mathematical ﬁctionalism (see Field 1989). I’ll discuss Field’s proposal shortly.

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the sense that it doesn’t rely on the existence of mathematical entities. These entities play no role in the epistemological story, and they might just as well be taken as useful ﬁctions. The point of the ﬁctionalist epistemology is to establish how we can have knowledge of mathematical entities ‘‘without coming into contact with the mathematical realm.’’ The key idea is to advance a primitive notion of consistency (or logical possibility), and argue that this notion provides all that is needed for our knowledge of mathematical objects. After all, given FBP, if a mathematical theory is logically consistent, then it is true of part of the mathematical realm. Thus, given FBP and the principle of closure for the knowledge operator, if we know that a mathematical theory is consistent, we know that it is true (of some part of the mathematical realm). In other words, as noted above, for the FBP-ist, knowledge of the consistency of a mathematical theory immediately yields knowledge of the theory’s truth (regarding some part of the mathematical realm).⁸ So, the FBP-ist only has to explain how we come to know that mathematical theories are consistent; knowledge of their truth will follow straightway. The strategy is ingenious, of course, in that it uses what is perhaps the most attractive feature of anti-Platonism—the fact that it doesn’t seem to make mathematical knowledge a mystery⁹—in order to overcome the weakest aspect of Platonism, the difﬁculty of accommodating mathematical knowledge given the postulation of abstract entities. It’s precisely at this point that Fitch’s paradox comes in. First, note that, given the epistemological strategy that underlies FBP, the proposal is committed to the knowability principle (KP) in the context of mathematics; that is, every true mathematical claim can be known. After all, suppose that M is a true mathematical claim. Then, as the FBP-ist acknowledges, M is consistent (for the FBP-ist doesn’t think that there are true contradictions). But if M is consistent, it’s possible to know that M is consistent. (After all, only M ’s inconsistency would prevent the logical possibility of knowing that M is consistent, and so, if it were logically impossible to know that M is consistent, then M would be inconsistent. By contraposition, we get the intended conditional.¹⁰) Thus, if it’s possible to know that M is consistent, then for the FBP-ist, it’s possible to know M , given that, on the FBP-ist picture, M ’s consistency entails M ’s truth. In other words, every true mathematical claim can be known. ⁸ Note that the FBP-ist is not committed to the existence of true contradictions. Given two independently consistent, but jointly inconsistent, mathematical theories, their conjunction, being inconsistent, is not going to be true of any part of the mathematical realm. However, since separately each mathematical theory is (by hypothesis) consistent, each theory is going to be true of some part of the mathematical world. These parts, however, on the FBP-ist picture, do not overlap. ⁹ Typically, for the anti-Platonist, mathematical knowledge is conceptualized as some form of logical knowledge (knowledge of what follows from what). Using a primitive notion of consistency (or possibility), we can express the consequence relation—B is a consequence of A—as it’s not possible that A is true and B is false. ¹⁰ Recall that the FBP-ist is operating with the notion of logical possibility as the underlying notion of modality.

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It might be objected that the FBP-ist is not really committed to (KP). After all, there might be consistent mathematical statements that are so intractable that it’s not logically possible to determine their consistency. It may initially seem plausible that there are such statements. After all, nothing seems to rule out their existence. However, on further reﬂection, it’s quite unclear that we have any reason to believe that such statements exist. At stake is the existence of consistent mathematical statements such that any attempt to show that they are consistent leads to a contradiction. Of course, inconsistent statements would have this property (assuming classical logic, any attempt to show that they are consistent entails a contradiction). But inconsistent statements are not the relevant candidates here, since the assumption made involves consistent statements. As an alternative, it might be thought that Gödel’s incompleteness theorems provide such statements. But that doesn’t seem right either. First, a Gödel sentence is not intractable; in fact, it can easily be seen to be true. More importantly, from Gödel’s second incompleteness theorem, we have it that, under certain assumptions—namely, that T is a formal recursively enumerable theory that includes basic arithmetic and certain provability conditions—if T is consistent, T cannot prove its own consistency. The trouble, however, is that Gödel’s result is not relevant here. The assumption made in the objection above is not that the consistency of a certain mathematical theory M cannot be proved in M, but rather that M’s consistency cannot be proved at all—and just as a matter of logic. Moreover, the only assumption made in the objection is that the mathematical statements in question are consistent. But in order for us to apply Gödel’s theorem, we would also need the assumption that the mathematical statements in question constitute a formal recursively enumerable theory including basic arithmetic and certain provability conditions. In other words, the objection requires a much stronger claim than anything that can be reasonably supported by Gödel’s theorem, and the assumption made in the objection is much broader than those needed for the theorem to apply. Here is an illustration of the central point. Although we cannot show the consistency of arithmetic in arithmetic (assuming that arithmetic is consistent), we can show the consistency of arithmetic in set theory. However, the situation considered in the objection above is one in which the consistency of the mathematical statements in question cannot be established under any circumstance, for purely logical reasons—a much stronger claim! The burden is now on the FBP-ist to show that there are statements of this sort. I don’t see how this can be done. And unless the FBP-ist can show that there are such statements—and it’s not clear that this is the case—the argument given above to the effect that the FBP-ist is committed to (KP) still stands. Now, given the FBP-ist’s commitment to (KP), does it follow that the FBP-ist is thereby committed to (FP)? Well, recall that the argument for (FP), given in Section 1 above, established an inconsistency between two principles: (KP) and (NonO). According to the latter, we are non-omniscient; that is, there is a truth

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that nobody ever knows; in symbols: ∃p (p ∧ ¬Kp). So, the question now is whether the FBP-ist is also committed to (NonO). However the FBP-ist answers this question, problems emerge. In fact, we have a dilemma. Either the FBP-ist is committed to (NonO), or she isn’t. If she is committed, then the FBP-ist is committed to an inconsistency, given that (KP) and (NonO) are inconsistent. Alternatively, if the FBP-ist is not committed to (NonO), then there are three options: (i) The ﬁrst is that the FBP-ist is not committed to (NonO) in virtue of being committed to the negation of (NonO), that is to (O)—the claim that all truths are actually known; in symbols: ∀p (p → Kp). But this means that the FBP-ist would then be committed to (FP), given her commitment to (KP) and to the truth of mathematical statements p.¹¹ (ii) Alternatively, the FBP-ist is not committed to (NonO) because (NonO) lacks truth-value. But, for this move to the plausible, the FBP-ist needs to motivate why (NonO) is truth-valueless. Prima facie, (NonO) seems to be a perfectly intelligible and meaningful statement to the effect that there is a truth that nobody knows, and it doesn’t seem to have any of the usual semantic defective features that motivate lack of a truth-value (such as vagueness or indeterminacy). Moreover, even if this move were somehow well motivated, to make sense of truth-value gaps, the FBP-ist would need to introduce a non-standard semantics. But this conﬂicts with the FBP-ist commitment to a standard semantics. In fact, the use of a standard semantics is taken to be a signiﬁcant advantage of Platonism over various nominalist views (see Balaguer [1998]). Finally, (iii) the FBP-ist is not committed to (NonO) because she doesn’t know whether (NonO) is true or false. But, once again, the plausibility of this move would need to be established. It’s unclear what justiﬁes or even motivates, from a FBP-ist perspective, the limitation of such a knowledge claim. What is the reason why the FBP-ist doesn’t know whether there is a truth that nobody knows? Furthermore, if (NonO) is true—as it is typically taken to be, since we are not omniscient after all!—then this FBP-ist move is incoherent. After all, there would be something true, namely (NonO) itself, that nobody knows (according to the FBP-ist). But in this case, the FBP-ist would know that (NonO) is true! In the end, any of the four options just outlined (the ﬁrst horn of the dilemma and the three options that emerge from the second horn) are unacceptable for the FBP-ist. The ﬁrst makes the view inconsistent. The second generates a paradox for FBP. The third requires a major revision, and a substantial weakening, of the view. Finally, the fourth option makes FBP implausible, unmotivated, and possibly incoherent. But why does a commitment to (FP) engender a paradox for FBP? Given the strategy developed by the FBP-ist to obtain knowledge, it looks as though ¹¹ In other words, all of the conditions required to establish (FP) would then be met (see the argument for (FP), p. 253 above).

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from the fact that a certain mathematical theory M is possible (that is, logically consistent), and from the fact that we know that M is possible, we can infer, on the FBP-ist picture, that we know M . This is just what Fitch’s paradox (FP) allows us to infer! After all, for the FBP-ist, if we know that M is possible (i.e., logically consistent), then it’s possible to know M . (For, on this picture, all it takes to know M is to know that M is possible: M ’s logical consistency is sufﬁcient for M ’s truth. And if you know M , then it’s possible to know M .) Moreover, given FBP, if M is logically consistent, then M is true. But this means that, given Fitch’s paradox (FP), M is actually known. In other words, we conclude that M is actually known based only on M ’s truth (in fact, its logical consistency) and the possibility of knowing M . What should we make of the move to the effect that if it’s possible to know a true mathematical theory M , then M is actually known? We can perhaps take this conclusion as an important feature of full-blooded Platonism, and wonder what exactly was paradoxical about Fitch’s result in the ﬁrst place. Prima facie, it does seem counterintuitive to claim that the sheer possibility of knowing a true result is sufﬁcient for actually knowing it. However, as the FBP-ist insists, this is counterintuitive just because we don’t take the ontology of the mathematical world to be a plenitude of abstract entities and relations. Suppose the FBP-ist is right in claiming that every logically consistent mathematical theory is true of some part of the mathematical realm. In this case, on the FBP-ist picture, if it’s possible to know that mathematical theory is true, then we actually know the theory’s truth. This leaves open, of course, the issue of how we can know whether FBP itself is true; that is, whether the mathematical world is indeed the rich plenitude that the FBP-ist takes it to be. However, it’s unclear how the FBP-ist—or anyone else for that matter—could have that knowledge. The sheer consistency of FBP (assuming, for the sake of argument, that it is a consistent view) cannot be taken as a sufﬁcient condition for knowing that FBP is true. Making this move seems to assume a principle to the effect that consistency is sufﬁcient from truth. But this principle is not true in general, and clearly, it isn’t true of contingent domains, such as the actual world. However, perhaps the principle would be true of non-contingent domains, such as mathematics (assuming that mathematics describes non-contingent states of affairs). But even in this case, it’s not clear that the principle would hold, unless the FBP-ist picture of the mathematical universe is true, and hence the plenitude of the mathematical universe would entitle the FBP to move from the possibility of mathematical theories to their truth. The problem is that whether this move is acceptable is precisely the point in question. So, without begging the question, it’s unclear how the FBP-ist could invoke the modal principle under consideration in support of FBP. This suggests that the fact that FBP seems to lead to Fitch’s paradox should be taken not as a virtue, but as a difﬁculty for the proposal. After all, it’s only if

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FBP is true that the inference underlying Fitch’s paradox would seem acceptable. But it’s unclear how one could establish the truth of FBP in the ﬁrst place. One obvious way of establishing FBP’s truth is to invoke the inference that underlies Fitch’s paradox (see (FP), above). But this move, as we saw, begs the question, and it’s unclear how else the truth of FBP could be established. Alternatively, perhaps the FBP-ist could deny the commitment to the knowability principle (KP), by resisting the claim to the effect that if a mathematical theory M is consistent, then it’s logically possible to know that M is consistent. (Let’s call the latter italicized conditional: (C).) As we saw above, (C) is central to the argument that establishes the FBP-ist’s commitment to (KP). If (C) is rejected, the FBP-ist can avoid being committed to (KP), and hence can resist Fitch’s paradox. This move, however, won’t work. If the FBP-ist denies (C), he or she no longer can claim to have solved the epistemological problem of mathematics. After all, if it were logically impossible to know that a consistent mathematical theory M is indeed consistent, it would be logically impossible to know, on the FBP-ist picture, that M is true. And so, it would be logically impossible to know M . As a result, without (C) in place, it is unclear that the FBP-ist would have any epistemological advantage over standard Platonism. Thus, the cost incurred by FBP of positing a luxuriant ontology would come without the accompanying beneﬁt at the epistemological level. In response, the FBP-ist could insist that FBP was only meant to provide an epistemological story to explain the mathematical knowledge that we do in fact have, not to explain how we can know all mathematical truths. Perhaps there are unknowable mathematical truths—that is, truths whose consistency cannot be known (and, hence, which are not actually known). But this doesn’t challenge the success of the FBP-ist response to the epistemological problem of mathematics. After all, what the FBP-ist was after was only to explain how we have knowledge of mathematical objects and relations without having access to them. In particular, the project was only to explain how we have the mathematical knowledge that we have, not to guarantee that we are in a position to know every mathematical truth. Explaining the former is troublesome enough. This response makes perfect sense. It’s always advisable not to overextend one’s goals. However, by making this move, the FBP-ist would be acknowledging an intrinsic limitation in the view—a limitation that emerges in response to Fitch’s paradox. In order to avoid the latter, the FBP-ist tries to reject (C). Assuming for the sake of argument that this can be done, the FBP-ist is now committed to the existence of a consistent mathematical theory for which it’s logically impossible to know that it is consistent. The FBP-ist has here a commitment to something we don’t seem to have reason to believe exists. On the other hand, given the FBP-ist epistemological account, the mathematical theory in question cannot be known (assuming there is such a theory). This is an intrinsic limitation in the epistemological story offered by the FBP-ist.

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As a result, it seems that the FBP-ist faces a troublesome dilemma regarding the epistemology of mathematics in the context of Fitch’s paradox. If the FBP-ist rejects (C), she is committed to the existence of something we have no reason to believe exists. Moreover, it’s unclear then that FBP can offer a complete account of mathematical knowledge, since, by the FBP-ist’s own light, some consistent mathematical theories cannot be known. Alternatively, if the FBP-ist endorses (C), the FBP-ist is committed to (KP), and hence, to Fitch’s paradox. But, as we saw, this is similarly problematic. In light of these considerations, the option of taking Fitch’s paradox as raising trouble for FBP and articulating a different epistemological account for mathematics is a natural move, particularly for those who adopt a standard form of Platonism.

Fi t c h’s Pa r a d o x a n d St a n d a rd Pl a t o n i s m According to standard (or traditional) Platonism, mathematical objects, their properties and relations exist independently of our linguistic practices and psychological processes, and they are abstract (that is, they are not causally active nor are they located in space or time). However, as opposed to what goes on with FBP, the mere consistency of a mathematical theory is not sufﬁcient to guarantee the theory’s truth. After all, to be true, a mathematical theory needs to be more than consistent: it must correctly describe the mathematical objects in question, their properties, and the relations that these objects bear to each other.¹² How can we have knowledge of such objects, properties, and relations according to the traditional Platonist? Mathematical knowledge emerges from (i) the formulation of appropriate comprehension principles about mathematical objects (such as sets, numbers, functions, groups, graphs, topologies, and categories), and (ii) the exploration of consequences that these principles have for the corresponding objects.¹³ With the introduction of such comprehension principles, ¹² I’m assuming here for the sake of argument, together with Platonists, that the mathematical world is ‘‘consistent,’’ in the sense that no inconsistent theory correctly describes that world. Given that Platonists typically adopt classical logic as the underlying logic of mathematical theorizing, this consistency requirement seems prima facie reasonable. The requirement, however, can be dropped if the underlying logic is taken to be paraconsistent. As a result, even inconsistent but non-trivial mathematical theories can be studied. In this case, instead of assuming consistency as the minimal desideratum for a mathematical theory, non-triviality would play that role. That is, as long as not every sentence in the theory’s language is a theorem, the theory can be fruitfully pursued (see da Costa, Krause and Bueno 2007; and Bueno 2002). Of course, non-triviality is only a necessary requirement, since the mathematical theories in question need also to be mathematically interesting: they need to be mathematically tractable and allow for sophisticated, unexpected results (see Azzouni 2004). ¹³ A very elegant and insightful framework in which these comprehension principles can be formulated is provided by Ed Zalta’s object theory (see, e.g., Zalta 1983 and 2000). It should be noted, however, that object theory itself doesn’t commit one to Platonism, since the theory also allows for a nominalist interpretation (see Bueno and Zalta 2005).

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and by drawing consequences from such principles, the standard Platonist has all that is needed to articulate an account of mathematical knowledge. In a nutshell, mathematical knowledge is obtained by identifying the consequences of the comprehension principles that are postulated. Of course, this still leaves it open how we can have knowledge of the comprehension principles themselves. One move that traditional Platonists have explored at this point is to introduce a special notion of intuition, which is meant to work similarly to perception and should explain our knowledge of the comprehension principles directly in terms of this special access to the truth of the corresponding principles.¹⁴ It’s unclear, however, that such a move succeeds, given that presumably one would need to provide an account of how intuition itself functions, and in virtue of which features it is able to yield reliable access to the truth of substantive comprehension principles. Given that mathematical objects are abstract, the intuition in question would have to be this very peculiar cognitive faculty that is somehow akin to perception, but which allows us to apprehend abstract objects without any (empirical) access to them. In the end, explaining how such an intuition supposedly operates is likely to be as problematic as explaining directly the knowledge of the comprehension principles themselves. Whatever is the fate of traditional Platonism on the epistemological front, it’s important to highlight that, in contrast with FBP, modal notions do not play any role in the standard Platonist’s account of mathematical knowledge. Besides needing to provide a reasonable account of how we have knowledge of comprehension principles, the other component of the standard Platonist’s epistemology is to invoke the notion of logical consequence in order to draw consequences from such principles. However, the notion of consequence is typically spelled out by Platonists in model-theoretic, rather than modal, terms. And so, modality does not enter the picture even at this point. It might be objected, however, that unless the comprehension principles are true, the standard Platonist is in no position to claim that he or she does have knowledge of the objects, properties and relations in question. After all, even granting a suitable faculty of intuition, how can the standard Platonist know that the comprehension principles correctly describe the objects under investigation? Presumably, on the standard Platonist’s picture, the existence of mathematical objects and the properties they have are independent of any theorizing about these objects, or any apprehension of such objects via intuition. For the Platonist, mathematical objects are independent of any theorizing about these objects, or of any intuitive access to them, in the sense that these objects would exist and have the properties they have even if no comprehension principles have ever been formulated. But in this case, it’s unclear how the Platonist can guarantee that suitable comprehension principles correctly describe the mathematical objects ¹⁴ This move is often attributed to Gödel. I don’t claim here that this attribution is warranted.

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under study. This seems to require that we have some access to these objects independently of any theorizing, so that the correctness of our judgments about them can be assessed. But how can one apprehend a set, say, without specifying which kind of set it is (for instance, without determining whether it is a Zermelo–Frankel set or a von Neumann–Bernays–Gödel set)? In response, the Platonist could insist that the comprehension principles in question provide the meaning of the mathematical terms they introduce, and so these principles will be true in virtue of their form alone. Let’s grant that this is indeed the case. But the Platonist still owes us an account that establishes the existence of abstract objects independently of the framework determined by the comprehension principles in question. Why should we believe that by simply asserting certain analytically true mathematical claims, we will hence be able to pick out objects, properties and relations that exist independently of such claims? How can we guarantee that the objects that are referred to in these comprehension principles are precisely those that the Platonist takes to exist independently of the principles themselves? The issue here is not how we can know that a principle of the form ‘‘ ‘sets’ refers to sets’’ is true. Principles of this form are obviously true given the meaning of ‘‘refers.’’ The problem is how can we know when we assert that ‘‘there are inﬁnitely many sets’’ which sets we are referring to (say, Zermelo–Frankel sets or von Neumann–Bernays–Gödel sets)? And if we specify which sets we have in mind, how can we know that such sets exist independently of our speciﬁcation of them? This is the sort of independence that is central to the standard Platonist’s picture. It was precisely to overcome this sort of problem that the full-blooded Platonist postulated a dramatic increase in the size of mathematical ontology. If every mathematical object that logically could exist actually does exist, then the truth of a mathematical theory—and, in particular, the truth of appropriate comprehension principles—is guaranteed by the mere consistency of that theory. As discussed above, this move does seem to provide some epistemological help for the FBP-ist. However, as we also saw, this particular use of modal notions in mathematical epistemology opens the FBP-ist to the charge of engendering Fitch’s paradox. And, clearly, this is a difﬁculty that the standard Platonist would gladly like to avoid. As it turns out, however, the standard Platonist does seem to have the resources to resist Fitch’s paradox. On the standard Platonist’s perspective, the possibility of knowing a true mathematical theory is not sufﬁcient for knowing that such a theory is true, given that the standard Platonist doesn’t make the additional assumption (made by the FBP-ist) that the mathematical universe is a plenitude. As a result, the standard Platonist can clearly block Fitch’s paradox, since in general on the standard Platonist view, it’s not the case that: p → (♦Kp → Kp); that is, the sheer possibility of knowing a true mathematical result p doesn’t imply that p is actually known. After all, on the standard Platonist’s view, to know p one would need to show how to derive p from suitable comprehension

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principles. And in order to do this, the sheer possibility of knowing p—that is, the possibility that there is a derivation of p from appropriate comprehension principles—is not enough. An actual derivation of p needs to be provided so that one can claim that p is known. This means that the standard Platonist can easily reject the knowability principle (KP) that underlies Fitch’s paradox; according to this principle, every truth can be known. For the Platonist, however, there might be truths about mathematical objects that we have no means of knowing—that is, proving in a particular formal system. Gödel sentences provide a clear example of this phenomenon: they are true sentences that cannot be proved within a certain formal system. If we can know that such sentences are true, it’s not by proving them in the system. (This is another place where intuition comes in on the Platonist’s picture.) Moreover, there might be truths that are mathematically intractable, and so we can never obtain a proof that establishes them. Of course, in such cases, we cannot generally know that the results in question are true. But, for the Platonist, given the independence of the mathematical objects and their properties from our theorizing about them, nothing precludes the existence of intractable mathematical truths that cannot be known. As a result, if the standard Platonist can reject (KP), Fitch’s paradox cannot get off the ground.¹⁵ (I’ll return, in Section 6 below, to the discussion of the plausibility of rejecting (KP).) So, standard Platonism seems to be in a better position than FBP with respect to Fitch’s paradox. However, for the reasons discussed above, the proposal doesn’t seem to be able to provide a complete account of the epistemology of mathematics, since knowledge of the comprehension principles themselves is ultimately left open. Is there a better alternative?

Fi t c h’s Pa r a d o x a n d Ma t h e m a t i c a l Fi c t i o n a l i s m According to mathematical ﬁctionalism (or nominalism), mathematical objects do not exist (see Field 1980 and 1989). After all, the mathematical ﬁctionalist can resist the only argument for Platonism that doesn’t beg the question against the nominalist: the indispensability argument. This latter argument is meant ¹⁵ Similar considerations also apply to structuralist views about mathematics, such as those articulated by Shapiro (1997) and Resnik (1997). Note that on both views modal notions are not invoked to account for mathematical knowledge. In Shapiro’s case, given his commitment to second-order logic and a general theory of structure, he requires that the relevant mathematical structures be coherent. But just as the supposed consistency of Zermelo–Frankel set theory is not sufﬁcient to guarantee the theory’s truth, the supposed coherence of Shapiro’s theory of structure is not sufﬁcient to guarantee its truth either. In the end, given the additional requirements they introduce for mathematical knowledge, these views also seem to be safe from Fitch’s paradox. Now, it’s of course a much more contentious issue whether structuralism provides a complete epistemology of mathematics. In particular, it’s far less clear whether Platonism sits well with a structuralist epistemology. But this is a point to be explored elsewhere.

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to establish that we ought to be ontologically committed to mathematical entities given that they are indispensable to our best theories of the world.¹⁶ Hartry Field, however, provided a strategy to resist the argument in a particular, but important, case by showing that it’s possible to reformulate a signiﬁcant scientiﬁc theory, Newtonian gravitational theory, without quantiﬁcation over mathematical entities (see Field 1980).¹⁷ After providing detailed support for this view, Field has explored a major consequence of his position for the nature of mathematical knowledge (see Field 1984). One of the advantages of Field’s ﬁctionalism is that it accommodates straightforwardly the difﬁculties that plagued the standard Platonist proposal. As we noted above, it’s unclear how standard Platonists can explain our knowledge of a realm of causally inert and inaccessible mathematical entities. However, if there are no mathematical objects, this puzzle does not even arise, since we would not expect to have access to such non-existent entities (see Field 1989: 252). But another issue about the nature of mathematical knowledge still remains. How should we distinguish a person who has lots of mathematical knowledge (a mathematician, say) from another who hasn’t (a lay person)? Certainly, the ﬁctionalist has to come to terms with this issue. Field acknowledges the point, and he has advanced an interesting proposal. Mathematical knowledge is ultimately either empirical or logical knowledge (Field 1984). The idea is that what distinguishes a person who has lots of mathematical knowledge from another who hasn’t is the knowledge that mathematicians accept certain principles (which is empirical knowledge), and the knowledge that certain mathematical statements follow from others (logical knowledge). This move can be traced back to the suggestions made by early logical-empiricists who also spelled out (mathematical) knowledge in either empirical or logical terms. The distinctive feature of Field’s view is the way in which logic enters into the picture. The standard characterization of logical consequence (namely, Tarski’s 1936), which is crucial for an account of logical knowledge, quantiﬁes over mathematical entities, and thus is not nominalistically acceptable.¹⁸ What is required here is a nominalist account of the notion of consequence, and, more generally, of the applicability of mathematics to metalogic. And it is at this stage ¹⁶ For a thorough discussion of the argument, which was ﬁrst formulated by Quine and Putnam, see Colyvan (2001). ¹⁷ Of course, to refute completely the indispensability argument, the nominalist would have to show that quantiﬁcation over mathematical entities is dispensable in all of our best theories of the world. Field is, of course, aware of that, and what he provided is a program of nominalization rather than a complete nominalization of science. There are serious worries, however, as to how far the program can go, and whether it can successfully provide a nominalization of quantum mechanics (see, e.g., Bueno 2003). ¹⁸ A sentence α is a logical consequence of a set of sentences if and only if in all models in which all elements of are true, α is true as well. Moreover, a set of sentences is logically consistent if there is a model in which all the elements of are true. Models are, of course, abstract objects, typically formulated in set theory.

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that Field’s program leads him to introduce modal notions. Roughly speaking, a sentence B is a logical consequence of A if (A ⊃ B), where ‘’ is a primitive modal operator of logical truth. So, where we were expecting to ﬁnd an account of a purely logical notion, Field puts forward a primitive modal operator. Instead of logic, we have here modality. But how is the proposal spelled out? Field’s idea (1984 and 1991) is to reduce the logical side of mathematical knowledge to two forms of claims:¹⁹ (i) we know that a body of mathematical claims whose conjunction is A is logically consistent, i.e., we know that ♦A; (ii) we know that a claim B follows from a body of claims whose conjunction is A, i.e., we know that (A ⊃ B) (Field 1984: 84–5). In this way, mathematical knowledge is ultimately logical knowledge (see also Field 1991: 5–6 and 11–17). But what is this primitive modal operator? According to Field, it can either be an operator of logical implication or of logical truth—any of them will do the job, and they are interdeﬁnable. For instance, if we take the logical implication operator (‘→’) as primitive, it can be used to deﬁne a one-place operator of logical truth (‘L ’) in the following way: ‘L A’ is deﬁned as ‘(A ∨ ¬A) → A’ (Field 1989: 34; see also Field 1991: 8).²⁰ Moreover, Field stipulates that ‘L ’ obeys the laws ‘L A ⊃ A’ and ‘L (A ⊃ B) ⊃ (L A ⊃ L B)’, and also suggests that ‘L A’ should be taken as a logical axiom whenever A is a logical axiom, as well as taking ‘L A ⊃ L L A’ as a logical axiom (ibid.). Having said that, he then observes: The laws we have just found for ‘L ’ have, of course, a recognizably modal character: they are the characteristic S4 laws for a necessity operator. (Field 1991: 8; italics added; see also Field 1989: 34)

This passage suggests that, instead of having been imposed, the laws for ‘L ’ were discovered by some sort of investigation. But, as we saw, the nominalist has not exactly provided grounds for taking these laws as basic. Stipulating that the modal operator obeys the laws above does not answer the obvious epistemological question about this operator: how do we know that these laws are true? Of course, the fact that they are characteristic features of S4 cannot provide any warrant for the nominalist. After all, S4 has a semantics that is typically formulated in set theory, and thus is ultimately unacceptable for the nominalist, given the commitment to sets. Moreover, as a purely proof-theoretic construction, S4 is still abstract, since it relies on a notion of proof that is not tied to any actual inscription (a sentence is provable in S4 even if no one has ever written down that proof ). Furthermore, as the nominalist would of course acknowledge, the fact that we are all familiar with S4 laws clearly does not provide grounds for the claim that these are the laws that an operator of logical truth should obey. If the ¹⁹ I’ll focus the discussion on the logical aspect of mathematical knowledge, which, in any case, is the key one. ²⁰ As Field points out, ‘A 1 , . . . , An → B’ is deﬁnable from ‘L ’ as ‘L (A 1 ∧ . . . ∧ An ) ⊃ B’. Therefore, it does not matter which primitive one assumes (ibid.).

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nature of mathematical knowledge is to be explained,²¹ the nominalist should tell us why mathematical knowledge has the features it has: why should these S4 laws be taken as basic? In other words, the nominalist cannot simply assume those properties of knowledge claims that should be in fact the result of an account of knowledge. Of course, every theory has its primitive concepts. The problem I am raising is that, philosophically, it is not enough to ground mathematical knowledge on logical knowledge and then, in order to avoid commitment to mathematical objects, spell out logical knowledge in terms of a primitive modal operator—unless we already have an appropriate epistemology of modal notions. After all, the epistemology of modality is by no means less problematic than the Platonist epistemology of mathematics (see Shapiro 1993). In this sense, it is unclear what is gained epistemologically with the move to a primitive modal operator.²² In response, one can argue that since Field is trying to provide an account of mathematical knowledge (at least in the restricted sense of explaining the difference between the knowledgeable in mathematics and the ignorant), he is surely trying to obtain an epistemological advantage over the Platonist. And, prima facie, it does seem easier to explain how we know that certain things are possible, than to explain how we know that abstract objects exist. However, with further reﬂection, it becomes clear that this impression is precipitate. Given that the possibilities to be known are the consistency of certain mathematical theories, a substantial amount of mathematical information is required. Typically, in a model-theoretic approach, the consistency of a mathematical theory T is established by the construction of a model for T , in a suitably stronger theory T . And we know that, if T incorporates arithmetic, to avoid inconsistencies, T has to be stronger than T . Again this is due to a mathematical result (Gödel’s theorem). However, given Field’s rejection of the model-theoretic account of consistency, it is unclear (i) what exactly is meant by the consistency of a mathematical theory, and (ii) how Field’s primitive notion of consistency is related to the model-theoretic one. To be fair to Field, he does indicate one kind of relationship between these two notions of consistency. This is done through the following two principles (Field 1989: 108): (MTP # ) If (NBG ⊃ there is a model for ‘A’), then ♦ A. (ME # ) If (NBG ⊃ there is no model for ‘A’), then ¬♦ A.²³ ²¹ And, of course, providing such an explanation is a central feature of an account of mathematical knowledge. ²² The ontological gain with the introduction of such an operator is clear: it allows the nominalist to avoid commitment to abstract objects when doing metalogic. But the issue we are considering here is epistemological. ²³ ‘NBG’ refers to von Neumann-Bernays-Gödel (ﬁnitely axiomatized) set theory, and ‘♦’ is Field’s primitive notion of logical possibility.

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In Field’s terminology, ‘MTP # ’ stands for model-theoretic possibility, and ‘ME # ’ for model existence. The symbol ‘ # ’ indicates that, according to Field, these principles are nominalistically acceptable. But this is exactly what is unclear. It could be argued that (MTP # ) and (ME # ) are acceptable to the nominalist, since they are modal surrogates for the corresponding Platonistic principles: (MTP) If there is a model for ‘A’, then ♦ A (ME) If there is no model for ‘A’, then ¬♦ A.²⁴ But what is the difference between these two groups of principles? First, the ‘nominalistic’ versions make explicit the particular set-theoretical context where the models for ‘A’ are formulated (namely, in NBG). Second, the antecedents of the ‘nominalistic’ conditionals are ‘necessitated’. But why are these features enough to make the #-formulations nominalistic? Note that in (MTP) and (ME) a particular set-theoretic context is assumed, even if this is not made explicit. After all, the models for ‘A’ referred to in the antecedents are typically formulated in set theory, which provides a broad framework for model construction.²⁵ However, even if we grant that set theory is not assumed, a convenient mathematical framework surely is—for such a framework is required for the formulation of models. In other words, the intelligibility of (MTP) and (ME) depends on the assumption that there is an underlying mathematical framework from the start. So, whether it is set theory or another setting, the difference between the two groups of principles does not lie in the mathematical framework. With regard to the second point, simply by necessitating a Platonistic claim, and putting it in the antecedent of a conditional, one does not make the resulting sentence nominalistically acceptable in general. For example, if (MTP) is not acceptable to a nominalist, why should (MTP∗ ) If necessarily there is a model for ‘A’, then ♦ A be any different? It may be argued that (MTP∗ ) does not entail the existence of a model for ‘A’, since it can be true, even if it is possible that there is no such model. In reply, the same conclusion holds for (MTP), for it can also be vacuously true. However, as opposed to (MTP∗ ), (MTP) is taken to be nominalistically unacceptable. The general point is that simply by reformulating each mathematical statement A in terms of a conditional of the form (MTP∗ ) is not enough to provide a nominalization strategy for mathematics. This strategy resembles the so-called ‘if-thenism’, and, as Field knows, this proposal is quite problematic. For, if nothing satisﬁes the antecedent of such a conditional (which will be the case if nominalism is right), the conditional will be true even if we replace the conditional’s consequent by its negation (see, for instance, Parsons ²⁴ For a discussion, see Field (1989: 103–9). ²⁵ Of course, category theory may also be used at this point.

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1990). In other words, as a translation scheme for mathematics, this strategy is hopelessly inadequate, since for a mathematical statement A, both A and its negation can be translated into a (vacuously) true nominalistic claim—unless we make sure that the antecedent is satisﬁed.²⁶ But perhaps this complaint is uncharitable. Field’s proposal is not to assume a reformulation of mathematical claims via the use of conditionals plus the necessitation of the antecedents. The point is to express that a given result follows from a particular mathematical theory, and this should be done in the object language, without recourse to the model-theoretic notion of consequence. In the case in question, (MTP # ) states, in nominalistic terms, that if it follows in NBG that there is a model for ‘A’, then ♦A. And this formulation is nominalistically acceptable, since whether a particular result follows or not from a given mathematical theory is (i) an ‘‘empirical’’ question, which (ii) does not commit one to the truth of this theory. In this sense, the reference to NBG in (MTP # ) is not essentially different from a claim about a ﬁction, and what holds in it. The problem with this suggestion is that, in a context where we are dealing with abstract objects, we don’t expect ﬁctions to tell us about what is possible and what isn’t. So, on what grounds can the nominalist claim that, because there is a model for ‘A’ in (the ﬁction) NBG, ‘A’ is logically possible? In other words, can the nominalist believe in (MTP # ) without violating nominalism? I think the answer is no. For either the existence of certain models in NBG tells us something about logical possibility (Field’s ‘♦’)—but then it is unclear that the nominalist can bracket away the resulting ontological commitments, since these models are mathematical entities—or these models are not tied to logical possibility, but then (MTP # ) is groundless. In a nutshell, a story has to be told as to why our knowledge that a mathematical theory T is logically consistent is less problematic than our knowledge of T . The problem, as we saw, is twofold. On the one hand, knowledge of consistency seems to depend upon a substantial amount of mathematical knowledge. However, not only is this move closed to the nominalist (since mathematical knowledge then relies on the existence of mathematical objects), but it would also make Field’s proposal circular, since his epistemological account depends on the claim that mathematical knowledge is ultimately knowledge of consistency (and, as we saw, knowledge of consistency depends, in turn, on mathematical knowledge). On the other hand, if knowledge of consistency is not tied to mathematical knowledge, one loses the grip on the notion of consistency used in this context. For how can we recapture the information about the consistency of certain mathematical theories (on nominalistic grounds)? Thus, it is not clear in which respect the ²⁶ Note that Field does assume that the antecedent of (MTP # ) is satisﬁed; or more explicitly, he assumes that ♦NBG (Field 1989: 109). But what I am considering here is (MTP∗ ), and its relationship to (MTP). (MTP # ) will be discussed shortly.

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introduction of the primitive modal operator is helpful for illuminating the epistemological problem of mathematics.²⁷ Note that, according to several authors, primitive modalities are not always problematic. On the modalist’s view, for instance, primitive modal notions are inevitable if we are trying to provide an account of modality itself (see Shalkowski 1994). My point here is that, if we are considering mathematical knowledge, it is not obvious in what respect the move to a primitive modal operator can be of help in the absence of an appropriate modal epistemology. In order for Field to make room for the difference between those who know lots of mathematics and those who know very little, he has to provide an account of modal knowledge. After all, in his picture, that is ultimately what mathematical knowledge amounts to. The problem, however, is that usual accounts of modal knowledge often presuppose mathematical knowledge. For example, David Lewis’s main argument for the existence of modal knowledge rests crucially on the existence of knowledge of mathematical objects. After all, in Lewis’s view, just as we do have mathematical knowledge, despite the fact that we never causally interact with mathematical entities, we also do have modal knowledge, even though we never causally interact with possible worlds (see Lewis 1986: 108–15). Of course, on pain of circularity, this is not a route open to Field. Otherwise, he would be explaining mathematical knowledge in terms of modal knowledge and modal knowledge in terms of mathematical knowledge. Furthermore, a characterization of modal knowledge in terms of conceivability won’t work in this context either. After all, in the case of mathematics, we are often guided by mathematical theories to conceive what is mathematically possible or impossible. But this presupposes that such theories are reliable. And this is exactly what we expect to obtain from an account of mathematical knowledge, rather than something we could simply assume that we have from the outset. Despite the suggestive title of Field’s 1984 paper, ‘‘Is Mathematical Knowledge Just Logical Knowledge?,’’ the strategy he has actually taken (ultimately ‘‘reducing’’ logical knowledge to modal knowledge) only leads to another question ‘‘Is Mathematical Knowledge Just Modal Knowledge?’’ But then, the answer to this question is still left open. Let’s grant, however, for the sake of argument, that the mathematical ﬁctionalist can somehow overcome these worries and provide an account of mathematical knowledge. How would that account bear on Fitch’s paradox? As we saw above, for the mathematical ﬁctionalist, mathematical knowledge emerges from two features: (a) knowing that certain mathematical claims are consistent, and (b) knowing what follows from such claims. This imposes very little constraints on what it takes for us to have mathematical knowledge, and it’s hard to see ²⁷ Note that it won’t help Field’s case to claim that we have a priori knowledge of the properties of the modal operator ‘L ’, since this kind of knowledge is also modal, and we are back to the difﬁculty.

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how the mathematical ﬁctionalist could avoid Fitch’s paradox. After all, since the ﬁctionalist denies the existence of mathematical objects, it’s unclear how he or she could deny (KP), the knowability principle, according to which every truth is knowable. The standard Platonist, as we saw, can deny that principle given the additional constraint he or she imposes on mathematical knowledge: to be taken as knowledge, a mathematical result has to be proven from suitable comprehension principles. But this route doesn’t seem to be open to the mathematical ﬁctionalist, given the latter’s skepticism about the existence of mathematical entities. How can one motivate that certain claims may not be knowable if one denies the existence of objects that such claims are supposed to be about? As we saw, the standard Platonist can insist that, since the mathematical facts do not depend on us, there might be some that, due to their complexity and mathematical intractability, we won’t be able to track. However, no such move seems available to the mathematical ﬁctionalist. But perhaps the mathematical ﬁctionalist can use a different strategy of defense from Fitch’s paradox, and motivate the rejection of (KP). Once a certain body of mathematical principles and a particular logic are adopted, it doesn’t depend on us what follows (or not) from such a body of claims. In particular, if we adopt, say, NBG as the set theory we work with and ﬁrst-order logic as the underlying logic, there will be an inﬁnitude of results for which we won’t know whether they follow or not from the theory. In fact, due to (simple extensions of) Gödel’s ﬁrst incompleteness theorem, inﬁnitely many true arithmetical statements cannot be derived from such a theory. Moreover, consistency claims about mathematics are notoriously hard to settle in general. And since even on the ﬁctionalist view the logical facts are independent of us, it’s perhaps not surprising that it’s quite difﬁcult to decide such consistency claims. As a result, it seems that the mathematical ﬁctionalist could deny (KP) after all, since several truths cannot be known, and avoid Fitch’s paradox. But there’s still a worry here. The ﬁctionalist might be able to motivate the rejection of (KP) in this way, but the risk is to end up embracing skepticism about mathematical knowledge. Recall that, on the ﬁctionalist view, one of the necessary conditions for us to know a certain mathematical theory, say NBG, is for us to know that such a theory is consistent. However, due to Gödel’s second incompleteness theorem, we cannot know that NBG itself is consistent—unless we assume the consistency of a more powerful set theory, whose consistency, in turn, is more questionable than NBG’s. Thus, on the mathematical ﬁctionalist’s account of mathematical knowledge, we cannot claim to have knowledge of the results that follow from NBG, since one of the conditions for mathematical knowledge (namely, knowledge of the consistency of the theory in question) cannot be met. In other words, claiming that we know that NBG is consistent given that, say, we proved its consistency in a stronger theory won’t do. After all, if we have worries about NBG’s consistency, we would have still more worries about the consistency of a theory stronger than NBG that would need to be used

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in order to run such a proof. Since it’s unclear how the mathematical ﬁctionalist could be in a position to know the consistency of signiﬁcant mathematical theories, such as NBG, and given that this is taken to be a requirement for knowledge in mathematics, the position seems to lead to skepticism about mathematics. This would be the case for all substantial mathematical theories, such as set theory, but also arithmetic and analysis, whose consistency cannot be proved in general due to Gödel’s second incompleteness theorem.²⁸ It might be argued that this worry isn’t justiﬁed. Consider a similar move against reliabilism. If knowledge requires reliable cognitive faculties, do we need to know that our faculties are reliable? The reliabilist would insist that the answer is negative, of course. So, why can’t we make a similar claim in the case of NBG? That is, we could claim that there’s no need to know that the theory in which we prove NBG’s consistency is itself consistent. Having the consistency proof for NBG in some theory or other is enough. But this move doesn’t work: it’s not clear that it offers a reliable strategy in the ﬁrst place. Proving the consistency of NBG in a stronger theory would settle the issue regarding NBG’s consistency only if we knew that the stronger theory is itself consistent. After all, it won’t be of much help to prove the consistency of NBG in an inconsistent theory! And if we don’t know that the stronger theory in which we prove the consistency of NBG is itself consistent, we won’t be in a position to assert that NBG is indeed consistent. So, the analogy with reliabilism is not quite apt, given that we are not offered a reliable strategy to begin with. It might be objected that this response doesn’t push the analogy with reliabilism far enough. The point of the reliabilist move is to insist that, in order to have knowledge, there’s no need to have any positive evidence in favor of the reliability of a given process—or, perhaps, all that is needed is the absence of any salient negative evidence for the unreliability of the process. Similarly, in the case of NBG, all that is needed is that NBG be in fact consistent, independently of whether we have any evidence for its consistency. The problem here is that if NBG turns out to be inconsistent, we wouldn’t be able to determine that by simply working with the theory—unless we happen to stumble into an inconsistency. Frege worked for a long time with an inconsistent system without realizing it and if not for Russell, it is unlikely that he would have found the inconsistency. So, not even something as weak as the absence of any salient negative evidence for the unreliability of a process—e.g. no sign that we ²⁸ The mathematical ﬁctionalist could claim that in the case of such substantive mathematical theories, such as various forms of set theory, our knowledge of their consistency is obtained inductively: by not being able (so far) to show that such theories are inconsistent. Of course, as the ﬁctionalist would certainly acknowledge, this doesn’t prove that such theories are consistent, since someone may show that the theories in question, despite the appearances to the contrary, turn out to be inconsistent. Moreover, this move requires the ﬁctionalist to give up the claim that knowledge of the consistency of mathematics is part of mathematical knowledge, since on the inductivist conception, that knowledge is ultimately an inductive matter that cannot be established by mathematical techniques.

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are working with an inconsistent system—is sufﬁcient to support the reliability of that process. The reliabilist may, of course, simply run the risk: if the system in question (NBG in this case) turns out to be inconsistent, despite the appearance to the contrary, so be it. The important point, the reliabilist insists, is that if the system is consistent, it will yield knowledge—whether we know the system’s consistency or not. The problem is that, short of ﬁnding an inconsistency, there’s no way of telling that the system in question is indeed inconsistent. And if the system turns out to be inconsistent, the results obtained with it won’t be reliable. After all, assuming classical logic (which is the logic adopted by the mathematical ﬁctionalist), the negation of all the results obtained would also be derivable. It’s not by chance that the version of mathematical knowledge provided by the mathematical ﬁctionalist explicitly required that we know the consistency of the system we use. Moreover, this reliabilist move puts mathematical ﬁctionalists in the awkward position of having to assert the consistency of NBG without having grounds for that—in fact, without even requiring that they have such grounds! This move violates a basic norm of assertion, according to which you should assert only that for which you have some defeasible justiﬁcation. Of course, there are those who argue that the norm of assertion needs to be even stronger, since the norm ultimately requires knowledge: you’re entitled to assert only what you know (see Williamson 2000a). On this conception, the reliabilist move would be incoherent. But perhaps the reliabilist would claim that the mathematical ﬁctionalist doesn’t need to assert that NBG is reliable. It’s enough that NBG be reliable. But this would be, again, inadequate. Without being able to assert the consistency of NBG and other mathematical theories, the mathematical ﬁctionalist wouldn’t be in a position to make sense of signiﬁcant aspects of mathematical practice, where consistency claims are widespread. Even to make sense of Gödel’s theorems, it’s crucial to be able to assert the consistency of certain mathematical theories. A view that is unable to accommodate this basic feature of mathematical practice is entirely implausible. For these reasons, it’s unclear that the mathematical ﬁctionalist can completely resist Fitch’s paradox. Or, if the proposal manages to resist the paradox, by rejecting (KP), it seems to engender skepticism about mathematical knowledge. This raises the issue as to whether there is some version of ﬁctionalism that can resist the paradox in a well-motivated way. I think there is. Fi t c h’s Pa r a d o x a n d A g n o s t i c Fi c t i o n a l i s m Mathematical ﬁctionalism is a fairly traditional form of nominalism. It denies the existence of mathematical entities, and tries to explain the usefulness of mathematics by highlighting the fact that mathematical theories help to shorten

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derivations—even though in principle quantiﬁcation over mathematical entities could be dispensed with. Now, if mathematical entities don’t exist, then existential mathematical claims are false. So, claims such as ‘‘there are inﬁnitely many prime numbers,’’ which are taken to be true by mathematicians, turn out to be false. The Platonist (whether standard or full-blooded), however, has no difﬁculty in taking such claims literally: since prime numbers exist independently of us, and there are inﬁnitely many of them, the claim above is literally true. This is a signiﬁcant advantage of Platonism. In order to get verbal agreement with the Platonist, the mathematical ﬁctionalist has to introduce a ﬁction operator: ‘‘According to theory M , . . .’’, where M is a mathematical theory suitable to the context in question (see Field 1989). The ﬁction operator will then turn false operator-free existential statements into true mathematical claims: ‘‘According to arithmetic, there are inﬁnitely many prime numbers.’’ However, this means that the syntax of mathematical statements has to be changed, and as a result, as opposed to what happens in the Platonist’s view, mathematical discourse is not taken literally. This is a problem particularly if we want to make sense of mathematical practice, rather than simply construct a philosophical discourse parallel to that practice. Is there a way of preserving the central advantage offered by Platonism—of taking mathematical discourse literally—without the drawback of being committed to the existence of mathematical objects? I think there is. But this means developing a more robust form of ﬁctionalism about mathematics. This version of ﬁctionalism, which I call agnostic ﬁctionalism (for reasons that will emerge in a moment), is meant to provide the advantages of Platonism without its corresponding costs—or, equivalently, the advantages of nominalism without its accompanying troubles. The central idea of agnostic ﬁctionalism is that mathematical practice is always tied to the formulation of certain mathematical principles that characterize and deﬁne the meaning of the terms used in that domain of mathematics. For instance, in order for us to consider whether sets exist, we need ﬁrst to specify which sets we are considering. The term ‘set’ is referentially indeterminate, and we need to disambiguate between various different extensions of this term: each set theory speciﬁes a particular extension, a possible way of specifying the meaning of the term ‘set’, and the corresponding properties that such sets have. The speciﬁcation is done, just as in the case of the standard Platonist view, by providing suitable comprehension principles for sets. Once again, different set theories do that differently: consider, for example, the differences between the axioms for Zermelo–Frankel and von Neumann–Bernays–Gödel set theories, and the fact that the latter, but not the former, quantiﬁes over proper classes besides sets. Each set theory introduces its respective comprehension principles as constitutive for sets, and assumes a particular underlying logic for the theory. (Usually, that logic is left tacit, and it’s typically, but not always, taken to be classical.) Set theorists then draw consequences from the principles that have

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been introduced, and in light of the theorems that have been established, they determine the properties of the sets under investigation. The description so far, although focused on set theory, could be just as easily applied to different branches of mathematics, such as analysis, geometry, or arithmetic. In all of these cases, one needs to determine the objects of investigation by specifying suitable comprehension principles. In the case of analysis, there are many different options, ranging from classical, standard analysis (the usual analysis done in second-order logic) through classical non-standard analysis (such as the systems articulated by Abraham Robinson 1974) to non-classical standard analysis (when standard analysis is developed using some non-classical logic). In each case, suitable comprehension principles need to be speciﬁed and explored, and without such principles, it’s simply not determined which objects we are talking about. Similarly, there are various different geometries, from classical Euclidean geometry to various non-Euclidean geometries. Furthermore, we obtain additional geometrical systems by applying non-classical logics to the previously mentioned systems. Even in the case of arithmetic, we still need to specify the appropriate comprehension principles. If we want to determine the nature of natural numbers (that is, what kind of objects they are), we obtain different answers: one can adopt a neo-Fregean construction of such numbers, or one of many possible reformulations of such numbers in set theory, or the traditional formulation of arithmetic following the Peano axioms. In each case, we obtain a different system, and a different answer to the question of the nature of natural numbers. For convenience, we can simply stipulate that all such formulations are adequate, and show that they are equivalent for certain purposes (basically, the same results about numbers can be obtained in each system). But this doesn’t change the fact that, strictly speaking, each particular formulation of arithmetic provides a different characterization of the nature of numbers. For these reasons, it’s crucial to be explicit about the comprehension principles that are introduced in a particular branch of mathematics. And this may all look very Platonistic. But it isn’t. First, the truth of mathematical statements is now tied to the particular comprehension principles that specify the meaning of the mathematical terms that are employed in a given context. (The context here is determined by the speciﬁcation of the comprehension principles in question.) So, although the agnostic ﬁctionalist can follow the Platonist in taking mathematical statements as literally true, the truth of such statements is always dependent on the comprehension principles in question. Such principles determine a particular, internal context in terms of which the relevant mathematical statements are assessed. Second, no claim is made about the existence of mathematical objects beyond the context determined by the comprehension principles in question. Within the context of such principles, it’s trivially true (constitutively true) that there are mathematical objects of the appropriate sort. But nothing is claimed (or can be claimed) beyond such contexts. After all, beyond the contexts of

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such principles, it’s not clear what is meant by the mathematical terms under consideration: one needs comprehension principles to specify their meaning. It might be objected that the existence of mathematical objects cannot be warranted by the simple introduction of comprehension principles. After all, the truth of such principles cannot be guaranteed simply by the implicit deﬁnitions they offer of the relevant terms. Given that mathematical objects exist independently of any description we may have of them, the introduction of an implicit deﬁnition is not sufﬁcient to guarantee the existence of the corresponding objects. In response, note that the proposal here is not Platonist, and has no intention of guaranteeing the existence of mathematical objects independently of comprehension principles that systematize the discourse about them. Once a comprehension principle is given, we have the resources to talk about certain objects in exactly the same way as once a story is written we can talk about certain ﬁctional characters. That there are objects to talk about in each of these cases emerges from the systematization provided by the principles involved (comprehension principles in the case of mathematics, descriptions of ﬁctional characters in the case of ﬁction). However, from the fact that there are objects to talk about it doesn’t follow that the objects in question exist independently of the context provided by the principles in question. We can certainly talk about Sherlock Holmes, but we don’t take that object to exist. So, the view here allows us to quantify over mathematical objects without the assumption that they exist. This is accomplished by drawing a distinction between the existential quantiﬁer and the existence predicate. As will emerge shortly, this is a perfectly natural distinction, and one that provides a helpful device to avoid overextending one’s commitments (see McGinn 2000, and Azzouni 2004). The usual understanding of the existential quantiﬁer ends up mixing two very different functions of this quantiﬁer (McGinn 2000). One function (let’s call it the quantiﬁcational role) is to indicate that, in a certain domain of discourse, we are considering only some objects, rather than all objects, in that domain. The other function (let’s call it the existential role) is to assert that the objects in question exist. These are, of course, very different functions, and are better kept apart. Otherwise, we would be unable to say things such as: (∗ ) There are objects (such as ﬁctional entities, or frictionless inclined planes) that don’t exist. By restricting the existential quantiﬁer to its quantiﬁcational role, and introducing an existence predicate to play the appropriate existential role, we avoid having a quantiﬁer with these functions mixed. We can also express very easily sentences that have the form of (∗ ): ∃x (Ox ∧ ¬Ex), where ‘O’ stands for the predicate: is an object, and ‘E’ is the existence predicate. The conditions that the existence predicate should meet vary depending on one’s views about existence. This is, of course, not the place to take on such a

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large issue. It will sufﬁce to say that, in order to avoid begging the question against the Platonist, it’s enough to provide sufﬁcient conditions for us to know whether certain objects exist; namely, that we can track the objects in question, that we can interact with them, and reﬁne our access to them. Now, mathematical objects don’t seem to satisfy these conditions. But since these are only sufﬁcient (and not necessary) conditions, this doesn’t mean that the objects in question don’t exist. However, we need not be committed to the existence of mathematical just because we quantify over them either. As a result, the view proposed here is indeed nominalist, at least in the minimal sense that the existence of mathematical objects independently of suitable comprehension principles is never asserted. Whether there are mathematical objects outside such contexts is a question that is not well speciﬁed enough to be answerable. The view is agnostic about this matter. Does agnostic ﬁctionalism engender Fitch’s paradox? I don’t think so. After all, similarly to what happens with Platonism, to have mathematical knowledge, the agnostic ﬁctionalist requires that one produces a suitable proof of the results under investigation from the relevant comprehension principles. So, the mere possibility of knowing a true result is not enough for one to know the result: an actual proof needs to be produced. The agnostic ﬁctionalist can then motivate very naturally the rejection of (KP), the knowability principle. Given some framework for mathematics, there are truths that cannot be known (they cannot be derived in the system). As we saw above, Gödel’s results illustrate such situations very clearly. And without a commitment to (KP), Fitch’s paradox doesn’t get off the ground. Does agnostic ﬁctionalism engender skepticism about mathematical knowledge? That is, is it impossible to have mathematical knowledge on the agnostic ﬁctionalist view? The answer, once again, is negative. As we saw, Field’s mathematical ﬁctionalism seems to engender skepticism about mathematical knowledge given the requirement that we know that mathematical theories are consistent, and the difﬁculty of satisfying this requirement in general. The agnostic ﬁctionalist, however, doesn’t impose such a requirement. After all, we can have knowledge of even inconsistent mathematical theories. In a paraconsistent set theory, for example, we can show that the Russell set, {x : x ∈ / x}, has certain properties and lack others (see, e.g., da Costa and Bueno 2001). The Russell set is obviously an inconsistent object, but it’s not a trivial object—in the sense that it satisﬁes every property. On the agnostic ﬁctionalist view, we can have knowledge of such an object in the same way as we can have knowledge of other mathematical objects: by specifying suitable comprehension principles and a suitable logic, and determining what follows from such principles. Knowledge of consistency is not a requirement. Rather, the agnostic ﬁctionalist avoids triviality: that everything follows from the comprehension principles introduced. But, in an inconsistent context, this can be done by adopting a suitable paraconsistent logic (see da Costa, Krause, and Bueno 2007). In this way, knowledge of inconsistent mathematical objects is not essentially different from knowledge of consistent ones.

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A Di a g n o s i s As we saw, the responses to Fitch’s paradox from traditional Platonism and agnostic ﬁctionalism both rely on the rejection of the knowability principle (KP). But is it really plausible to reject this principle? Prima facie, aiming to know all the truths in a given domain seems to be a perfectly reasonable goal of inquiry. Clearly, such a goal may have informed substantial parts of scientiﬁc and mathematical research, particularly on a realist construal of these activities. In fact, for the realist—either about science or about mathematics—to establish the truth, or the approximate truth, about a certain domain is taken to be the aim of inquiry. This seems to presuppose, at least in principle, the possibility of knowing such truths. So, (KP) may be integral to the realist’s enterprise.²⁹ However, it’s unclear to me whether the goal of knowing all the truths in a given domain is sensible, and so whether the presupposition that it’s possible to know all such truths should be accepted. First, as opposed to the realist’s claim, ²⁹ It might be argued that the aim of inquiry for the realist is not to establish all truths, but to establish all knowable truths. I’ll call the view that takes as the goal of inquiry the establishment of all knowable truths epistemic realism. It turns out, however, that thinking of realism in epistemic terms faces several difﬁculties. First, it turns, by ﬁat, a radical skeptic who denies that we can know any truths into the most successful epistemic realist. For the skeptic could insist that we have established all knowable truths: it just turns out that there are none! Second, leaving the skeptic aside, the restriction to knowable truths makes the epistemic realist view extremely hard to implement. How could the epistemic realist know when he or she has achieved the goal of establishing all knowable truths? Well, when he or she knows that all knowable truths have been established. To know that, the epistemic realist would need to know what are all the knowable truths. But this is precisely what needs to be established in the ﬁrst place. Finally, presumably the epistemic realist would need to provide some principled account to distinguish knowable and unknowable truths. However this is spelled out, it would be hard to distinguish the practice of this epistemic type of realist from that of an anti-realist who denies that there are evidence-transcendent truths. The difference between the two views seems to be only verbal. Both views would agree on the knowable truths (those truths for which we have evidence), and both would dismiss the other truths (the unknowable ones). Although the epistemic realist would claim that there are such unknowable truths and the anti-realist would deny that, nothing in their practice would distinguish what they do when they conduct their research. It should be noted that none of these worries apply to a realist who characterizes her view in terms of searching for the truth (rather than knowable truths), since the view is not formulated in epistemic terms. First, the radical skeptic who claims that no truths can be known wouldn’t be a successful realist, since, as the realist would certainly point out, the fact that we can’t know whether something is true doesn’t entail that there is no independent truth to be found. Whether such independent truths can be known or not, the realist is after them. Second, the realist view can be implemented, by articulating better mechanisms of access to the truth. Since the realist doesn’t know the truth, it’s fallible which of these mechanisms will actually work. But progress is made by devising and testing such mechanisms. (As will become clear shortly, this doesn’t mean that the realist is problem-free by invoking truth as an aim of inquiry. I’ll return to this point below.) Third, the difference between realism and anti-realism is not purely verbal. The realist insists that there are truths that transcend our evidence, but this doesn’t preclude us from trying to forge better evidential mechanisms to ﬁnd the truth. I conclude that the realist is better off characterizing her view in terms of truth, as is typically done, rather than via knowable truths.

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it’s not so obvious that that goal has informed particular research programs. After all, in some cases the guiding principle may not be truth, but something weaker and indistinguishable from truth in the context of observable entities, such as empirical adequacy (see van Fraassen 1980). The same point applies to mathematics, where goals other than truth for mathematical activity have been entertained, such as conservativeness (see Field 1989). Moreover, there are several limits to what can be known, given the boundaries of human cognitive abilities. This ranges from clear limitations to what we can perceive, given the sensory faculties we have, the amount of information we can process, and how reliably we can process that information, even when we extend our capacities using various sorts of instruments (from microscopes to computers). There are also intrinsic limitations to what certain formal systems can yield, and Gödel’s theorems provide a clear example of that. In retrospect, from the point of view of realism, given the impossibility of establishing that we have reached the goal of knowing all the truths about a certain domain, it’s hard to see why (KP) should have been taken even as a plausible, regulative presupposition to begin with.³⁰ It is a reasonable requirement for a goal of inquiry that the participants should be able to know when that goal has been reached. But this is precisely what cannot be established in the case of knowing all the truths about a domain. Now, this raises a difﬁculty for the presupposition of such a realist goal, namely, that it’s possible to know all such truths, as (KP) stresses. After all, what grounds do we have to accept such a possibility, given the well-known limitations to human knowledge? One of the signiﬁcant lessons from Fitch’s paradox is to make this point transparently clear, by showing that (KP) is incompatible, as it should be, with the claim that we are non-omniscient. And so, a natural interpretation of the paradox is to take it as providing a reductio of (KP): there are truths that we cannot know. In the end, we can know many things, but there’s a lot that we can’t. The excursion above in recent philosophy of mathematics drives this point back home. Conclusion Although Fitch’s paradox poses an unexpected problem for accounts of knowledge, it’s possible to resist the paradox’s conclusion depending on the details of ³⁰ Of course, those in the intuitionist tradition have much to say in support of (KP). And, as a result, they try to block Fitch’s paradox by other means. But as I noted in the introduction to this paper, I’m exploring here the implications of Fitch’s paradox to the philosophy of mathematics. To do that, I have assumed, for the sake of argument, that Fitch’s paradox is a genuine paradox, as it seems to be in the context of classical logic. If it turns out that the inference leading to the paradox can be legitimately blocked, for example by changing the underlying logic, then, according to those who advocate such a change, there is no paradox. The issue then becomes how independently wellmotivated are the moves to block the relevant steps in the derivation of Fitch’s result. But this issue, although of course important, is entirely different from the one I’m concerned with in this paper.

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the epistemology one adopts. In particular, the paradox doesn’t seem to threaten a traditional form of Platonist epistemology that is based on the introduction of suitable comprehension principles. Since these principles introduce additional demands on what is required from mathematical knowledge, the traditional Platonist doesn’t face the paradox. However, as we saw, it’s still not clear that the traditional Platonist can successfully account for mathematical knowledge, at least if Platonism is true and mathematical objects do exist independently of our ways of describing them. After all, it’s then unclear how the traditional Platonist can account for knowledge of the comprehension principles themselves. In response to this epistemological worry, a more robust form of Platonism, full-blooded Platonism (FBP), was developed. The view may in principle be better situated to provide an epistemology of mathematics than traditional Platonism, given the increase in the ontology and the corresponding use of modal notions. But, as it turns out, the proposal is particularly susceptible to Fitch’s paradox. In the end, this poses a dilemma for Platonist epistemologies: If these epistemologies provide a full account of mathematical knowledge (such as FBP), they seem to be open to Fitch’s paradox. If they don’t provide a full account of mathematical knowledge (such as standard Platonism), they don’t seem to be open to Fitch’s troublesome conclusion, since they can then reject (KP). But then they would hardly be adequate, given their failure to yield a comprehensive account of mathematical knowledge. In either case, additional work is needed on the Platonist’s front. The situation seems more promising on the nominalist’s side. Although mathematical ﬁctionalism still faces some worries on the epistemological front, it might be able to resist Fitch’s paradox. However, as we saw, the response provided doesn’t seem to be as well motivated as it should be. As an alternative, agnostic ﬁctionalism has been proposed as a way of having the beneﬁts of Platonism (taking mathematical discourse literally) without the costs associated with nominalism (having to yield a parallel discourse for mathematics). In particular, agnostic ﬁctionalism provided a well-motivated response to Fitch’s paradox, by indicating why the knowability principle (KP) shouldn’t be accepted. In the end, we seem to have here a view that seems to offer the best of both worlds.

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17 Performance and Paradox Michael Hand The knowability paradox, or Fitch’s paradox, is thought to threaten semantical (Dummettian) antirealism. Here I suggest that the lesson of the paradox concerns the theoretical location at which to impose the antirealist’s ‘‘epistemic’’ constraints on truth, i.e., on the ‘‘central notion’’ of the antirealist’s meaning theory.¹ In particular, the knowability principle —that every truth is knowable—is not a successful way of capturing the antirealistic insight that truth is epistemically conditioned. I try to sharpen the intuitive feeling that the ‘‘paradoxical’’ Fitch conjunction ϕ & ∼ Kϕ is subject to a sort of ‘‘recognition-transcendent’’ truth that is of no interest for the realism/antirealism dispute, being instead an instance of a tame and well-understood pragmatic (not semantic) phenomenon. I suggest that antirealism’s epistemic constraints on the notion of truth are only mistakenly believed to motivate a global pragmatic constraint on ‘‘veriﬁcation procedures’’ and performances of them—which is the effect of the knowability principle—and are rightly taken to be local semantic constraints on the structures of such procedures. At the end (almost), in view of a not uncommon conviction among antirealists that the appropriate response to the paradox is to seek a restricted knowability principle to enforce the desired epistemic constraints on truth while avoiding paradox, I argue that no restricted knowability principle can serve the crucial meaning-theoretical purpose that antirealists reserve for their notion of epistemic truth. T h e Pa r a d o x The argument of the paradox is well known. The knowability principle (KP), (KP) ∀ϕ (ϕ → ♦Kϕ ), I am much indebted to Jon Kvanvig and Joe Salerno for comments on earlier drafts. ¹ In fact, the central notion of antirealistic meaning theory is not truth (canonical veriﬁability) but canonical veriﬁcation. To keep the current presentation simple, I sometimes talk as if truth itself is the central meaning-theoretical notion. The points I make are easily converted into a stricter statement in terms of canonical veriﬁcation procedures; the sections below on truth’s ‘‘recognition-transcendence’’ are especially pertinent.

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is incompatible with the common-sense claim that some truths remain always unknown, (CS) ∃ϕ (ϕ & ∼ Kϕ ). Thus what might appear to be a principle that is at least philosophically defensible, the thesis that for every truth it is possible that it be known at some time by someone, turns out to commit its proponents to a claim most semantic antirealists would reject (or at least, would deny being committed to simply in virtue of their antirealism), the claim that every truth does eventually become known.² The argument is simple. If ϕ is an always unknown truth, then this very fact, ϕ & ∼ Kϕ, is unknowable, for to know it requires knowing ϕ as well as knowing that ϕ is never known. Thus, if all truths are knowable, then there are no such truths as ϕ & ∼ Kϕ. For a true ϕ, then, we have ∼∼ Kϕ, and opponents of antirealism who countenance classical principles of inference rejoice in concluding Kϕ. Antirealists should have a difﬁcult time swallowing even the weaker conclusion, since it yields the unpalatable result ∀ϕ (∼ Kϕ →∼ ϕ ), i.e., only falsehoods remain forever unknown. Here is a formalization of the argument. The epistemic operator K is subject to two rules of inference: factivity and distributivity. (fact) Kϕ ϕ (dist) K(ϕ & ψ) Kϕ K(ϕ & ψ) Kψ First, note the inconsistency of K(ϕ & ∼ Kϕ ). K(j&∼Kj) dist K∼Kj ∼Kj

K(j&∼Kj) fact

dist Kj ⊥

Call this the key reductio. There are various ways to get from the key reductio to the result of the paradox. For example, ϕ & ∼ Kϕ, taken as an assumption for ∃-elimination on CS, together with ϕ & ∼ Kϕ → ♦K(ϕ & ∼ Kϕ ), from KP by ∀-elimination, yield ♦K(ϕ & ∼ Kϕ). The key reductio then permits the conclusion ♦ ⊥ from those two assumptions by means of obvious rules for ♦. This is as good as ⊥ itself for reductio purposes, which fact can be codiﬁed, for ² Some theorists have taken to calling ∃ϕ(ϕ & ∼ Kϕ) a nonomniscience claim, indicating that they think its denial, equivalent (classically) to ∀ϕ(ϕ → Kϕ), expresses omniscience. The claim that for each truth there is time when it is known—monotonicity of knowledge being assumed—does not entail that there is a time when all truths are known.

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instance, in a rule ♦ ⊥⊥. Now, by ∃-elimination, we have a reductio on the assumptions KP and CS. Fitch himself already observed that such a result is available for any factive distributive operator. Moreover, factive distributive operators are a dime a dozen, and a bit of familiarity with a few lessens considerably any sense of surprise at the result for K.

In d e x i c a l s a n d A s s e r t i ve Se l f - d e f e a t I follow Kaplanian orthodoxy in the semantics of indexicals. Some indexical sentences are such that although they express propositions (relative to contexts) that are not necessarily true, the sentences cannot be asserted falsely. For instance, ‘‘I am here now,’’ whenever asserted, is true. Indeed, ‘‘I am here now’’ is, following Kaplan’s use of the term, analytic. A sentence is analytically true if and only if for every context c, it expresses a true proposition relative to c. On the other hand, ‘‘I am speaking English’’ is true relative to any context in which the speaker says it (as well as any context in which the speaker—actually it’s better to avoid the term ‘‘speaker’’, since the ‘‘speaker’’ of a context need not be speaking; let us say the ‘‘agent’’—says anything in English at all), though it is not analytically true. It is false relative to contexts wherein the agent speaks in a different language, as well as those wherein the agent says nothing. An assertion of ‘‘I am speaking English’’ is self-fulﬁlling —it cannot be asserted falsely—though the sentence is not analytically true. ‘‘I am here now’’ is an extreme example of such a self-fulﬁller, for it manages to be true relative to any context, whether asserted therein or not, simply in virtue of the interaction of its indexicals. Analyticity is a semantical phenomenon. A sentence’s analyticity explains its self-fulﬁllment: because it is true at any context, it is true at any context in which it is said. More pertinent to present purposes are ‘‘I am speaking English’’ and similar examples, e.g. ‘‘I sometimes speak English,’’ which are merely self-fulﬁlling. (S is asserted in a context if and only if the context’s agent asserts S at the context’s time, etc.) From a purely semantical point of view, mere self-fulﬁllment is not interesting: although analyticity is a semantical phenomenon, mere self-fulﬁllment isn’t.³ Every contingent proposition requires for its truth that the world be (nonvacuously) a certain way. Some propositions require for their truth that the world be a certain way concerning what propositions are expressed therein, by whom, when, where, and how. Some propositions may require the world to be some way ³ This distinction between analyticity and mere self-fulﬁllment depends on the semantical principle of the ‘‘totality of character’’ in the semantics of indexicals: a sentence expresses a proposition relative to contexts wherein it is not uttered as well as those wherein it is. See Tsohatzidis (1992); Hand (1993); Tsohatzidis (1993); Zimmerman (1997); Kuppfer (2001, 2004); and Hand (2005: manuscript, still in preparation) for discussion.

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involving that very proposition’s being asserted or denied, or in some way used or not used, somehow or other—most generally, it may require something contingent of the used sentence itself (e.g., that it is eventually known). An asserted sentence may express, relative to its context, a proposition that is true in virtue of having been asserted: ‘‘I sometimes speak in English.’’ An asserted sentence may express, relative to the assertion’s context, a proposition incompatible with the assertion’s having been made in the context: ‘‘I never speak English.’’ The pragmatic interest of this phenomenon is the fact that a given use of a sentence may make the world a way that sufﬁces for the truth (or falsity) of its expressed proposition. A helpful example of this phenomenon can be seen in the relationship of (1a) and (1b). (1a) Je ne te tutoie jamais. (1b) Je ne vous tutoie jamais. Relative to any context, these say the same thing.⁴ Still, (1a) cannot be asserted truthfully, while (1b) can. Although they say precisely the same thing, what they say is about how they say it. (1a) says it one way; (1b) says it a different way. If I put to you what they say, and it is true in the context, then I have not used (1a) to do so. This does not mean that (1a) and (1b) say different things, but only that what they say, they say differently, and what they say concerns how I say things to you, and thus in particular how I say to you what (1a) and (1b) both say. An assertion of (1a) falsiﬁes itself, but at the same time it falsiﬁes (1b) as well (relative to the same context). Assertion of (1b) does not falsify itself, nor does it falsify (1a). They are equivalent, but the world is different if I use (1a) than if I use (1b), relevantly different as to whether members of the pair are true or false. Note that self-fulﬁllment and self-defeat are pragmatic properties of assertions. We may call a sentence itself self-fulﬁlling or self-defeating when, relative to any context in which it is asserted, its assertion self-fulﬁlls or self-defeats. Consideration of such derivative properties of sentences will shed light on the knowability paradox.

Pr a g m a t i c Se l f - d e f e a t i n Ge n e r a l Self-defeat of the above sort is a special case of a more general phenomenon of self-defeat. Consider ⁴ This pair is discussed repeatedly in the papers mentioned in the preceding note. (1a) and (1b) exploit the fact that French has two second-person singular pronouns, the informal ‘‘tu’’ and the formal ‘‘vous’’ (which also serves as the second-person plural, as does ‘‘you’’). Zimmerman 1997 prefers English examples, and introduces two English forms of singular ‘‘you’’: ‘‘youinf ’’ and ‘‘youform ’’. (1a) says, roughly, ‘‘I never address you informally,’’ using the informal pronoun to do so, and (1b) says the same, but uses the formal pronoun.

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(2) ϕ is true but I don’t know that ϕ is true. (2) can be true, yet I cannot know (2). That is, (3) is impossible. (3) I know that [ϕ is true and I don’t know that ϕ is true]. For (3) to be true, I must know that ϕ is true and at the same time know that I don’t know that ϕ is true. (3) is inconsistent in the manner of the paradox: the operator ‘‘I know that . . .’’ is factive and distributive. (2) is interestingly analogous to the earlier self-defeating sentences. Those can be true, but cannot be asserted truthfully. (2) can be true, but I cannot know it. Fitch’s ϕ & ∼ Kϕ can be true, but it cannot be known at all. We might naturally say that (1a) is a self-defeater with respect to assertion, that (2) is a self-defeater with respect to ﬁrst-person knowledge, and that ϕ & ∼ Kϕ is a self-defeater with respect to knowledge simpliciter. John Mackie (1980) gives another helpful example of a self-defeater. Surprisingly (in the way that the knowability paradox is surprising), the principle that every truth can be written truthfully in green ink entails that every truth is written in green ink.⁵ Let ϕ be a truth that is not written in green ink. Then there is the further, conjunctive truth (4), (4) ϕ is true but not written in green ink, which is incompatible not only with every true sentence’s being written in green ink, but with the assumption that every true sentence can be written truthfully in green ink. To write (4) in green ink is to write a false sentence in green ink, because to write it in green ink requires writing ϕ that way, thus falsifying the second conjunct. If, therefore, it is assumed that every truth is writable truthfully in green ink, then there is no truth not written, sooner or later, in green ink. No such difﬁculty arises for the unparadoxical principle that every sentence can be written in green ink, but only the claim that every true one can be written truthfully in green ink, i.e., can be written in green ink and still be true; i.e., that every truth is compatible with its being written in green ink. In light of the foregoing, it is hardly surprising that the principle fails. Some propositions are about (or entail something about) how they are used (in a very broad sense of ‘‘used,’’ meant to include such features as being known, or being written truthfully in green ink, or or indeed pretty much anything about the propositions and expressions of them) so as to ensure that they cannot be used that way and be true. The epistemic self-defeat seen in ϕ & ∼ Kϕ is a species of what we may call generally self-defeat with respect to (the operator) O. When O is factive and distributive, the claim that a non-O truth ϕ is a non-O truth, ⁵ Preserving the analogy with the knowability paradox, take the principle to assert that every true sentence can be written in green ink by someone at some time; similarly the conclusion is that every true sentence is sooner or later written in green ink.

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ϕ & ∼ Oϕ, cannot itself be an O-truth. Some truths are incompatible with being O. That a given truth is non-O is such a one. Since the key reductio for a given operator O hinges only on the factivity and distributivity of O, the fact that ϕ & ∼ Oϕ is subject to the reductio is not always a strictly pragmatic phenomenon. Truth itself is factive and distributive, but the deduction of ∼ Tr(ϕ & ∼ Trϕ ) indicates a semantic fact, that ϕ & ∼ Tr(ϕ ) cannot be true. Any factive distributive operator not equivalent to truth is stronger than truth (since it is factive). When the additional force of O involves pragmatic matters concerning what language-users do, such that ϕ and ∼ Oϕ are compatible, then the key reductio demonstrates the pragmatic self-defeat of ϕ & ∼ Oϕ. Thus we have a recipe for pragmatic self-defeat. Pick any distributive operator A involving pragmatic matters—expressed in English, written in green ink, known—such that ϕ and ∼ Aϕ are compatible, and deﬁne a new operator Oϕ =def ϕ & Aϕ. Then O is factive and distributive. So ϕ & ∼ Oϕ is pragmatically self-defeating with respect to O; thus Fitch self-defeat. The further conclusion ∼ ♦O(ϕ & ∼ Oϕ ) may surprise us when we attribute to all truths the possibility of possessing some stronger (factive, distributive) pragmatic property O that we expect is not possessed by all truths. We simply overlooked the fact that the operator affords a case of Fitch self-defeat. Reﬂection on such examples should minimize the surprise that often attends ﬁrst exposure to a speciﬁc case of Fitch self-defeat. I introduced the assertive self-defeat of ‘‘I am not speaking’’ and (1a) to show that there are non-Fitch forms of self-defeat that are simple and well understood, thereby (I hope) easing the lesson of Fitch self-defeat in particular: for any factive distributive operator O, ϕ & ∼ Oϕ is a Fitch self-defeater. Here is an especially vivid example due to Kvanvig (Chapter 13 of this volume). Let our factive distributive pragmatic operator O be ‘‘it is true and wished for . . . ’’. One might expect that for any truth ϕ, it is possible that ϕ be both true and wished for. Yet again O(ϕ & ∼ Oϕ ) is inconsistent, and ϕ & ∼ Oϕ a Fitch self-defeater: it cannot be true-and-wished-for that ϕ is true but not true-and-wished-for, for this requires both that ϕ is true-and-wished-for and that it is true-and-wished-for that ϕ is not true-and-wished-for, and the latter entails that ϕ is not true-andwished-for. Thus if there is a truth not wished for, then not all truths can be wished for. One more example. Say that knowledge that ϕ is shared when each of at least two individuals sooner or later knows that ϕ. The principle of shared knowability—that for every truth ϕ, it is possible that there are at least two individuals and two times such that one of the individuals knows ϕ at one of the times and the other at the other—is incompatible with the proposition that some truth is never known by more than one person. The principle entails that Wright’s (1987a) example ‘‘Thatcher is a master criminal’’ is false, for if it is true, it is known by at most a single person, the criminal Thatcher herself.

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Our response to the knowability paradox ought to be that this is hardly a big deal. There is nothing perplexing about it, nor, now, surprising. K is just another run-of-the-mill factive distributive pragmatic operator. If all truths are K-able, then there are no truths of the form ϕ & ∼ Kϕ, and so there are no truths ϕ such that ∼ Kϕ.⁶ The notion of pragmatic self-defeat stems immediately from reﬂections on the semantics of indexicals. It is hard to see how this semantic apparatus might be uncongenial to antirealists simply in virtue of their stance on the metaphysical issue separating them from realists. It is prima facie neutral as to whether the involved notion of truth is the realist’s epistemically unconstrained one or the antirealist’s epistemically constrained one. The whole account of self-defeat, and in particular the account of Fitch self-defeat for factive distributive operators, seems so remote from the philosophical issues at stake, and from that lofty perspective so mundane, that the real paradox of the knowability paradox consists in the idea that a single unremarkable instance of Fitch self-defeat would have such overpowering force as is sometimes attributed to it against a metaphysical view on the nature of truth. If antirealism has been previously formulated in such a way that this phenomenon does profoundly undermine it, then antirealists are at worst guilty of an oversight, a mistake in formulation that should be easily correctable in a transparent way. (Is it really the case that it would have been reasonable for Dummett, say, to abandon his project straightaway had the phenomenon of Fitch self-defeat come to his attention in semantic antirealism’s early days?)

A n t i re a l i s t i c Tr u t h a n d Re c o g n i t i o n -t ra n s ce n den ce Antirealism opposes any conception of truth that permits truth to obtain ‘‘recognition-transcendently.’’ Opposition to recognition-transcendence motivates the antirealist’s rejection of the principle of bivalence, because this principle, which accords to every (meaningful) sentence a determinate truth-value, embodies a notion of truth that outstrips any intelligible idealization of our epistemic capacities for determining truth-values. Presumably, not every sentence is ‘‘decidable’’ with respect to some ﬁnite extension of our abilities to recognize truth. Most notably, quantiﬁcation over inﬁnite domains introduces undecidability into our language even if all our epistemically basic sentences are decidable. On this model, epistemically nonbasic sentences are taken to be logical compounds built inductively from basic ones. ⁶ I am not claiming that K is a conjunctive operator obtainable by the recipe above. This would amount to the extremely contentious claim that knowledge can be analyzed as truth plus some further nonfactive property, e.g. belief+justiﬁcation, or belief+justiﬁcation plus an anti-Gettier feature. The plethora of proposals in recent epistemology concerning the last of these inspires no conﬁdence, and some theorists reject the very project of analyzing knowledge in anything like this way.

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Dummett’s famous arguments that provide the ‘‘philosophical basis of intuitionistic logic’’ are taken to show that a theory of meaning adequate for explaining our understanding of our sentences cannot employ an epistemically unconstrained ‘‘central notion.’’ Antirealism thus rejects the classical conception of truth, and denies that distinctively classical rules of inference are valid. The antirealistic idea of a meaning theory for a language is roughly as follows. Sentences that possess truth-values are associated with methods for ascertaining those truth-values, and among these methods are so-called canonical ones. A canonical method for recognizing ϕ’s truth-value, when ϕ is not epistemically basic, is something like an intuitionistic normal form deduction from literals (or at least the notion is thus inspired, and said to have certain attractive features also found in normal form proofs). It is an oversimpliﬁcation to say ϕ’s truth-value depends merely on the values of its immediate subformulas, which depend in turn on theirs, and so on down to epistemically basic sentences, but it will sufﬁce for present purposes. Oversimplifying again (and again because it sufﬁces for present purposes), we may simply posit that the canonical methods of recognizing the truth-values of epistemically basic sentences are null, thus we assume that the basic sentences are all epistemically decidable.⁷ Canonical methods for ascertaining truth-values are meaning-constituting. A language-user’s grasp of ϕ’s canonical truth-recognitional procedure is what her understanding of ϕ consists in.⁸ Other, noncanonical procedures for ϕ are justiﬁed by reference to ϕ’s canonical one, just as nonnormal natural deduction proofs in intuitionistic logic are reducible to normal form ones. A grasp of noncanonical procedures for ϕ is not required for understanding ϕ. In this way an antirealistic theory of meaning avoids truck with a notion of recognition-transcendent truth. When ϕ is true, it has a canonical procedure traversing at most a ﬁnite number of more basic sentences, terminating at a ﬁnite number of epistemically basic ones. Each step of traversal is epistemically constrained —this is the point of barring distinctively classical rules of inference and other objectionably inﬁnitary steps, e.g. allowing the procedure to take a quantiﬁed sentence to an inﬁnite number of instances all of which must be traversed. ⁷ The meaning-theoretical story gets especially complicated concerning conditionals; ϕ → ψ requires an effective method for converting any proof of ϕ into a proof of ψ. Such a method is paradigmatically a proof, from the open assumption ϕ, of ψ. Such a canonical procedure can thus be seen to proceed from the assumption toward the periphery where the basic sentences lie, then changing direction and moving away from the periphery toward the conclusion. Certain difﬁcult meaning-theoretical matters arise in this connection. Because a conditional’s truth seems to depend only on the structures of canonical procedures for its antecedent and consequent, it is hard to see how its truth-value depends differentially on the values of epistemically basic sentences. This complexity is not pertinent to our discussion of ϕ& ∼ Kϕ, so we ignore it. ⁸ Because these methods are determined inductively and parallel the logical structure of their sentences, such a theory of meaning is also able to account for a grasp of an undecidable ϕ as well as a decidable one, thus avoiding the old-fashioned veriﬁcationist’s conclusion that only epistemically decidable sentences are meaningful.

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Such a procedure has the structure of a ﬁnite tree whose undischarged leaves (assumptions) are epistemically basic sentences.⁹ Truth-values of the leaves, by hypothesis none of which requires introduction of a recognition-transcendent notion of truth, ﬁx the truth-values of (some, and in the present case the relevant) nonbasic sentences in the tree by means of antirealistically acceptable, epistemically constrained rules of stepwise traversal (e.g., intuitionistic rules of inference). The involved notion of truth in general is thus epistemically constrained. As we shall see, this need not entail that all true nonbasic sentences are thereby knowable. Let us simplify further. The operator K abbreviates a doubly quantiﬁed sentence, but this complication too is irrelevant to a successful antirealistic response to the knowability paradox. We shall avoid all issues pertaining to the semantical treatment of sentences with K dominant by treating them as basic sentences (!), with various conditions imposed on available distributions of truth-values among them; factivity requires that ϕ be assigned truth when Kϕ is, and distributivity requires that Kϕ and Kψ be assigned truth when K(ϕ & ψ) is. This is all that will matter in the following. The antirealistic meaning theory must involve an epistemically constrained ‘‘central notion.’’ Its associated epistemology of understanding, i.e., its account of what it is to understand a sentence, is a rather ﬁne-grained ‘‘truth-conditional’’ one. To know a sentence’s meaning is to know, in a very particular way, its truth-condition: ϕ is true when there is a canonical procedure for ϕ that issues in the value true. This procedure embodies a partial ordering such that ϕ’s daughters, the sentences immediately preceding ϕ in the ordering, are those a grasp of which are needed for a grasp of ϕ itself. There are only a few ways that ϕ’s daughters may relate to ϕ, and these are the epistemically constrained relationships amounting (in our idealization) to the rules of intuitionistic logic.¹⁰ Our very simpliﬁed picture of a meaning theory ignores the inconclusive, defeasible but nonetheless justiﬁed attributions of truth that are common in empirical discourse. Besides, the whole idea of conclusive reasons for judging an empirical claim true is problematic. The picture also abstracts entirely from the question of the meanings of epistemically basic sentences, not to mention the question of whether there are such sentences in empirical discourse. Moreover it ignores matters pertaining to the meanings of sentences with K dominant, consigning them to the class of basic sentences and capturing only K’s factivity and distributivity by the artiﬁcial device of imposing constraints on assignments of values to basic sentences. Still, this picture manages to embody ⁹ It is especially important to recognize that the complications arising in connection with conditionals and universal quantiﬁcations are irrelevant here, despite the classical appearance of this picture. ¹⁰ Some supplementation of this picture is needed in order to get from basic false sentences to their negations that are needed as open leaves in the tree, but this can be left aside. The proper treatment of falsity is a delicate matter: see my 1999 and Tennant’s 1999.

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everything in an antirealistic theory of meaning that we need in order to explore how the knowability paradox bears upon antirealism. That is, it sheds sufﬁcient light on how an antirealistic theory of meaning avoids objectionably recognition-transcendent truth for us to see why Fitch self-defeat does not threaten antirealism.

Ve r i ﬁ c a t i o n Pro c e d u re s a n d t h e Ac q u i s i t i o n o f K n ow l e d g e It is surely possible that although we all grasp ϕ’s tree, no one ever performs it. Indeed, we can be sure that there are plenty of epistemically decidable sentences that are true, and which we all understand, but whose canonical procedures no one has bothered to perform and never will, nor any indirect procedures for them. This is, on reasonable assumptions, nothing more than the antirealistic rendition of the common-sense claim CS: there are truths never known. Can an antirealist avail herself of this construal of CS? To do so commits her to a potentially troublesome attitude toward these trees. ϕ’s tree exists, in some sense appropriate for the existence claim, and the involved basics are true and false in a distribution of values with respect to which, according to the semantical relationships embodied by its tree, ϕ is true. In other words, these relationships sufﬁcient for ϕ’s truth obtain despite the failure of anyone ever to note that they obtain. Should antirealists grant to ϕ’s tree whatever sort of existence has just been asserted?¹¹ Note that I endorse what a paragraph ago seemed obvious: (i) ϕ is true: it stands in the right semantical relationships to its involved basics and the intervening sentences occurring in its tree, and so its tree exists, just as such abstract entities as, say, natural numbers and functions on them exist; and (ii) ϕ is never known to be true: no one ever performs ϕ’s canonical procedure (its tree) nor any noncanonical procedure for it. There are, then, two distinct issues, belonging to distinct levels of linguistic explanation. First, there is ϕ’s truth, to be characterized meaning-theoretically in terms of the structural features of ϕ’s tree, here conceived as an abstract entity akin to other abstractions that antirealism countenances. Second, and posterior, there are matters of performance of recognitional methods pertaining to ϕ’s truth. The ﬁrst level is a properly semantic one. The second, lower level is a pragmatic one, presupposing strictly semantical notions in its explanations of pragmatic phenomena. ¹¹ There is much to be said concerning this question. See the helpful discussion in Raatikainen. The main advocate of this attitude toward ‘‘veriﬁcation procedures’’ has been Prawitz; Dummett has typically opposed it. (See Raatikainen for references.) After this paper was written, Cesare Cozzo’s 1994 was brought to my attention, wherein he proposes a solution to the knowability paradox that rests, as does the present one, on this conception of these procedures. Comparison with my proposal must await a future opportunity.

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We have seen that an antirealistic theory of meaning avoids recognitiontranscendent truth by imposing epistemically motivated constraints on the structures of canonical procedures, rejecting distinctively classical rules of inference as well as certain other epistemically troublesome ones. Earlier I mentioned as an example of the latter a rule requiring ‘‘traversal’’ of an inﬁnite number of daughters in the determination of a sentence’s value. ‘‘Traversal,’’ however, invites a performance-level reading, while in fact the antirealistic rejection of the rule belongs to the higher, semantical level. To be sure, such a rule is rejected on epistemic grounds, but the antirealist’s meaning-theoretical constraints can be characterized abstractly and belong to the semantic level of the theory. In fact the entire realism/antirealism issue ought to be conceived as belonging to this semantical level.

Un p r o b l e m a t i c “ Re c o g n i t i o n - t r a n s c en d en c e” There are sorts of ‘‘recognition-transcendence’’ of truth to which antirealism need not be allergic. An important aspect of empirical truth is that opportunities to discover it come and go. A truth may have been discoverable at a past time but no longer. ‘‘Whip sneezed downstairs at 3:00 this morning’’ may be true, and had I been awake downstairs at that time I could easily have known it. But I was asleep upstairs, and he sneezes softly. No sneeze-traces remain. Shall we count the truth of the sneeze claim as exhibiting an antirealistically objectionable sort of recognition-transcendence? If so, we are reduced to Ayer’s view that the only true claims about past events are those that we can now establish as true. Antirealists should admit that some truths ‘‘transcend’’ our epistemic capacities merely in virtue of our temporal or spatial remoteness or other incidental, though now permanent, inabilities to perform the procedures needed for discovery. We should hold that our merely missing the opportunity to evaluate the sentence does not count against its truth. Our having missed our only opportunity to perform the tree is not a semantical fact about the tree at all, but only enters our theorizing at the performance level. By present lights, such ‘‘lost opportunity’’ cases are easily explainable in terms of the distinction between our two theoretical levels: decidability, truth, entailment, contingency, logical compatibility, etc., are semantical notions. Lost opportunities are matters of performance, not entering into an account of truth itself. The knowability paradox seems to introduce recognition-transcendence of a related sort. The truth of Fitch’s ϕ & ∼ Kϕ is in some way recognition-transcendent. If antirealism’s opposition to recognition-transcendent truth is correctly codiﬁed in the knowability principle, then the recognition-transcendent truth of ϕ & ∼ Kϕ is thereby assimilated to the realist’s distinctive recognition-transcendent truth

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as opposed to the antirealistically unobjectionable recognition-transcendent truth of lost-opportunity cases. ¹² T h e Re c o g n i t i o n -t ra n s ce n den t Tr u t h o f ϕ & ∼ Kϕ It is common sense that ϕ can be true but never known to be true, never investigated. It is a certainty that ϕ & ∼ Kϕ cannot be known. This is a third, distinctive sort of recognition-transcendent truth: ϕ & ∼ Kϕ can be true, but its truth ‘‘outstrips’’ our epistemic capacities. What sort of recognitiontranscendence is this? Recall (1a): ‘‘Je ne te tutoie jamais.’’ It can be true, but it cannot be asserted truthfully. The mere assertion of it makes the world such that the assertion is false. It can only be true when unasserted. At the heart of this assertive self-defeat is the fact that our language has resources for talking about what is asserted, and these can be slyly exploited to produce such self-defeaters. Fitch self-defeat is parallel: ϕ & ∼ Kϕ can be true, but it cannot be known. It cannot be investigated and found to be true. In particular, no performance of its tree can result in knowledge of its truth. By coming to know ϕ, we make the world be such that ∼ Kϕ is false. At the heart of this epistemic self-defeat is the fact that our language has resources for talking about what is known, and these can be slyly exploited to produce such cases of self-defeat. (1a)’s assertive self-defeat is a ‘‘pragmatic’’ matter posterior to an account of truth. The phenomenon of assertive self-defeat emerges straightforwardly from our semantics together with the fact that our language has the right expressive resources. Fitch self-defeat is likewise a ‘‘pragmatic’’ matter. The phenomenon of Fitch self-defeat emerges straightforwardly from our semantics together with the fact that our language has the right expressive resources. The parallel is close. In both cases, the explanation lies in the language’s possession of sufﬁcient linguistic resources to construct sentences that cannot have a certain ‘‘pragmatic’’ property. The parallel seems to me to tug rather strongly against the thought that antirealism must ﬁnd the Fitch self-defeat of ϕ & ∼ Kϕ troublesome. (And let us not forget Mackie’s claim that Fitch self-defeat ‘‘should be no more surprising than the fact that while I may be saying nothing at t 1 , I cannot say truly at t 1 that I am saying nothing at t 1 ’’. Indeed.) ¹² See Crispin Wright’s suggestive discussion in his 2003. He suggests that the Fitch problem is of the same species of innocuous ‘‘recognition-transcendence’’ as lost-opportunity cases, being matters of ‘‘contingencies of epistemic opportunity’’ as opposed to ‘‘necessities of limitation.’’ He does not explain the assimilation, however. The issue is not simple: in lost-opportunity cases, we take the sentences to be knowable now in virtue of having been knowable then. Fitch’s ϕ & ∼ Kϕ is not a lost opportunity case: there is no then. I agree that the recognition-transcendence of ϕ & ∼ Kϕ is innocuous, but for a reason very different from that of lost opportunities.

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T h e Pe r f o r m a n c e Pr i n c i p l e Rules of inference are the ‘‘building blocks’’ of canonical procedures. Antirealists hold that these must be epistemically constrained. That is, each must be performable in the appropriate sense. It does not follow from this that any procedure built from them is likewise performable under any distribution of values to basics. Due to the presence of linguistic resources permitting talk about these procedures themselves, a language may fail to meet the Performance Principle: if the steps (inference rules) that make up canonical procedures are performable with epistemically basic ingredient sentences—i.e., are performable when they constitute a sentence’s entire procedure—then every (nonbasic sentence’s) procedure built from them is performable too. In other words, if each step in a procedure is performable in the basic case, then the whole procedure is performable. The principle entails, for instance, that if ϕ and ψ have performable canonical procedures, then so does ϕ & ψ. Note that this principle is not a semantical principle. Epistemic constraints on procedural steps are imposed by the antirealist upon the very structures of these procedures—rules of inference embodying a nonepistemic notion of truth are barred. Whether the Performance Principle holds for a language is a further, performance-level issue posterior to imposition of the antirealist’s constraints on truth. The antirealist need not move to this postsemantical level and impose a new pragmatic constraint. Yet the knowability principle is just such a further constraint. The knowability principle says nothing explicit about performability of procedures, but it entails something about them. Given that no truth can be known without having been discovered, and that discovery requires the (abstract, nontemporal) existence of that truth’s canonical procedure, the knowability principle amounts to the Performance Principle. An antirealist’s ready adoption of the knowability principle amounts to acceptance of the Performance Principle without proper attention to the expressive power of the language at hand. This is the mistake underlying the antirealist’s attraction to the knowability principle. For a language containing K, the principle fails. The set of truths with performable procedures is not closed under procedural extension by means of procedural steps that are already antirealistically acceptable. Even conjunction does not always preserve performability. Nonetheless, conjunction preserves ‘‘epistemic decidability,’’ understood as a structural feature of procedural trees themselves. The present language of modal epistemic logic within which we phrase the knowability principle has sufﬁcient resources to formulate a Fitch selfdefeater, ϕ & ∼ Kϕ. Yet we have assumed that antirealism’s epistemic constraints are in force. The knowability paradox demonstrates that these constraints, imposed on the steps of canonical procedures, do not ensure performability

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of all truths’ procedures. Since these procedures already satisfy the antirealist’s epistemic constraints on truth, the Performance Principle’s failure is of no great metaphysical signiﬁcance. That principle is a substantive pragmatic principle not entailed by the antirealist’s epistemic constraints on truth. But is this weaker understanding of the antirealist’s slogan Truth is epistemic (or Truth is epistemically constrained or Truth cannot outstrip our epistemic capacities) really strong enough to be the lesson of the antirealist’s arguments? I must leave the issue for another time. To make the case would require a careful re-examination of versions of those arguments, with an eye toward whether their conclusion can, in light of the Performance Principle’s failure, be rephrased as a claim about the structures of canonical procedures, or whether the conclusion requires formulation in pragmatic terms like knowability and performability. Related antirealistic concerns, involving such ‘‘proof-theoretic’’ matters as harmonious rules of inference and whether a logical constant’s inclusion in a language yields a conservative extension of that language, are prima facie compatible with the spirit of the present proposal. In a nutshell, truth of ϕ consists in there being a properly constrained canonical procedure for ϕ which, under the circumstances, yields for ϕ the value true. Since the language at hand has sufﬁcient expressive resources for the expression of the Fitch self-defeater ϕ & ∼ Kϕ, the Performance Principle fails, and in particular the procedure for the truth ϕ & ∼ Kϕ is not performable.¹³ When the conjunction is true, it is not knowable. This sort of ‘‘truth-transcendence’’ is of an unobjectionable sort, being a pragmatic phenomenon independent of the antirealistic imposition of epistemic meaning-theoretical constraints operative at the semantic level, unlike the recognition-transcendence to which antirealism takes exception and in this way like that of lost-opportunity truths. The very term ‘‘veriﬁcation procedure’’ is a misnomer, when the language under discussion is one for which the Performance Principle fails. Epistemically constrained procedures for truths are not always performable. This is no reason to think that an epistemically constrained central notion of a theory of meaning for the language is unavailable. It is no reason to think that the attendant notion of truth is thereby antirealistically unacceptable for being insufﬁciently constrained epistemically.

A g a i n s t t h e Re s t r i c t i o n St r a t e g y I hold that antirealism’s epistemic constraints do not commit the antirealist to the knowability principle. The knowability paradox is a diverting but irrelevant sideshow in pragmatics. A different response to the paradox is to impose a ¹³ Note that ∼ Kϕ is not epistemically basic. In reality, of course, even the sentence of which Kϕ is an abbreviation is not a basic one.

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restriction on the knowability principle so as to avoid the key reductio by precluding instances of the principle that would otherwise give rise to it, while maintaining the principle’s status as the fundamental epistemic constraint on the antirealist’s theory of meaning. According to this restriction strategy, truth’s epistemicity consists in its knowability under the imposed restriction conditions. A good deal of attention has been focused on the pros and cons of restrictionstrategic proposals.¹⁴ I need not address details of particular ones, since my comments apply to any restricted principle and the role it can play in articulating the conceptual structure of antirealism, that is, whether it can serve as the antirealist’s fundamental epistemic constraint on the antirealistic meaning theory’s central notion. No such proposal can save antirealism, as familiarly understood, from the paradox, for no antirealist who takes knowability to be antirealism’s fundamental epistemic constraint on truth can abide a restriction on it. Rather, the antirealist must formulate an epistemic constraint on truth that is unrestricted, and presumably some restricted knowability principles would be derivative, pragmatic consequences. (The latter approach is not a restriction-strategic one as I am using the term.) It is already clear from the paradox that antirealists cannot hold the unrestricted knowability principle to be true, regardless of the theoretical role they might have liked it to play. There are of course restrictions that successfully prevent emergence of the paradox. On my own view, for instance, performability (and knowability) can be expected to be preserved in most cases, indeed in all cases except self-defeaters. If this is right, then Neil Tennant’s (1997) restricted knowability principle is one I will endorse. I will maintain it as a pragmatic thesis, however, derivative upon across-the-board epistemic constraints at the semantic level. The ‘‘restriction strategy,’’ on the other hand, seeks both to avoid the paradox and to maintain the meaning-theoretical centrality of the new, restricted principle. Both Tennant (1997) and Michael Dummett (2001) are restriction strategists in this sense. For example, in ‘‘The Philosophical Basis of Intuitionistic Logic,’’ Dummett writes, The argument told in favor of replacing, as the central notion for the theory of meaning, the condition under which a statement is true, whether we know or can know when that condition obtains, by the condition under which we acknowledge the statement as conclusively established, a condition which we must, by the nature of the case, be capable of effectively recognizing whenever it obtains. (Dummett 1978: 226–7)

That is, truth, as the realist conceives it, cannot serve as the ‘‘central’’ meaningtheoretical notion, and must be replaced by a property whose presence we ¹⁴ Tennant’s proposal is criticized by Jonathan Kvanvig and myself (1999), to which he responds in Tennant (2001b), and by Timothy Williamson (2000b), to which he responds in Tennant (2001a). Tennant (2002) criticizes Dummett’s proposal, and Berit Brogaard and Joseph Salerno (2002) helpfully investigate both Tennant’s and Dummett’s proposals.

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can always effectively discern. So truth, antirealistically conceived, is knowable—always. This conception of truth, unlike the realist’s, satisﬁes the requirements placed upon it by its role in a theory of meaning. Whatever plays this role in a theory of meaning must be epistemic. Tennant, in discussing Dummett’s disavowal of ‘‘global antirealism’’ in favor of a more piecemeal approach, considers the apparent generality of Dummett’s arguments and writes of the meaning-theoretical principles of Manifestationism, Molecularism, and Compositionality, These are principles of very wide scope, which are not to be thought of as standing or falling depending on the discourse in question. There is therefore certainly enough depth and substance in the antirealist’s initial thoughts about meaning for it to be quite in order to represent him as putting forward a global antirealism. In particular, before he even considers what is peculiar to any one discourse, the anti-realist will be committed to the tenet that truth is in principle knowable. (1997: 50)

He adds later in the same work, concerning the force of the knowability paradox against the antirealism in question (i.e., one that balks at concluding that no truths remain unknown), that the antirealist will still want to maintain that all truths are knowable, even if not actually known. That, after all, is what makes him an anti-realist. (1997: 265)

I read this as indicating not merely that some (restricted or unrestricted) knowability principle is a necessary commitment of antirealism, perhaps following from some deeper antirealistic thesis about truth, but rather that knowability is again the antirealist’s fundamental epistemic constraint on truth. Thus do our restriction strategists take the epistemic character of truth to be the constitutive claim of their antirealism, and to equate truth’s epistemicity with its knowability. For such antirealists, restricting the knowability principle is not an option. They escape the reductio, but forfeit the equation of truth’s epistemic nature with its now restricted knowability. Restriction strategists grant that the paradox demonstrates the untenability of the knowability principle. They fail to realize that a restricted knowability principle cannot serve as the key epistemic feature of truth that antirealism requires, no matter how the restriction strategist proposes to restrict it. If the fundamental issue separating antirealism from realism is whether truth must invariably be subject to some interesting epistemic constraint due to its meaningtheoretical role, then this feature is not knowability, restricted or otherwise. Whatever the relationship of truth to knowability, the latter must be at best an occasional consequence of truth’s being as antirealism distinctively requires. Antirealism’s fundamental epistemic constraint on truth cannot amount to a knowability principle, restricted or unrestricted, though it can be expected to entail some restricted ones. That truth is epistemically conditioned is a claim about the very nature of truth, motivated by its role in the antirealist’s meaning theory. It precludes truth’s

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ever obtaining nonepistemically. Epistemic truth is the notion in terms of which the antirealistic meaning theorist explains what it is to know the meaning of any (meaningful) sentence.¹⁵ If truth’s epistemic conditioning consists in nothing more than the knowability of some but not all truths, then the antirealist’s meaning theory is a lost cause. It endeavors to explain, strictly by means of its epistemic conception of truth with respect to some sentences, what it is to know the meaning of a sentence, even one that falls outside the restriction and whose truth need not be ascertainable. To assert that the epistemic nature of truth amounts to its knowability—that knowability is the fundamental antirealistic constraint on truth—and then respond to the paradox by restricting this knowability defeats the primary meaning-theoretical purpose of the notion of truth. Antirealism imposes epistemic constraints on truth generally, not merely in special cases. A restricted knowability principle has the form ∀ϕ (ϕ → (Rϕ → ♦Kϕ ) ), and if R is relevantly interesting then the realist and antirealist will disagree on the principle. As Tennant observes concerning his own proposed R, ‘‘To claim that every such truth is in principle knowable is still to forswear metaphysical realism’’ (1997: 275). But let condition R be the complement of R among sentences; the restricted principle says nothing at all about truth of R -sentences. It locates the epistemicity of truth in an epistemic feature of R-truth while the strategist hastens to add that not all R -sentences are false. (Indeed, the restriction strategy is designed precisely to permit its proponents this addendum, lest they succumb to the paradox, afﬁrm that no propositions outside the restriction are true, and conclude that no truths remain unknown.) This issue, whether R-truths in particular are all knowable, cannot be the fundamental meaning-theoretical one concerning truth that divides antirealism from realism. If it is, then the realism/antirealism issue is not a general one about truth. If antirealism is a distinctive meaning-theoretical view about the nature of truth, then it must inform us even about the truth of paradox-generating propositions. Antirealists may be unable to produce such a truth, but nonetheless their view must not require them to treat R -propositions so dismissively as to be helpless against the suggestion that these are subject to a nonepistemic notion of truth (or else are all false). To enshrine a restricted knowability principle as the fundamental epistemic constraint on truth reduces the issue to one concerned only with propositions satisfying the restriction, as if antirealism allows the metaphysical chips to fall where they may when it comes to the remainder, not all of which are false. Is nothing left of the dispute when realists and antirealists are asked to comment on the meanings of R -propositions? Are R -propositions ¹⁵ As mentioned earlier, the matter is made somewhat complicated by the fact that not all meaningful sentences possess a truth-value, on the antirealistic account. This does not mean that our understanding of such a sentence is not explained in terms of recognitional abilities regarding truth. Indeed, it is because antirealistic truth is conditioned by our epistemic resources that it is not subject to bivalence.

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simply irrelevant to the dispute? They are certainly not meaningless. A theory of meaning must bring its ‘‘central notion’’ to bear on them despite their peculiarity, yielding (if truth is not itself the central notion) an epistemically constrained notion of truth applicable to them as well as to well-behaved others. The antirealist cannot say that ‘‘Truth is epistemically conditioned’’ expresses only an insight into the nature of R-truth, while refusing to assert that R-truth is all the truth there is. This is not to deny that some restricted knowability principles hold, but such a principle cannot be the fundamental epistemic constraint on truth. The antirealist’s afﬁrmation of it cannot be, as Tennant says, ‘‘what makes him an antirealist,’’ for surely antirealists must deny that truth is ever nonepistemic. And yet restriction strategists do not hold that truth is always knowable. At this point, our antirealist must produce an account of truth’s epistemicity that applies across the board, and must explain by means of this account why failure of the unrestricted knowability principle has no bearing on the antirealist’s insistence that truth is epistemically constrained. To do this is to give up the restriction strategy. T h e Id e a l i s m Pr o b l e m Tennant has drawn attention to another self-defeater with respect to knowledge, here phrased as (I). (I) There are no epistemic agents. (By ‘‘epistemic agents’’ I mean individuals able to know things.) Even the antirealist would like to grant that (I) is possible, but (I)’s truth precludes its being known. It is thus unknowable. Is this ‘‘recognition-transcendence’’ a sort to which the antirealist need object? Must the antirealist judge (I) to be necessarily false? (I call this problem the ‘‘idealism problem’’ because the antirealist wishes to avoid this typically idealistic claim.) The antirealist is not required to hold that (I) is necessarily false, on my view, for truth of (I) would consist in the fact that the canonical procedure for (I) takes, under imagined circumstances, the value true. The fact that there is no one around to perform the procedure does not threaten the antirealist’s implementation of her epistemic constraints, but rather is just another case of pragmatic self-defeat. Just as ‘‘I never speak English’’ can be true only when unuttered, ‘‘There are no epistemic agents,’’ like ϕ & ∼ Kϕ, can be true only when unknown, i.e., when its procedure goes unperformed. Fi n a l C o m m e n t The proposal at hand has some worrisome parts. It assumes that a canonical procedure for a true sentence can be conceived by the antirealist as an abstract

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nontemporal formal object that may never be instantiated and may not even be instantiable. It imports a semantics/pragmatics distinction and applies it in a way that may turn out to be inappropriate. It endangers the Dummettian idea that we manifest our grasp of a truth by recognizing a canonical veriﬁcation of it when presented with one. In these ways and perhaps others it may involve unhappy concessions to realism. Even so, it strikes me (right now, at any rate) as a principled and indubitably antirealistic response to the knowability paradox, which can be seen as a bitter reminder of certain linguistic phenomena that antirealists must not ignore.

18 The Mystery of the Disappearing Diamond C. S. Jenkins

In t ro d u c t i o n : Tw o Pu z z l e s The proof now often known as the ‘paradox of knowability’ was originally formulated in print by Fitch in his 1963 (and previously suggested to him by Church in a referee’s report—see Salerno, Chapter 3 of this volume). It presents a challenge to a claim which is commonly associated with certain forms of global anti-realism. ‘Global anti-realists’, as I use the term, are those who are sympathetic to some version of the claim that reality, in its entirety, is dependent in some signiﬁcant way upon ourselves (usually upon our minds and our ways of thinking). This sort of world view often gives rise to the thought that, because of its minddependent nature, all of reality is epistemically accessible to us. And this thought in turn is often taken to amount to the claim that all true propositions are knowable. The Church–Fitch argument purports to show that, provided we accept only a couple of uncontroversial principles about knowledge, the claim that all true propositions are knowable commits one to the apparently much stronger claim that all true propositions are known. This (for all but the most extreme) is an obviously undesirable consequence. As is customary, in this discussion I shall use ‘Kp’ to mean ‘It is known by some being at some time that p’, hiding the two existential quantiﬁers. I shall begin by presenting what I think is the clearest exposition of the paradox argument, that found in Williamson (2000a) (except that, for the sake of simplicity of presentation, where Williamson uses quantiﬁcation over propositions I shall use schematic letters ‘p’, ‘q’ etc. to stand for arbitrary propositions). I am grateful for helpful comments and suggestions made by Jc Beall, Eline Busck, Lars Gundersen, Jon Kvanvig, Aidan McGlynn, Daniel Nolan, Joe Salerno, Kim Stebel and members of the NAMICONA research centre at the University of Århus who attended a seminar where a version of this paper was presented in August 2005. I would also like to repeat my thanks to those acknowledged in Jenkins (2007) (Nick Denyer, Dominic Gregory, Michael Potter and an anonymous referee), since the work represented there provides a foundation for the current paper in many respects.

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The argument relies on the factivity of knowledge: FACT: (Kp ⊃ p) and the claim that knowledge necessarily distributes over conjunction: DIST: (K(p & q) ⊃ (Kp & Kq) ) to show that what Williamson calls ‘weak veriﬁcationism’: WVER: (p ⊃ ♦Kp) entails what he calls ‘strong veriﬁcationism’: SVER: (p ⊃ Kp). The argument runs: (1) (2) (3) (4) (5) (6)

(K(¬Kp) ⊃ ¬Kp) (K(p & ¬Kp) ⊃ (Kp & K(¬Kp) ) ¬K(p & ¬Kp) ¬♦K(p & ¬ Kp) ( (p & ¬Kp) ⊃ ♦K(p & ¬Kp) ) ¬(p & ¬Kp)

by FACT by DIST from (1) and (2) equivalent to (3) by WVER from (4) and (5)

Note that the only things we rely upon in order to derive (6) are WVER and the uncontroversial principles FACT and DIST. Of course, (6) isn’t quite the same as SVER, but SVER is a trivial consequence of (6) in classical logic. And rejecting classical logic in favour of intuitionistic logic won’t help us much, because it is worrying enough for defenders of WVER if it commits you to (6), the claim that no true proposition is unknown. Other, more radical, departures from classical logic could be used to block the reasoning in one way or another.¹ For current purposes, however, I shall assume that more conservative approaches are to be preferred where available. Two interesting sets of questions have been raised in connection with the Church–Fitch argument. One familiar set, which I shall call ‘the Classic Puzzle’, concerns the relevance of the argument to the question of whether global antirealism is true. It might be tempting to regard the argument as a straightforward refutation of that doctrine. But even those who (like myself) are inimical to global anti-realism tend to feel that such an easy victory is a bit cheap. It seems unlikely that the deep issues which divide the realist and the anti-realist can be settled by a six-line proof. Questions which I take to form part of the Classic Puzzle, and which have received a good deal of attention in the literature, include: • •

Does the Church–Fitch argument really refute global anti-realism? If it does not, is this because the argument is fallacious, or because anti-realists are not in fact committed to WVER? ¹ Thanks to Jc Beall for pressing me to clarify this point here.

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If anti-realists are not committed to WVER, how should their doctrine of epistemic accessibility be expressed?

Although I think all these questions are important and interesting, in this paper I shall be primarily concerned to address a second set of questions, of which Jonathan Kvanvig (2006 and Chapter 13 of this volume) has recently stressed the importance. According to Kvanvig, the preoccupation with the relevance of the Church–Fitch proof to anti-realism has led philosophers to neglect the fact that, regardless of whether one is an anti-realist or not, there is something deeply surprising about the fact that SVER follows from the apparently weaker WVER. There is a sort of modal collapse: the diamond in WVER just disappears by the time we get to SVER. This is the ‘mystery of the disappearing diamond’ of my title. I take the following questions to belong to the second set, which I shall refer to as ‘the New Puzzle’: • • •

Does this apparent modal collapse really occur? If it does not, where does the Church–Fitch proof go wrong? If it does, what satisfying explanation can we give of this collapse?

(Note that, although there is some overlap between the questions in the ﬁrst set and those in the second, the focal points of the two enquiries are rather different.) I am inclined to think that the best response to the Classic Puzzle is that anti-realists are not (or at least should not be) committed to WVER. At worst, they might be committed in virtue of their epistemic accessibility thesis to a different modal claim, one which does not commit them to SVER. In the next section, I shall outline my take on the Classic Puzzle. In the remainder of the paper, I shall try to get clear about what exactly the New Puzzle is and how (if at all) we can solve it. Although I shall begin by adding a question to the set which comprises the New Puzzle, I shall eventually propose that my favoured response to the Classic Puzzle provides resources for addressing the New Puzzle too.

C h a n g e - t h e - c l a s s a n d C h a n g e - t h e - c l a i m² One classic response to the Classic Puzzle is to adopt what’s sometimes known as a ‘restriction strategy’. (The name must be used with some caution, for reasons to be described shortly.) That is, to somehow reformulate the epistemic accessibility claim of the anti-realist so that it does not share the undesirable consequences ² An alternative way of categorizing the various strategies described in this section can be found in Brogaard and Salerno (2004). They describe Edgington’s manoeuvre as a ‘semantic’ restriction strategy, and Tennant’s and Dummett’s as ‘syntactic’ restriction strategies. A semantic/syntactic distinction need not match up extensionally with the distinction I draw between change-the-claim and change-the-class manoeuvres. For instance, it could be proposed that we limit the class of relevant propositions to those which meet some semantic criterion.

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of WVER. A common way of doing this is to say that not all true propositions are supposed by the anti-realist to be knowable, but only some. (‘Restriction strategy’ is a good name for these views.) Tennant (1997, chapter 8), for instance, has suggested that what anti-realists should say is that for every true proposition p such that it is consistent to assume that p is known, p is knowable. This prevents the anti-realist having to accept line (5) of the above proof. Dummett (2001) argues that anti-realists should say that basic propositions are knowable if true, which also prevents acceptance of line (5). Both of these are what I shall call ‘change-the-class’ strategies: they work by changing the class of propositions p for which the anti-realist holds p ⊃ ♦Kp. Another way of thinking about them is as changing the antecedent of WVER from ‘p’ to ‘p & q’ for some q; for instance, Dummett’s ‘q’ is ‘p is basic’. There has been a good deal of debate concerning the acceptability of these change-the-class strategies, which I won’t go into here. I’ll just note a (commonly shared) intuitive response, which is that such strategies can seem rather ad hoc. They have a whiff of monster-barring: we want to adhere to the general principle WVER, but we don’t like the consequences of certain particular instances of it, so we explicitly rule out those instances. The hard work to be done if this is your preferred tack is to dispel this whiff. A different sort of response to the Classic Puzzle is to change, not the class of propositions p for which the anti-realist holds p ⊃ ♦Kp, but rather the claim that the anti-realist makes about all propositions. One can think of this type of strategy as changing the consequent of WVER from ‘♦Kp’ to something else. A ‘change-the-claim’ strategy may or may not amount to something which could sensibly be called a ‘restriction’ strategy. For the resulting anti-realist claim may be strictly weaker than WVER, in which case the term ‘restriction’ would seem appropriate. Alternatively, it may be weaker in some ways (enough to avoid the paradox) and stronger in others, in which case the word ‘restriction’ is potentially misleading. In any case, it is important to note that the species of restriction involved in change-the-claim strategies is logical weakening rather than (the more speciﬁc notion of) delimitation of the class of propositions to which the knowability claim applies. My own preferred strategy with respect to the Classic Puzzle, which I’ll describe in a moment, is a change-the-claim strategy. Another is due to Cozzo (1994), who suggests that an anti-realist understanding of truth need only lead one to accept that if p is true then there is an ideal argument for p. This is intended as a claim that is in some ways weaker than WVER since, on Cozzo’s conception, an ideal argument for a true proposition p may exist without p’s being knowable. Hence it is consistent to suppose that all true propositions have ideal arguments although some are unknowable. But Cozzo’s anti-realist position might also be thought to be stronger than WVER in some respects. For all Cozzo says, it may be that some true proposition is knowable although no ideal argument for it exists. If that is so, then it is

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consistent to suppose that all true propositions are knowable although some lack ideal arguments;³ in which case Cozzo’s anti-realist position has strength which WVER lacks. Edgington (1985) has proposed a change-the-class-and-the-claim strategy. She suggests that the anti-realist should say that for all p, if actually p is true then actually p is knowable, or in symbols: Ap ⊃ ♦KAp. I’ll follow Williamson in calling Edgington’s proposed anti-realist principle WAVER. Edgington’s approach might be described as a ‘restriction’ strategy in two senses, in that WAVER is strictly weaker than WVER (for it is simply WVER applied to truths of the form Ap) and the class of truths mentioned in its antecedent is restricted. (Notice that WAVER is importantly different, with respect to its modal status, from the pure change-the-claim strategy which adopts as its anti-realist principle p ⊃ ♦KAp. The latter is false at some worlds where WAVER, and indeed WVER, are true.⁴ Thus the pure change-the-claim strategy generates something which is in some ways stronger than WVER, while WAVER is strictly weaker.) Using WAVER instead of WVER we block the paradox argument by changing the consequent of (5) to ♦K(A(p & ¬Kp)), which does not contradict (4). I shall not go into the merits of Edgington’s approach, except to mention that two problems with it are that it does not seem quite true to the spirit of anti-realism, because it only claims that certain necessarily true propositions are knowable (actually p is necessary if p is true), and that it might be difﬁcult for beings in other possible worlds to know propositions of the form actually p, since reference from within non-actual worlds back to the actual world is problematic. (See Williamson 2000a, chapter 12, for a discussion of these objections. R¨uckert attempts to respond to them in his 2004, but I query the success of these responses in Jenkins 2007.) My own view concerning the nature of mind-dependence anti-realism suggests two ways in which the global anti-realist might try to avoid a commitment to WVER. Firstly, I argue in Jenkins (2005) that anti-realism should not be characterized in modal terms. I don’t think anti-realism is the view that it’s impossible for there to be a truth which is not appropriately related to our mental lives. Instead, I suggest that anti-realism is (some form of) the view that what it is for something to be true (or, more carefully, what it is for something to be the case) is for it to be appropriately related to our mental lives. You might think the latter implies the former, but there are some who would dispute that. Our assessments of other possible worlds take place within this ³ This supposition is not consistent with the further assumption that some truths are unknown, of course. But that is irrelevant to the question of relative strength which is at issue here. ⁴ Consider some contingent falsehood f and a world w at which f is true, such that there is an omniscient being at w. Everything which is true at w is known (and hence knowable) at w. At w, therefore, WVER is true (and hence so is WAVER). But f ⊃ ♦KAf is false at w. For f is true at w, but it is not possible at w (or anywhere else) for someone to know Af. Af is false—hence unknown—at every world because f is false at our actual world.

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world. So you might think that, although what it is for a sunset to be beautiful is for us to think of it as beautiful, there is a possible world where a sunset is beautiful although we don’t exist at that world and hence don’t think of it as beautiful at that world. The reason that sunset is beautiful, even though what it is for it to be beautiful is for us to think it is beautiful, is that we in our actual world think of it as beautiful when we are assessing this possible world. (See my 2005: 202–4 for further discussion of this point.) However, even if you’re not persuaded by this line of thought (and I myself am not sure how persuasive it should be taken to be), there are reasons to doubt whether anti-realists are committed to the particular modal claim represented by WVER. As I argue in Jenkins (2007), the most the anti-realist is committed to is the view that reality is epistemically accessible. And that, I claim, amounts to the view that: WVER∗ : For any true p, the state of affairs S which at the actual world makes p true is recognizable.⁵ This doesn’t imply that p is knowable, because it may be that at any world where the state of affairs S is recognized, S does not make p true.⁶ In order to motivate this claim, I require that there be a difference (at the actual world) between knowing p and recognizing the state of affairs which renders p true. For if there is not, then there are no possible worlds where people do the latter without doing the former. States of affairs must be, to some extent, extensionally individuated, so as to allow the requisite distance between recognizing states of affairs and knowing the corresponding propositions. That is, the identity of a state of affairs must be somewhat independent of the proposition by which it is picked out, such that, even if recognizing a state of affairs always amounts to knowing some proposition or other, it need not be that recognizing the state of affairs which actually makes p true necessarily amounts to knowing the proposition p. It may be that, at some non-actual worlds, knowledge of a different proposition which picks out the same state of affairs sufﬁces for recognition of that state of affairs. What I recommend in Jenkins (2007) is that, for these purposes, we think of the states of affairs corresponding to true propositions p as extensional as regards the objects referred to in p and the ranges of quantiﬁer phrases in p, but hyperintensional⁷ as regards the properties ascribed to those objects or collections ⁵ I assume a one–one correlation between true propositions and states of affairs which make them true. ⁶ Some claim that the existence of a truthmaker necessitates any proposition it makes true; see e.g. Armstrong (1997). I am not tempted by truthmaker necessitarianism myself. For current purposes, however, I am equally happy to deny that the states of affairs I’m interested in are ‘truthmakers’ in the sense Armstrong has in mind. The idea behind the envisaged notion of truthmaking is that for every true proposition p, there is a state of affairs the existence of which supplies a certain kind of explanation of p’s truth. Explanation does not require necessitation. ⁷ Jenkins (2007) says ‘intensional’, intending the (somewhat old-fashioned) usage on which this simply means non-extensional. Current usage prefers to reserve ‘intensional’ for that which

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of objects. (So that, for instance, the state of affairs which renders true the proposition All cats purr can be represented as an ordered pair consisting of the class of all cats and the property of purring. The range of the quantiﬁer phrase ‘all cats’ is then treated extensionally, since classes are extensionally individuated, but the state of affairs retains a hyperintensional aspect, since the property of purring is not deﬁned by its extension or intension.) As with Cozzo’s proposal, it is not perspicuous to describe my strategy as a ‘restriction’ strategy. Like Cozzo, I change the claim that the anti-realist makes concerning all true propositions, and there are some respects in which the new proposal is weaker and some respects in which it is stronger. It is weaker because recognition of the state of affairs which actually makes p true can occur (at non-actual worlds) without knowledge of p. Hence it may be that every true proposition is such that the state of affairs which actually makes it true is recognizable, even though some of those propositions are unknowable. Hence WVER∗ avoids a commitment to WVER and the Church–Fitch argument does not commit WVER∗ ’s defenders to SVER. But WVER∗ is also stronger in some respects than WVER (for reasons related to those described in note 4 above). Consider some contingent falsehood f and a world w at which f is true, such that there is an omniscient being at w. Everything which is true at w is known (and hence knowable) at w. At w, therefore, WVER is true. But WVER∗ is false. For it is not possible at w (or anywhere else) for someone to recognize the state of affairs which at the actual world makes f true: there is no such state of affairs at any world.⁸

W h a t E x a c t l y i s t h e Ne w Pu z z l e ? So much for the Classic Puzzle. The New Puzzle is supposed to make the Church–Fitch argument interesting regardless of our solution to the Classic Puzzle, and regardless of whether we are realists or anti-realists. So let us now turn our attention to it. Why should realists care about the Church–Fitch argument, given that they can just deny WVER? Well for one thing, realism as I deﬁne it in Jenkins (2005) is consistent with WVER. We can accept that all true propositions are knowable (WVER) while denying that what it is for a proposition to be true is for it to be knowable (i.e. while remaining realists by my lights). We might simply want to combine realism with optimism about our epistemic capabilities. So some is non-extensional and non-hyperintensional. In any case, hyperintensionality is more speciﬁcally what is intended. ⁸ It is for this kind of reason that anti-realists should not regard WVER∗ as necessary; see Jenkins (2007), note 2. (A related principle involving the appropriate relativization to worlds will, however, presumably be regarded as necessary.)

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realists may feel tempted to accept WVER. These realists are obliged to interest themselves in the paradox of knowability, at least to the extent that anti-realist defenders of WVER are so obliged. And even realists who are not tempted to accept WVER might ﬁnd it surprising that a commitment to WVER commits one to SVER, and want to know more about why this is so. Moreover, even if realists try to free themselves of the Church–Fitch problem by denying WVER, realism does not itself supply a good enough explanation of why WVER is false, as Douven (2005: 63–4) has pointed out. Certain kinds of realist view might give us reasons to doubt whether it is feasible, or physically possible, to know certain truths. But taking the diamond in WVER to indicate only metaphysical possibility, realist thinking typically gives us no good reason to reject WVER. In fact, however, the primary way in which I think the Church–Fitch argument is equally important to realists and anti-realists alike is that the argument is simply interesting in its own right and regardless of whether we accept WVER. WVER is a thought-provoking thesis and it is interesting to think about what sorts of consequences it has. Moreover, the appearance of modal collapse between WVER and SVER is somewhat surprising. To quell this surprise we need either an explanation of what’s wrong with the reasoning from WVER to SVER, or else some satisfying explanation of the collapse. It might sound even more impressive to put the explanans this way (as Kvanvig does in his 2006: 55 and elsewhere): WVER and SVER are shown by the Church–Fitch proof to be logically equivalent, so we seem to have lost a ‘logical distinction’ between the two. SVER obviously commits us to WVER, and now Church–Fitch shows us that WVER commits us to SVER too. Given that FACT and DIST are both true, WVER is true iff SVER is. And given that FACT and DIST are both necessary, WVER is true in exactly the same worlds as SVER. However, the claim of logical equivalence is potentially misleading. It is far from obvious that FACT and DIST are both logical truths, but the proof of SVER from WVER uses them⁹ (see Jenkins 2006). Kvanvig acknowledges the non-logical nature of FACT and DIST upfront in his paper for this volume (Chapter 13). However, many remnants of his 2006 formulation remain (sometimes, but not always, qualiﬁed as ‘loose’ or ‘careless’ statements of the problem). And, as we shall see on p. 314 below, the question of whether or not the equivalence is logical is potentially important when considering what is the correct approach to the New Puzzle. So it is worth stressing again, to avert any potential confusion, that logical equivalence is not on the cards. Even Kvanvig’s most ‘careful’ statement of the supposed problem says that there is a ‘lost logical distinction between actuality and possibility with respect ⁹ Williamson (1993) points out that some arguments against WVER and related principles might be constructed without relying on DIST (section IV). But, as he points out, DIST is still required for the formal proof of the equivalence.

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to what is known’ (p. 211). This is very misleading. Setting aside the point about non-logicality, there is clearly a distinction ‘between actuality and possibility with respect to what is known’ as that phrase is most naturally understood. For the known truths are a proper subset of the knowable truths. Formulations like this risk conﬂating something genuinely alarming but not established by Church–Fitch with something established by Church–Fitch but not genuinely alarming. Another alarming-sounding but, on a little reﬂection, misleading formulation of the supposed problem is also offered: that there is ‘no logical distinction between universally knowable truth and universally known truth’. This sounds impressive enough. Except that the only explication offered of this claim is that it means that there is an equivalence (given FACT and DIST) between WVER and SVER. But that is the familiar equivalence which Church–Fitch obviously brings to light. Kvanvig is supposed to be telling us what is so alarming about this equivalence, not just attaching a new label to it. He is claiming to have noticed a ‘lost logical distinction [which] is part of a ﬁrmly entrenched understanding of the nature of the modalities of necessity, possibility and actuality’ (p. 222). But it is far from ‘ﬁrmly entrenched’ that WVER is not equivalent to SVER in the presence of FACT and DIST. Many people think it is. This is a mere quibble, however. One thing it is much more important to note is that what we do not have on our hands here is a case of complete collapse of ‘♦Kp’ into ‘Kp’, where complete collapse would mean that we could replace ‘♦Kp’ with ‘Kp’ wherever we liked. It’s just that (in the presence of FACT and DIST), we can make such a substitution within the consequent of this one conditional: ‘p ⊃ ♦Kp’. It’s thus misleading to say that the Church–Fitch proof threatens us with the conclusion that there is ‘no . . . distinction between actuality and possibility in this way’ (this volume, p. 208), or to say, however ‘carelessly’, that it suggests that ‘possible knowledge implies actual knowledge’ (p. 208). The difference in strength between ♦Kp and Kp is not undermined by the Church–Fitch proof, since there are still plenty of contexts where the latter cannot be substituted for the former, even given FACT and DIST. This fact should already go some way towards lessening any surprise we may feel at learning that WVER commits one to SVER. For this sort of thing, i.e. modal ‘collapse’, or other similar strenthening, within the consequent of one particular conditional, happens in many other cases too—cases where it is clearly nothing to be concerned about. For instance, it’s not surprising that we can replace ♦p with p in the consequent of a material conditional that has a necessarily false antecedent and end up with something that is equivalent to what we started with. Similarly, p ⊃ ♦p is equivalent to p ⊃ p, but I take it this is not surprising either. Neither of these equivalences does anything to ‘threaten the logical distinction between possibility and actuality’ in this area, i.e. the distinction between ♦p and p.

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One might think that the reason these cases aren’t surprising is that we’re dealing in logical truths, and it’s never surprising that two logical truths are equivalent. I’m not sure how good this response is. Insofar as it is ‘surprising’ that you can replace ‘♦Kp’ with ‘Kp’ in WVER without changing the circumstances under which it is true, you might think that it should be equally ‘surprising’ that when you can replace an occurrence of ‘♦p’ with ‘p’ in one of the above contexts without changing the circumstances under which it is true—i.e. without transforming the initial logical truth into something which is not a logical truth. But in any case, there are other parallel cases that do not deal in logical truths. For instance, ¬p ⊃ (p v q) is logically equivalent to ¬p ⊃ q. I’ll call this the case of the disappearing disjunct. I take it that the fact that the consequent here can be strengthened from (p v q) to q without changing the circumstances under which the proposition is true isn’t especially alarming or paradoxical, provided we have a good grip on how the material conditional works. Certainly it does not do anything to threaten the logical distinction between (p v q) and q. Moreover, further examples are available where the equivalence is not even logical (for an even closer analogy with the Church–Fitch case). For instance, given that it is a necessary, but non-logical, truth that all jade is either nephrite or jadeite, ‘(¬X is jade) ⊃ (X is nephrite or X is jadeite)’ is equivalent to, i.e. true at all the same worlds as, ‘(¬X is jade) ⊃ (X is nephrite)’. Again, nothing paradoxical is going on here; the distinction between ‘X is nephrite or jadeite’ and the stronger ‘X is nephrite’ is not under threat just because the latter can be substituted for the former in the consequent of this one conditional without changing the circumstances under which the conditional is true. This is another disappearing disjunct case from which nothing alarming follows. So I think the New Puzzle is best understood slightly differently from the way Kvanvig suggests. He encourages us to be surprised that the strengthening from ♦Kp to Kp in the consequent of WVER makes no difference to the circumstances in which the conditional is true. But it can’t be the mere strengthening that’s surprising, because that sort of thing (strengthening within the consequent of one particular conditional) happens all the time.¹⁰ It must be something else. What is it? I think answering that question will be half the battle of solving the New Puzzle. In fact, I think it will probably be almost all the battle. I think one important question which properly belongs to the New Puzzle is: •

Why are we surprised by the Church–Fitch proof?

¹⁰ Kvanvig’s comments about ‘multiple contexts’ at pp. 213–19 (targeted on Mackie’s ‘syntactic’ approach to lessening the surprise of the Church–Fitch result) might be thought to address this kind of point. But to take them that way would be to say that what the analogies I am drawing here actually do is ﬂag up ‘a more general paradoxicality’. That is to say, it would suggest that one really does ﬁnd the case of the disappearing disjunct, and like cases, paradoxical. Of course, if it’s that easy to come up with paradoxes, paradoxicality is not very worrying. Certainly no one should be looking to revise any of their beliefs just because they throw up the case of the disappearing disjunct.

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Once we know the answer to this question, we will be able to see how to give an explanation of why the proof works which satisfactorily removes this surprise. So l v i n g t h e Ne w Pu z z l e Consider a simple explanation of why the Church–Fitch argument works, and hence of why WVER commits one to SVER: E: Nothing of the form (p & ¬Kp) is knowable. That’s obvious. But given WVER, if something of that form is true, then it is knowable. That’s why, if WVER is true, nothing of the form (p & ¬Kp) is true. And that’s why if WVER is true then it follows that SVER is true too. Kvanvig thinks there is something deeply surprising about the Church–Fitch proof; he must, therefore, think there is more to the surprise the proof engenders than the kind of surprise we can get over just by thinking carefully about how the proof works. The proof is paradoxical, on his view, because we somehow cannot bring ourselves to accept that it works, even after we have seen it and fully understood it. It would be inappropriate to acknowledge the truth of E and just get over it. If that’s right, there must be something inadequate about the simple explanation E. But what? One thing Kvanvig feels is in need of explanation is that the Church–Fitch proof shows WVER and SVER to have the same modal status (this volume, (p. 211). We are supposed to be surprised when the proof forces us to accept this, because according to Kvanvig we would previously have thought that WVER ‘if true, is . . . necessarily true [since] it is a purported implication of a proper understanding of the nature of truth’, whereas SVER is supposed to be contingent. However, on the assumption that WVER is false, there is no reason to suppose it is non-contingent. So realists, at least, might well believe, before thinking about Church–Fitch, that both WVER and SVER are contingently false (contingently because presumably they will think there is a possible world where everything is known, at which both WVER and SVER are true). And they can, of course, continue to hold this after thinking about Church–Fitch. So the proof tells them nothing new about the respective modal status of the two claims. Since Kvanvig wants us to focus on an explanatory challenge which faces everyone alike, realist or anti-realist, this can’t be part of it.¹¹ ¹¹ He may intend to make a similar point when he says (p. 213) that the challenge of Church–Fitch is that ‘we are told that what looks like a modal truth is logically equivalent to what looks like a non-modal truth’. If not, it is unclear what point this passage is making. Obviously, both WVER and SVER are non-modal in the sense that neither is governed by a modal operator. It’s true that one contains a diamond and the other doesn’t, but that is true of a great many obviously and unsurprisingly equivalent pairs (e.g. p → p and p → ♦p).

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So what can it be that is wrong with E? Do we need further explanation of one or more of the claims it draws upon? I don’t think so, but even if we do it could surely be given (see, for example, my comments in footnote 13 below). If my characterization of the New Puzzle is accurate, one thing that the simple explanation E leaves out is an explanation of why, before encountering the Church–Fitch proof, we feel that WVER shouldn’t commit us to SVER. But again, however, there is a simple explanation available. What’s behind this fact is that when someone innocent of Church–Fitch hears the claim All true propositions are knowable, she just doesn’t think about true propositions of the form (p & ¬Kp). These aren’t exactly the kinds of things that spring to mind when this kind of general claim about true propositions is made, if one hasn’t been exposed to the Church–Fitch proof. According to the simple explanation E, it is attention to these cases that reveals why 1 and 2 are equivalent (or rather, why the surprising direction of the equivalence holds). The explanation of why we feel strongly beforehand that WVER shouldn’t commit us to SVER is simply that we haven’t thought hard enough about the full implications of WVER—we haven’t thought about what it will mean for propositions of the form (p & ¬Kp). Even if that’s right, though, it may be thought that there is still something lacking in these simple explanations which prevents our just getting over it. It might be said that they don’t really explain the disappearance of the possibility operator; they just explain why SVER follows from WVER. This is an interesting kind of worry. What counts as a good explanation of a fact does plausibly depend on (among other things) the way the fact is presented. The thought here would be that presenting the implication of SVER by WVER as a case of apparent modal collapse makes the simple explanation offered above inadequate (even if it is a good-enough explanation of the same fact under a different description—e.g. when it is described as the fact that WVER commits us to SVER). I am not unsympathetic to those who think the simple explanation E is explanation enough of the fact in question under either guise, and who think a sufﬁcient explanation of why we are surprised when we ﬁrst encounter the Church–Fitch proof is that we just hadn’t thought hard enough about all the instances of WVER. I am sympathetic, that is to say, to those who think the best response to the New Puzzle is basically an injunction to get over it. At any rate, I don’t think someone who has this view can fairly be dismissed as ‘living in logical denial’ (Kvanvig, this volume, p. 313¹²). ¹² As an aside, I note that on p. 313 Kvanvig shifts between two very different responses to Church–Fitch: on the one hand, denying that it is a paradox, and on the other, accepting that SVER is a necessary truth. The latter response is a proper subspecies of the former. Since Kvanvig mentions myself and Williamson in a footnote during this passage as advocates of the view he is discussing, it is worth pointing out that neither Williamson nor I accepts SVER, let alone accepts that it is necessary. We both think the proof is non-paradoxical for other reasons, which Kvanvig does not describe.

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Be that as it may, what’s required of a satisfactory explanation of a fact can depend, not just on how the fact is presented, but also on who the explanation is for. So, while some may reasonably ﬁnd the simple explanations adequate, others may reasonably demand to hear more before their puzzlement is resolved. For such people, I think more can be said, and that is what I shall try to offer in the rest of this section. First, let me make a quick remark concerning Kvanvig’s suggestion as to what kind of explanation we should be looking for. Kvanvig’s paradigm of a satisfying explanation of modal collapse is the Kripke-style semantic explanation of why what is possibly necessary is not logically distinct from what is necessary (the characteristic commitment of modal logic S5). If we accept a possible worlds semantics for the modal operators (and if we make the required assumptions about accessibility) it becomes obvious why ‘possibly necessarily p’ is equivalent to ‘necessarily p’. By thinking about the semantics for the operators involved in these two propositions we can see why the two are equivalent in strength. Two things are noteworthy about this. One is that it is not clear that an explanation that appeals only (or primarily) to semantics is to be expected or desired in the Church–Fitch case, where the equivalence is, as I stressed above, not establishable through logic alone but only with the aid of substantive nonlogical principles about knowledge, namely FACT and DIST. Compare the jade example on p. 311 above. A big part of the explanation of this equivalence is presumably the empirical fact that there are exactly two kinds of jade: nephrite and jadeite.¹³ The second point to note is that there is another respect in which the modal collapse that characterizes S5 is very different from the modal ‘collapse’ revealed by the Church–Fitch proof. The former, but not the latter, is a complete collapse. That is to say, with S5 we get intersubstitutability in all contexts between ♦p and p, whereas with Church–Fitch we get intersubstituability between ♦Kp and p only within the consequent of a certain conditional. For these two reasons I think that anyone who thinks it appropriate to seek a semantic explanation of the disappearance of the diamond in WVER, along the lines of the possible-worlds explanation of why ♦p implies p, probably has the wrong sort of target in his sights. But what sort of thing should we be looking for? I am tempted to think that some light might be shed on this matter through comparison (not formal, but psychological) between what happens when we think about the Church–Fitch argument and what happens when we think ¹³ Of course, semantics may also be part of, or may help underwrite, the explanation; for instance, the rigidity of ‘jade’ is presumably part of the explanation of why it is necessary that all jade is nephrite or jadeite. But for that matter, semantics may be made part of, or may help underwrite, explanation E—some appeal to the meaning of ‘knows’, in particular, the fact that ‘knows’ is factive, might be included in an explanation of E’s initial claim that nothing of the form (p & ¬ Kp) is knowable.

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about Russell’s set-theoretic paradox. Russell’s paradox proves that a seemingly innocent universal claim (‘Every property determines a set of objects with that property’) leads to contradiction. The Church–Fitch argument proves that a seemingly innocent universal claim (‘Every true proposition can be known’) leads to contradiction of a patent truth (that some true propositions are unknown). Understood correctly, Russell’s ‘paradox’ is not really paradoxical. It’s just a proof that the seemingly innocent claim that every property determines a set is in fact false, and less innocent than it seemed. To see why, we are invited to consider a speciﬁc (and initially unobvious) case. The paradox argument shows that, although being non-self-membered is a perfectly good property for sets to have, there is no corresponding set of non-self-membered sets. It can be correctly called a paradox only insofar as one is tempted by the thought that there really should be such a set, or that the general claim should be true, or perhaps simply that it should not be possible to disprove the general claim by this sort of method. Similarly, the Church–Fitch proof is not really paradoxical. It’s just a proof that the seemingly innocent claim that every true proposition is knowable is in fact false (assuming, that is, that not all true propositions are known), and less innocent than it seemed. To see why, we are invited to consider its implications in a speciﬁc (and initially unobvious) case. The Church–Fitch proof shows that, although there are perfectly good true propositions of the form (p & ¬Kp), there is no possibility of knowing a proposition of this form. It can be called a paradox only insofar as one is tempted by the thought that it really should be possible to know things like this, or that the general claim should be true, or that it should not be possible to disprove the general claim by this sort of method. I think that, in both cases, what happens is that an innocent-sounding (but in fact far from innocent) universal claim gets put forward in a confused attempt to express a similar, genuinely innocent, claim. In the case of Russell’s paradox, what was intended is better captured by the axioms of standard iterative set theory.¹⁴ And, I believe, in the Church–Fitch case, what was intended is better captured by the claim that if p is true then the state of affairs which at the actual world makes p true is recognizable. ¹⁴ It has been suggested to me that, in fact, the notion of set characterized in the axioms of iterative set theory is very different from that which was supposed to be characterized by naïve comprehension. While there are of course some differences, I think there are also enough similarities to allow us to describe the axioms of (say) ZFC as an attempt to capture what was previously supposed to be characterized by naïve set theory, the most important of these being that both are theories of how a number of things can be collected together to form a new entity which is something over and above its members. ( Thanks to Aidan McGlynn for raising this interesting issue in online discussion at . McGlynn here also raises the interesting possibility that something akin to what I think is happening in the case of the Church–Fitch paradox may also be happening when we feel tempted to accept the major premise of the Sorites paradox.)

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In both cases, the paradox arguments seem bafﬂing, and cry out for substantive explanation, if we mistake the apparently innocent claim for the genuinely innocent one. For we can’t understand how the innocent claim we meant to express could have such objectionable consequences as are being attributed to the claim we have actually expressed. However, once we realize that the claim we intended to make is not the same as the one we actually made, it is no longer so bafﬂing that the claim we actually made has these undesirable consequences. In short, then, there is no mystery about the disappearance of the diamond in the Church–Fitch proof. The diamond disappears because WVER has (admittedly unobvious) strength, which it should not have if it is to serve as an expression of one’s commitment to the epistemic accessibility of reality. Note that nothing comparable is going on in the case of the disappearing disjunct (see p. 311 above), which is why we don’t ﬁnd ourselves experiencing any comparable bafﬂement about that case.

C o n c l u d i n g Re m a rk s A couple of further points are worth mentioning. One is that, as is well-rehearsed in the literature, Church–Fitch-style arguments can be constructed using other factive operators on propositions,¹⁵ not just for ‘K’. (Kvanvig, this volume, pp. 214–19, offers a discussion of this point.) For instance, if we assume that every true proposition can be truly believed, we will be able to derive that every true proposition is truly believed. Insofar as these proofs are surprising in the same way that the Church–Fitch proof is surprising, it would be nice to be able to offer the same diagnosis of that surprise. And in some cases this is no problem. For instance, we might propose that when we say that any true proposition can be truly believed, we don’t really intend to express p ⊃ ♦TBp, but rather something like: ‘Every true proposition is such that the state of affairs which makes it true can be correctly taken to obtain.’ However, it is important (and interesting) to note that some comparable claims involving other operators will not be amenable to treatment along quite these lines.¹⁶ Consider, for instance, the claim that any true proposition p is such that the state of affairs S which makes p true can be recognized by me now. For any true proposition q of the form (the state of affairs S which makes p true obtains now and is not now recognized by me), it is not possible that I recognize now the state of affairs T which makes q true. (Because, plausibly, recognizing T ¹⁵ And, indeed, some non-factive operators, such as ‘It is rationally believed that’ (see Mackie 1980). ¹⁶ I am indebted to Kim Stebel for online discussion of this point at .

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now will involve recognizing S now, but q must be false if I recognize S now.) Yet some proposition of the form of q is surely true. Therefore it is not the case that any true proposition p is such that the state of affairs S which makes p true can be recognized by me now. Moreover, the way we show that this is not so is Church–Fitch-like, and might therefore be expected to engender exactly the same kind of puzzlement as other Church–Fitch-style arguments. Yet we won’t be able to explain that puzzlement by saying that what we really meant to express was something else—something involving recognition of states of affairs rather than knowledge of propositions.¹⁷ One thing to note about this kind of case is that any surprise engendered by the new Church–Fitch-style argument can’t have much to do with the prior plausibility of the claim that any true proposition p is such that a proposition expressing the state of affairs S which makes p true can be recognized by me now. For that claim has very little prior plausibility. However, it might seem that there is still a salient similarity with the original Church–Fitch argument concerning the claim that all true propositions are knowable. Admittedly, in this case we are not at all surprised that the claim is false, but we are nonetheless surprised that it should be disprovable in this way. If it is true that this aspect of the new argument is surprising in the same way as the corresponding aspect of the original argument was surprising, then it is not clear how my explanation of the latter surprise could be correct. For in the new case we have exactly the same kind of surprise, but cannot give the same explanation of it. However, I am inclined to think that the surprise engendered in the new case is not entirely comparable to that engendered in the original case. In the new case, I would suggest, all the surprise is engendered by the fact that we just haven’t thought about the problem cases. When we do think about them, and when we understand how the Church–Fitch-like proof works, we understand why the apparently weaker claim actually commits one to the apparently stronger one. And our feelings of surprise should be thereby resolved. We should get over it. I am prepared to grant that, in the original case, some people do encounter a deeper and more robust surprise than this—a kind of surprise that is not fully resolved merely by thinking carefully about the problem cases and understanding the workings of the Church–Fitch proof. That deeper surprise, I think, is to be explained by showing that there is potential for confusion between the claim that all true propositions are knowable and the claim that reality is epistemically ¹⁷ Note that this is not just because what we actually expressed was something involving recognition of states of affairs rather than knowledge of propositions, but because the way I appeal to states of affairs to block the original Church–Fitch argument is by appealing to the ranges of the quantiﬁers appearing in ‘Kp’ (see Jenkins 2007, section IV). There are no quantiﬁers in the new operator, however, so this manoeuvre cannot be used here.

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accessible (the latter of which, we are right to think, does not commit us to thinking that all true propositions are knowable). Some closing comments are perhaps in order to make explicit the relationship between my favoured approaches to the Classic Puzzle and the New Puzzle. In addressing the Classic Puzzle, I offer mind-dependence anti-realists a defence against the charge that the Church–Fitch proof shows their view to be untenable. My response to the Classic Puzzle is to argue that anti-realism is best understood as commitment to a claim that is not prone to the Church–Fitch argument. Whereas my discussion (indeed, any discussion) of the New Puzzle is an exploration of the way we respond when thinking about propositions which are prone to the Church–Fitch argument. The latter, while interesting in its own right, is strictly speaking tangential to a discussion of whether anti-realism is true, if I am right about what that doctrine amounts to. Nevertheless, my discussion of the Classic Puzzle offers us resources with which to address the New Puzzle. For it enables us to argue that, insofar as any deep surprise is engendered by the Church–Fitch proof, that surprise is due to the confusion of WVER with a claim to the effect that reality is epistemically accessible to us. It seems likely that other approaches to the Classic Puzzle will also generate resources for addressing the New Puzzle. For instance, if Edgington is right that the anti-realist epistemic accessibility claim should really be WAVER, it could be argued that the reason we are especially surprised by Church–Fitch is that we mistook WVER for WAVER and hence were surprised when WVER turned out to have consequences which the anti-realist epistemic accessibility claim should not have. Other change-the-class or change-the-claim strategies could be put to similar use. Kvanvig (Chapter 13 of this volume) considers the application to the New Puzzle of Hand’s (2003) response to the Classic Puzzle. This involves pointing out a ‘structural interference’ between the operators and connectives in K(p & ¬Kp): knowing the ﬁrst conjunct entails that the second conjunct is false. Pace Kvanvig, I think something like this may well be (at least part of) a good explanation of the Church–Fitch result for some audiences. (For instance, it could be used to provide explanatory background for the ﬁrst two sentences of my simple explanation E above.) Kvanvig’s rejection of this manoeuvre seems to rest on his not ﬁnding it sufﬁciently analogous to one that can be made in defence of the ontological argument for the existence of God (p. 220–2). This is very puzzling. It might be that the ‘structual interference’ manoeuvre works quite differently, in a way that is not analogous to this—or any—defence of the ontological argument. (For instance, it may function simply as a source of explanatory background for part of E). Note also that Kvanvig thinks that in order for the manoeuvre to work in the same way as the relevant defence of the ontological argument (and hence, apparently, to work at all) it must establish that SVER is necessarily false (p. 221). But no defence of the Church–Fitch equivalence should have to establish that

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SVER is necessarily false. Even once the Church–Fitch reasoning is understood and accepted in its entirety, SVER seems to be contingently false, as I pointed out on p. 312 above. For there are worlds where both WVER and SVER are true. Thus Kvanvig’s understanding of the way the manoeuvre is supposed to work if it works at all seems questionable. Some responses to the Classic Puzzle involve no change-the-class or changethe-claim strategy. Some, for instance, believe that WVER is a fair interpretation of the anti-realist’s epistemic accessibility claim, but deny that the derivation of SVER from WVER goes through in their preferred logic (see, e.g., Beall, Chapter 8 of this volume, for a discussion of logics which block the inference). This view can also provide resources for addressing the New Puzzle: it offers us grounds for denying that the puzzling equivalence is genuine. Others respond to the Classic Puzzle by holding that WVER is a commitment of anti-realism and the equivalence is genuine enough, and that anti-realism is thus undermined. Such a person might choose to respond to the New Puzzle by saying that all we need do is think hard enough about how the Church–Fitch proof works to enable ourselves to get over any initial surprise it generates. We should not be dismissive of this view, even if we think there is in fact more to say in response to the New Puzzle. It is often important to take seriously the possibility that a purported puzzle is no puzzle at all.

19 Invincible Ignorance W. D. Hart

There are truths that cannot be known. For suppose that all truths can be known. Then all truths actually are known. Otherwise, we may suppose for some p that p but it is not known that p. Then it can be known that p but it is not known that p. But when it is known that thus and such, it is known that thus and it is known that such. So it could be known that p and known that it is not known that p. But what is known is true. So it could be known that p and not known that p. But that is a contradiction, and no contradiction can be true. So all truths are actually known. But it is not in fact known whether the number of hairs on Caesar’s body at the instant Brutus’s dagger ﬁrst penetrated was odd. Hence, even if that number could be known, still, as promised, not all truths can be known. Fred Fitch, who ﬁrst published the argument in 1963, credited it to an anonymous referee of a paper Fitch submitted to the JSL in 1945 but never published. The respect due to provenance attaches Fitch’s name to the argument, and we might call the result that there are truths that cannot be known Fitch’s Formula. To call it Fitch’s Paradox is tendentious. But many do, so let us explore the tendency. On ﬁrst acquaintance with the formula, some ask for an example of a truth that cannot be known. The request is a confusion. For to answer it, one should present the requester with a truth, and the requester should not be satisﬁed without a demonstration that it is a truth. But then the requester need only note his or her mastery of the demonstration in order to recognize that he or she now knows the truth presented. Since it is thus actually known, it is out of the question that it cannot be known. In the jargon of the philosophy of mathematics, there could not be a constructive proof of Fitch’s Formula, that is, a proof that displays (or gives an algorithm for displaying) an example of what it claims to exist. A taste for constructive existence proofs is associated with various irrealisms about the objects (like numbers, sets, or vector spaces) mathematics describes, and a hostility to Fitch’s Formula suggests a less than robust realism. A realist about, say, numbers thinks that numbers, and how they are, do not depend on

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what we or anyone else might say, think, prove or know about them. The mode of dependence here denied is modal. So the realist thinks numbers can be as they in fact are even if it cannot be known how they are. Fitch’s Formula is of a piece with a robust realism. How might irrealism’s neurasthenia show up? A simple symptom would be belief that all truths actually are known. The traditional form is belief in a know-it-all god. Religion is often a refuge for the fearful. Since the orthodox can be dangerous when roused, let us confer only with those who admit there is ignorance. Thus we agree amongst ourselves that it is an absurdum that all truths are known. Fitch’s argument reduces the supposition that all truths can be known to this absurdum. So let us look now at the premisses of that argument. These are of two sorts, modal and epistemic. The modal logic involved is pretty minimal. We use what is called Gödel’s axiom, namely, that modus ponens transmits necessity as well as truth. If one is going to take modality seriously, it is hard to see how one would deny Gödel’s axiom. We use what is called the rule of necessitation, namely, that necessitations of theorems are theorems. If we take care that our axioms are necessary and that our other rules transmit necessity, then the rule of necessitation should hold. Gödel’s axiom and the rule of necessitation are all the purely modal machinery we need. On the epistemic side we use the premiss that what is known is true. That premiss is about as close to analytic as anything remotely philosophically interesting gets; a belief does not count as knowledge unless it is true. It would be heroic, even quixotic, to deny it. The other purely epistemic premiss we use is that when a conjunction is known, so are its conjuncts. This premiss is an instance of the claim that all logical consequences of what is known are known, and even a little familiarity with sophisticated deductive theories is enough to confute the general claim (though an interesting sorites remains here). But the instance we use is so transparent. If an explicit conjunction is known, it should be understood and recognized as a conjunction, whence commitment to the conjuncts seems inevitable. It seems more rational to afﬁrm that conjuncts of the known are known and to deny that all truths can be known than the reverse. That what is known is true and that conjuncts of the known are known are all the purely epistemic machinery we need. Is there a refuge left for the irrealist? Typing is a perennial favorite in solving, or suppressing, paradoxes. As long ago as 1903, Russell considered typing in The Principles of Mathematics to deal with paradoxes like his own and the liar. Zermelo’s (1908) revision of set theory (Zermelo 1967), seen through the axiom of foundation as the iterative hierarchy,¹ pictures sets as arranged in ascending ranks. Soon after Ramsey in 1925 simpliﬁed Russell’s theory of types so layering ¹ Boolos, G. 1998. ‘‘The Iterative Conception of Set,’’ in Logic, Logic, and Logic. Cambridge, MA: Harvard University Press: 13–29.

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only sets in types is to treat paradoxes like Russell’s (Ramsey 1960: 1–61), Tarski in 1931 used levels of languages to handle the liar paradox (Tarski 1956: 152–278).² Tyler Burge (1984: 83–118)³ and Anil Gupta (1988–89: 227–46) continue the device of stratifying language to proscribe semantic paradoxes. Alonzo Church in his referee’s report on the paper Fitch submitted to the Journal of Symbolic Logic in 1945 suggests typing knowledge as a way around Fitch’s Formula.⁴ Bernard Linsky proposes a type-theoretic reply to Fitch in his contribution to this volume (Chapter 11). In this spirit we might try typing knowledge. The basic idea is of a bottom type 0 of propositions (or sentences) in which knowledge (or the verb ‘‘to know’’) is absent, and then for each type n of propositions (or sentences) to be followed by a type n + 1 of propositions (or sentences) expressing knowledge (or predicating the verb ‘‘to know’’) of propositions (or sentences) of type n. Knowledge of any type would still be true, and knowledge of any type of a conjunction would be knowledge of that type of its conjuncts. The crucial difference would be that a proposition (or sentence) of type n could always be known but only of type n + 1. In private correspondence Hans Kamp and Charles Parsons showed independently that propositions (or sentences) not in fact known are consistent with this system. We have just imagined a system in which knowledge fractures into knowledge of type 1, knowledge of type 2, and so on. In this last sentence we used arabic numerals like proper names for types of knowledge. In the preceding paragraph we also used, in addition to these speciﬁc numerals, the predicate ‘‘is a type’’ and variables of quantiﬁcation ranging over types. If only because there is no clear end to the conﬂux of types, it is at least very hard to see how we could describe them without that predicate and those variables. So a forthright theory of types of knowledge should include, in addition to names for types, a predicate speciﬁc to types and variables of quantiﬁcation ranging over them. In such a forthright system, Fitch’s Formula recurs. Suppose that any truth of type n can be known at type n + 1. We will show that any truth actually is known at some type. For otherwise we may assume that p but it is known at no type that p. The assumption that p but it is known at no type that p must be of some type t; that is the point of type theory. Then it can be known at type t + 1 that p but it is known at no type that p. Thus, as before, it is possible that it is known at type t + 1 that p and it is known at type t + 1 that it is known at no type that p. So, again as before, it is possible that it is known at type t + 1 that p and it is known at no type that p. Hence, by universal instantiation, it is possible that it is known at type t + 1 that p and it is not known at type t + 1 that p. But ² I defend Russell’s and Poincaré’s assimilation of the set theoretic and semantic paradoxes which Ramsey separated in my 1984: 193–210. ³ In ‘‘Frege on Truth,’’ Burge says, ‘‘The notion of truth cannot be adequately represented in terms of a truth predicate that lacks some sort of stratiﬁcation.’’ See his 2005: 131. ⁴ Private communication from Joseph Salerno.

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contradictions still cannot be true. So any truth actually is known at some type. So, given the actuality of ignorance, some truths cannot be known at any type. Type theory that acknowledges itself violates itself. Suppose we try to say that all knowledge must be of some speciﬁc type or other, and that there can be no utterly general knowledge. We are claiming to know what we say, but our claim is a counter-example to what we said. Much the same sort of thing happens with the Liar Paradox; see my 1989–90: 161–5. It was pretty clear even from the early days of type theory that type theory self-destructs.⁵ If we say that every set is of some type (or rank), then no set can be the range of the variable in the universal quantiﬁer ‘‘every,’’ so sets are inadequate for semantics; they won’t let ‘‘all’’ mean all.⁶ The closest thing to a philosophical certainty may be that there are no philosophical certainties. But as it is a platitude of humility before a world beyond our control that not everything is known, so it might be a reasonable hypothesis of reason before a world independent of reason that not everything can be known. ⁵ At 3.332 in his 1961: 31, Wittgenstein wrote, ‘‘No proposition can make a statement about itself . . . (that is the whole of the theory of types).’’ 3.332 is obviously a counter-example to 3.332. I argue in my 1971: 271–88 that this paradox is the center of the distinction between saying and showing in the Tractatus. The Tractatus was published in 1919, but it would be impertinent to suppose that Russell had not seen long before that statements of the theory of types violate it. ⁶ If ZFC, Zermelo-Fränkel set theory with the axiom of choice, is true, then since it denies there is a set of all sets, there is no set of all sets. The model theory Tarski taught us says the domain of a model is a set, so no model is the world ZFC describes and ZFC is not true after all. If there is no set of everything, it is not there to be the domain of a model in which ‘‘Everything is self-identical’’ is true. Tarski taught us to say instead that it is valid, that is, that it comes out true in every non-empty domain (set) in which ‘‘is identical to’’ is interpreted as the identity relation restricted to that domain. But when we wrote ‘‘every set’’ just now, we used a quantiﬁer that, according to the doctrine we are expounding, does not have the interpretation we meant. The received solutions to the set theoretic and semantic paradoxes do not work. The late Raúl Orayen used to stress these points, but he died too early to publish them.

20 Two Deﬂationary Approaches to Fitch-Style Reasoning Christoph Kelp and Duncan Pritchard

0 . In t ro d u c t i o n Frederic Fitch (1963) famously argued that the thesis that all truths are knowable (henceforth, the knowability principle), in conjunction with a handful of apparently highly plausible logical and epistemic principles, entails the obviously absurd claim that all truths are known. This argument has become known as the paradox of knowability. Of course, it is only a paradox if one ﬁnds the knowability principle highly plausible in the ﬁrst place, since a basic prerequisite of an argument qualifying as a paradox is surely that it involves a highly contentious—indeed, unacceptable —conclusion which validly follows from highly plausible premises. Moreover, there is good independent reason to think that such a principle is not so plausible. It seems that the principle knowability is naturally understood as applying to cognizers like us; that is, to subjects with ﬁnite cognitive capacities and a ﬁnite lifespan. At the same time, it is plausible that there are some propositions that are too large to be grasped by such cognizers—for instance, some disjunctions with inﬁnitely many disjuncts. If these ﬁnite cognizers cannot grasp such propositions, however, then they cannot know them either. In consequence, for cognizers like us, some propositions must remain unknown. Provided that the principle of knowability is naturally understood as applying to cognizers like us, then it is not plausible that it is true. If so, however, the ‘paradox’¹ of knowability,² then, isn’t strictly speaking a paradox at all. We are grateful to Brit Brogaard, Tony Brueckner, Laurence Goldstein, Patrick Greenough, Allan Hazlett, Stephen Maitzen, Aidan McGlynn, Alan Millar, Paul O’Grady, David Papineau, Sven Rosenkranz, Peter Sullivan and to an anonymous referee from Oxford University Press. Special thanks also go to the editor of this volume, Joe Salerno, for all his (considerable) help. ¹ The thesis that the ‘paradox’ of knowability is not really a paradox has also been defended by Williamson (2000a, ch. 12). ² Of course, one might ﬁnd it independently puzzling that it is even possible to derive the conclusion that all truths are known from the premise that all truths are knowable. Fitch’s argument

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Nevertheless, there are (as we will see in a moment) substantive philosophical grounds in favour of the knowability principle, and thus even if Fitch’s argument does not point to a paradox as such it may still be thought to be a potential reductio of those philosophical views which feel the theoretical need to incorporate this principle. Accordingly, if there are such theoretical views then its defenders had better have something compelling to say in response to Fitch’s argument. In this paper, we will look at one—perhaps the only—theoretical view to which, on the face of it, the knowability principle is of central importance. We will then consider two deﬂationary responses to Fitch’s argument on behalf of defenders of this view. What we mean by a ‘deﬂationary’ response to the argument is a proposal which proceeds by weakening, on a principled basis, one of the principles essentially employed by that argument. The motivation for this strategy is this: ceteris paribus, if one can accommodate the considerations which prompt adoption of a certain principle by advancing a version of that principle which is (perhaps only slightly) logically weaker, then one ought to do so. If one can further show that the Fitch argument is blocked once the weaker ‘deﬂated’ version of the principle is adopted, then one will have succeeded in offering a deﬂationary response to the argument.³ The ﬁrst deﬂationary response that we will consider proceeds by weakening the factivity principle for knowledge. We will argue that this strategy does not stand up to closer inspection. Nevertheless, we claim that there are good grounds for holding that the second deﬂationary response that we consider—which rejects the principle of knowability in favour of a weaker principle—is effective at resolving the problem posed by Fitch’s argument.

1 . Se m a n t i c A n t i - re a l i s m The view to which the knowability principle is, on the face of it, of central importance, is often labelled ‘semantic anti-realism’. Semantic anti-realism is the rejection of realist theories of meaning (i.e., semantic realism). Indeed, semantic anti-realists often explicitly motivate their position by pointing to defects in realist theories of meaning. In this section, we will outline the problems which, according to the semantic anti-realist, beset realist theories of meaning and show how accepting the knowability principle can potentially avoid these problems. would then be philosophically interesting even if one did not ﬁnd the knowability principle plausible. Still, what is making the argument—taken in isolation—philosophically interesting is not that it poses a paradox. ³ This deﬂationary strategy—applied to epistemological issues—is explored and defended at greater length in Pritchard (2004). See also Greenough (2002).

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To begin with, let us look at the credentials of realist theories of meaning. Realist theories of meaning are commonly construed as having the following two features: (1) The meaning of a statement is identiﬁed with its truth-conditions. (2) There are evidence-transcendent truths.⁴ From these features of realist theories of meaning it follows that some statements have evidence-transcendent truth-conditions as their meanings. A general constraint on a theory of meaning is that it should at least be compatible with a theory of understanding—that is, for a theory of meaning to be satisfactory it must be compatible with an account of what a competent speakers’ linguistic understanding consists in.⁵ Semantic anti-realists suspect that realist theories of meaning will be unsatisfactory on just this score because they are incompatible with a satisfactory account of our understanding of statements with evidence-transcendent truth-conditions. Semantic anti-realists base their suspicion on a challenge to realist theories of meaning which arises from what they consider to be an important Wittgensteinian insight into the nature of understanding—namely, that understanding a concept consists in a set of practical abilities rather than in a state of mind. Certainly, if one is to be credited with a given practical ability then one must be able to manifest that ability in one’s behaviour. For instance, a child will be credited with the ability to swim only if she is able to manifest swimming behaviour in suitable circumstances. Hence, if the Wittgensteinian insight is to be taken seriously—that is, if understanding is to be conceived of as a set of practical abilities—then understanding must be manifestable in behaviour too. Presumably, the kind of behaviour in which understanding must be manifestable is linguistic behaviour (i.e., language use). According to the semantic anti-realist, however, what would count—at least minimally—as a manifestation by a speaker of her understanding of a statement in use is that the speaker is able to evaluate her own and other people’s use of the statement and, if circumstances render it appropriate, to adjust her use of it accordingly.⁶ Given that we understand what counts as manifestation of understanding in use in this way, however, it is hard to see how understanding of statements with evidence-transcendent truth-conditions could be manifested in use. After all, the truth-conditions of such statements are evidence-transcendent. As a result, there aren’t any circumstances that would provide the basis for an evaluation ⁴ Cf. Wright (1993a: 250). We take it that for present purposes this is an adequate representation of Wright’s statement of realism: ‘Realism about a given discourse, for the purposes of the Manifestation Challenge, is simply the combination of views (a) that the proper account of our understanding of its statements is evidence-transcendent truth-conditional, and (b) that the world on occasion exploits, so to speak, this understanding—does on occasion deliver undetectable truth-conferrers to such statements’ (ibid.). ⁵ Cf. Wright (1993a: 47). ⁶ Cf. Wright (1993a: 247).

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of one’s own or other people’s use of such statements. And, similarly, there aren’t any circumstances in the light of which one would adjust one’s use of such statements. If there aren’t any such circumstances, then understanding of statements with evidence-transcendent truth-conditions cannot be manifested. And, if understanding of such statements cannot be manifested, then it does not consist in a set of practical abilities after all—contrary to what the Wittgensteinian insight suggests. Accordingly, the challenge that semantic anti-realists pose to their realist opponents is to provide an account of understanding of statements with evidence-transcendent truth-conditions that is both faithful to the two core realist theses and respects the Wittgensteinian insight. Their suspicion is that this cannot be done.⁷ A related challenge that semantic anti-realists pose to semantic realists focuses on the acquisition of our understanding of statements with evidence-transcendent truth-conditions. Since if we accept a truth-conditional theory of meaning we acquire our understanding of a type of statement by bringing to bear evidence on the truth-values of instances of it, semantic anti-realists argue that it is hard to see how we could so much as acquire an understanding of statements with evidencetranscendent truth-conditions. Accordingly, semantic anti-realists challenge their opponents to provide an account of how we acquire our understanding of statements with evidence-transcendent truth-conditions.⁸ These two challenges were ﬁrst advanced by Michael Dummett (1978) and have become known as the manifestation and the acquisition challenge, respectively.⁹ Although there are further anti-realist arguments, these challenges—and the manifestation challenge in particular—appear to be the most common reason offered by semantic anti-realists as to why they ﬁnd realist theories of meaning problematic.¹⁰ Accordingly, semantic anti-realists have proceeded to deny at least one of the two core theses of realist theories of meaning. Initially, semantic anti-realists were tempted to deny the realist’s ﬁrst core claim—i.e., the commitment to a truth-conditional theory of meaning—and replace it with a theory that identiﬁes the meanings of statements with their assertibility conditions. The rationale for this is obvious, since by tying the meaning of a statement to its assertibility conditions (which are held not to be evidence-transcendent) rather than its truth-conditions, the anti-realist avoids the problems posed for a theory of meaning by allowing evidence-transcendent truths. More recently, however, this option appears to have become less appealing to semantic anti-realists. Instead, they have tended to reject the realists’ second core ⁷ Cf. Wright (1993a: 247–8). ⁸ Cf. Wright (1993a: 87). ⁹ See also Dummett (1993a). ¹⁰ For two further anti-realist arguments, see Wright (1993a), who outlines a challenge that proceeds from the normativity of meaning, and Putnam (1981), who adduces the so-called ‘modeltheoretic’ argument. Most contemporary anti-realists appear to accept the manifestation challenge. For examples, see Dummett (1978, 1993), Wright (1993a) and Tennant (1997).

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claim—i.e., that there are evidence-transcendent truths.¹¹ It ought to be clear that accepting the possibility of evidence-transcendent truths entails accepting the existence of unknowable truths, at least if one accepts the further (highly plausible) claim that in order to know a proposition one must have evidence in favour of it. Accordingly, if one holds, with the knowability principle, that there cannot be any unknowable truths, then it follows that one must reject the idea that there are evidence-transcendent truths as well. Given the foregoing, there is clearly a large theoretical pay-off in rejecting this key realist claim, since it avoids the worries just noted regarding our understanding of such truths. If there aren’t any such truths, then the fact that it is doubtful whether an understanding of them can be manifested—or acquired for that matter—won’t be a problem for the semantic anti-realist. Given that our primary interest is the Fitch argument, it is this second strand of semantic anti-realist thought—which, like the Fitch argument, has the knowability principle at its heart—that is our concern here. Henceforth, when we talk of ‘semantic anti-realism’ we will have this speciﬁc variety of semantic anti-realism in mind.

2 . Fi t c h’s A r g u m e n t Fitch’s argument clearly poses a fundamental challenge to semantic anti-realism. Indeed, given that it is an undeniable truth that we are not omniscient, unless the semantic anti-realist can ﬁnd some way to block this argument then she is faced with a reductio of her position. Since, in order to be able to discuss some options the semantic anti-realist may have to block Fitch’s argument, it will be a good idea to look at how the argument proceeds in a bit more detail. First, we will formalize the knowability principle in the following way:¹² (KP)

(∀P ) (P → ♦(∃s, t ) (Ks, tP ) )

Now we assume, for reductio, that one is not omniscient—i.e., that there is some truth (we’ll call it ‘P 1 ’) which is unknown: (1) P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) Given (KP), however, one can straightforwardly derive (2): (2) ♦(∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) ) ¹¹ For a good contrast of the two anti-realist approaches, compare the early and later essays collected in Wright (1993a). ¹² We will take quantiﬁcation over propositions (P , P 1 etc.), subjects (s, s 1 etc.), and times (t , t 1 etc.,) for granted. For a statement of the argument that does not rely on substitutional quantiﬁcation, see Kvanvig (1995).

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An essential feature of Fitch’s argument at this point is a sub-argument to the effect that (2) is false. This proceeds by ﬁrst assuming, for reductio, that the statement within the scope of the possibility operator at line (2) is true: (3) (∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) )) Plausibly, knowledge distributes across conjunctions, such that if a conjunction is known, then so are both of the conjuncts: (4) (∃s 2, t 2 ) (Ks 2, t 2 P 1 ) & (∃s 2, t 2 ) (Ks 2, t 2 ¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) Most will also agree that knowledge is factive, such that if one knows a proposition, then that proposition must be true. We can thus conclude (5): (5) (∃s 2, t 2 ) (Ks 2, t 2 P 1 ) &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) This is, of course, a contradiction. Since the assumption of this sub-argument leads to contradiction, we can therefore infer the negation of this assumption: (6) ¬(∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) ) Moreover, since this result has been derived based on no assumptions, we can also conclude that it is a necessary truth: (7) ♦¬(∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) ) Using standard modal logic, however, we can infer (8) from (7): (8) ¬♦(∃s 2, t 2 ) (Ks 2, t 2 (P 1 &¬(∃sl, t 1 ) (Ks 1, t 1 P 1 ) ) ) Now (8) is obviously inconsistent with (2). It therefore follows that the original assumption—that we are non-omniscient—must be denied. The knowability principle, at least when combined with some very basic epistemic and modal logic, is therefore inconsistent with non-omniscience such that if we retain this principle then we must, it seems, accept the absurd conclusion that all truths are known.¹³ Fitch’s argument therefore poses a serious problem for semantic anti-realism. In the remainder of this paper we will explore two deﬂationary approaches that the semantic anti-realist could pursue in order to evade this argument. 3 . A De ﬂ a t i o n a r y Ap p r o a c h t o Fi t c h’s A r g u m e n t I : We a k e n i n g t h e Fa c t i v i t y Pr i n c i p l e The ﬁrst deﬂationary proposal that we will be exploring considers the prospects of offering an anti-realist response to Fitch’s argument which denies the factivity ¹³ See Williamson (1988b, 1992) for an argument to the effect that the conclusion just canvassed—that all truths are known—will not follow within intuitionistic logic from Fitch’s reasoning, even though it does follow that non-omniscience is false. For an excellent overview of the debate regarding Fitch’s puzzle, see Brogaard and Salerno (2004).

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of knowledge. It should be quite obvious that once the factivity of knowledge is denied, the argument that leads to the paradox, at least in its present form, will no longer go through since the step from (4) to (5) will no longer be valid. Of course, it is easy to say that one does not accept factivity and that, therefore, one isn’t impressed by Fitch’s argument. However, factivity seems to play an important—indeed, indispensable—role in any plausible theory of knowledge. In particular, it is one of the central guiding intuitions regarding knowledge that one cannot know falsehoods. That is, the very idea that there could be a case in which an agent knows a proposition and yet that proposition is false, just seems plain incoherent. Now of course one might claim that even the most deeply entrenched intuitions could be called into question on theoretical grounds. Even if this is so, however, it remains that any theory which denied factivity and thereby held that it was possible to know falsehoods would face a pretty severe uphill struggle when it came to gaining widespread acceptance. Nevertheless, there may be some room for manoeuvre here. After all, as we will now see, there is a potential logical gap—at least by semantic anti-realist lights—between the claim that there cannot exist any cases in which an agent knows a falsehood and the factivity claim that knowledge entails the truth of the proposition known. If this is right, then the semantic anti-realist can exploit this logical gap in order to motivate a weakened version of the factivity principle which can nevertheless retain the core guiding intuition behind factivity that there cannot exist cases of false knowledge. In order to see this, let us ﬁrst state factivity more formally: (FAC) (∀P ) (∀s) (∀t ) (Ks, tP → P ) Furthermore, let us state explicitly the intuition that is meant to drive adoption of (FAC)—viz., that there are no cases of false knowledge:¹⁴ (∗ ) ¬(∃P ) (∃s) (∃t ) (Ks, tP &¬P ) Now from (∗ ) we can derive (∗∗ ): (∗∗ )

(∀P ) (∀s) (∀t )¬(Ks, tP &¬P )

And from (∗∗ ) we can derive (∗∗∗ ): (∗∗∗ )

(∀P ) (∀s) (∀t ) (Ks, tP → ¬¬P )

From (∗∗∗ ) it might seem like a very small move indeed to get to (FAC), since all one needs to do is introduce the double negation equivalence rule (DNE) to eliminate the double negation in the embedded consequent. Crucially, ¹⁴ Note that to avoid unnecessary complications, we have here expressed the intuition in a slightly weaker form—i.e., that there are no cases of false knowledge, rather than that it is impossible for there to be cases of false knowledge. Nothing in what follows turns on this.

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however, intuitionistic logic does not contain (DNE), and yet it is precisely this logic that semantic anti-realists typically endorse. Accordingly, it follows that an anti-realist can accept the intuition guiding adoption of (FAC)—which we have expressed as (∗ )—without being compelled to endorse (FAC) itself. Instead, this guiding intuition merely entails the weaker claim which we have expressed as (∗∗∗ ), but which it is open to the semantic anti-realist to argue is itself a respectable version of factivity. We will call this weakened version of factivity, (FAC∗ ): (FAC∗ )

(∀P ) (∀s) (∀t ) (Ks, tP → ¬¬P )

This line of reasoning seems deﬂationary in just the right sort of way, since it shows that there is, at least by the lights of a particular theoretical outlook, a way of properly responding to the core intuition motivating (FAC) which results in a logically weaker principle. If this logically weaker principle can help the semantic anti-realist block Fitch’s argument, then this would thus be an extremely attractive way of resolving the situation. Unfortunately, however, closer inspection reveals that the present proposal is ultimately unsuccessful. True, on the face of it, (FAC∗ ) blocks the move from line (4) to line (5) in that it only gives us (5∗ ): (5∗ )

(∃s 2, t 2 ) (Ks 2, t 2 P 1 ) & ¬¬¬(∃sl, t 1 ) (Ks 1, t 1 P 1 )

Crucially, however, this triple negation collapses into a single negation, even within an intuitionistic logic, and thus one will be able to derive line (5) of the paradox of knowability anyway, even without having to appeal to (FAC). So one won’t solve the paradox of knowability by rejecting the factivity of knowledge and replacing it by the ever so slightly weaker (FAC∗ ). In order to get this line to work one would have to replace the factivity principle with something that is weaker even than (FAC∗ ). The difﬁculty facing such a proposal, however, is that it will not be able to do full justice to our intuition that one cannot know falsehoods. In this way, it is highly doubtful whether the present deﬂationary strategy can ultimately be successful.¹⁵ ¹⁵ Interestingly, in some recent (and unpublished) work Finn Spicer and Allan Hazlett have independently argued that there are good grounds for rejecting (FAC) outright. In particular, they argue for the plausibility of the claim that knowledge is best understood as reliable true belief. Thus, since one can have a reliably formed false belief, it follows that (FAC) must go. If such an argument could be made compelling, then it would hold out the prospect that the semantic anti-realist could exploit this proposal in order to block the Fitch argument on non-factivity grounds. Notice, however, that such a suggestion is not within a deﬂationary spirit. That is, unlike the deﬂationary proposal explored here, the move is not towards offering a logically weaker formulation of factivity which can nevertheless do justice (by the broader lights of that theory at any rate) to the intuitions which drive acceptance of factivity. Instead, this line of argument involves the straightforward rejection of those intuitions. This feature of the proposal makes this sort of manoeuvre dialectically problematic, and certainly ensures that it is not relevant for our purposes here.

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4 . A De ﬂ a t i o n a r y Ap p r o a c h t o Fi t c h’s A r g u m e n t I I : We a k e n i n g t h e K n ow a b i l i t y Pr i n c i p l e Although the proposal to deny factivity will not do the trick, there is a second proposal available that is in the same deﬂationary spirit and which is much more promising. The thought is that instead of rejecting one of the epistemic principles which are employed within Fitch’s argument, one instead rejects the very principle that is the target of that argument—i.e., the knowability principle itself. In its stead is then put forward a slightly weaker principle which can nevertheless accommodate the guiding motivation behind the knowability principle. Informally, the weakened principle that we have in mind is as follows: for all true propositions, it must be possible to justiﬁably believe them. More formally: (JP)

(∀P ) (P → ♦(∃s, t ) (JBs, tP ) )

In order to see why this principle of justiﬁed believability, as we will call it, suits the purposes of the semantic anti-realist it is important to ﬁrst notice that it accommodates the semantic anti-realists’ worries regarding realist theories of meaning. Recall that the semantic anti-realist argued that realist theories of meaning will have a problem explaining how we can acquire and manifest an understanding of the meanings of statements with evidence-transcendent truth-conditions. Recall, furthermore, that we saw that accepting the knowability principle will avoid this problem. Given the plausible additional assumption that one knows a proposition only if one also has evidence for it, it follows that if all truths are knowable then it must also be possible to have evidence for them. If it must be possible to have evidence for all true propositions, however, then there can be no evidence-transcendent truths. In this way, the semantic anti-realist can resist the realist’s second core claim that there are evidence-transcendent truths by accepting the knowability principle. Notice, however, that a parallel argument will show that accepting the justiﬁed believability principle will do the job just as well. After all, it is also plausible that one justiﬁably believes a proposition only if one has evidence for it. That means, however, that if, for all truths, it must be possible to justiﬁably believe them, then it must also be possible to have evidence for them. But if it must be possible to have evidence for all true propositions, then there can be no evidence-transcendent truths. In this way, the semantic anti-realist can resist the realist’s second core claim by accepting the justiﬁed believability principle. In short, this principle will do the job for the semantic anti-realist just as well as the knowability principle. Notice that while replacing the knowability principle with the justiﬁed believability principle will allow the semantic anti-realist to avoid the conclusion

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that we are omniscient—after all, justiﬁed belief is not knowledge¹⁶—but that does not mean that the semantic anti-realist is no longer susceptible to refutation by a Fitch-style argument. After all, a parallel argument for justiﬁed belief threatens to show that the justiﬁed believability principle entails that all statements are justiﬁably believed. And that, it would seem, is almost as bad for the semantic anti-realist as the original conclusion of Fitch’s argument. So there is still work to be done. One might think, however, that even if there is work to be done, it is not much work. After all, justiﬁed belief, as opposed to knowledge, is not factive. That is, one can justiﬁably believe a falsehood. For instance, one might reliably and conscientiously form the belief that there is a barn over there—and thereby have a justiﬁed belief in this proposition—even though this belief is nonetheless false because, unbeknownst to you, what you are in fact looking at is a barn facade. Accordingly, since Fitch’s argument relies on the factivity of knowledge, it follows that it will not go through if the knowledge operator is replaced by a justiﬁed belief operator. Hence it would seem as though all the semantic anti-realist has to do is to replace the knowability principle with the justiﬁed believability principle in order to avoid the conclusion of Fitch-style reasoning. 5 . Pro b l e m s w i t h t h e Se c o n d De ﬂ a t i o n a r y Ap p ro a c h t o Fi t c h - s t y l e Re a s o n i n g There are, however, problems on the horizon for this line of reasoning. In particular, one might think that the conclusion just canvassed is either false or uninteresting. To take the ﬁrst horn ﬁrst, one might think that it is false because even granted that justiﬁed belief is not factive, the following reﬂection principle does, nonetheless, hold: if, at a certain time, one justiﬁably believes that one does not at that time justiﬁably believe a proposition, then one does not at that time justiﬁably believe that proposition. More formally, we can express this principle as follows: (RP)

(∀P ) (∀s) (∀t ) (JBs, t (¬JBs, tP ) → ¬JBs, tP ) )

The signiﬁcance of this principle is that the relevant ‘factivity’ move in a Fitch-style argument employing the justiﬁed belief operator would be from (4 ) to (5 ): (4 ) (∃s 2, t 2 ) (JBs 2, t 2 P 1 ) & (∃s 2, t 2 ) (JBs 2, t 2 ¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) ) (5 ) (∃s 2, t 2 ) (JBs 2, t 2 P 1 ) &¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) ¹⁶ At least on standard views of justiﬁcation at any rate. If one held that justiﬁcation was factive, then there would be scope to contend that there is no logical gap between justiﬁed belief and knowledge. Such a theory of justiﬁcation would be highly revisionary, however, and so we can legitimately set this possibility to one side here.

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If the justiﬁed belief operator were factive, then that would straightforwardly license this inference. It ought to be clear, however, that even if the justiﬁed belief is not factive, then this inference will go through just so long as (RP) holds. So one might object that the mere fact that justiﬁed belief is not factive does not get the semantic anti-realist off the hook, since it is still plausible that justiﬁed belief satisﬁes the reﬂection principle which, it would seem, sufﬁces to generate the Fitch result. On the other hand, one might object that the result is uninteresting because it has long been established that the semantic anti-realist can resist the conclusion of Fitch-style argument by stating the epistemic constraint on truth in terms of justiﬁed believability. J. L. Mackie makes the point in the following passage: Suppose we read K [the knowledge operator in Fitch’s argument] as ‘It is justiﬁably believed at t 1 that . . .’ This will distribute over &, but we might expect the argument now to fail at step 4 [to 5 in the above statement of the argument], since this K is not truth-entailing. But step 4 [to 5] still goes through. If it is justiﬁably believed that p at t 1 that p is not justiﬁably believed at t 1 , then p is not justiﬁably believed at t 1 . On the other hand, if we read K as ‘It is justiﬁably believed at some time that . . .’, then step 4 does not go through. It does not follow that if it is justiﬁably believed at any time that p is not justiﬁably believed at any time, then p is not justiﬁably believed at any time. It might be justiﬁable at t 1 to think that p is false and never has been and never will be justiﬁably believed and yet there might be some other time t 2 at which p was, or will be justiﬁably believed. So the argument does not enable us to reject the principle that what is true can be justiﬁably believed at some time. (Mackie 1980: 91–2)

In this passage, Mackie distinguishes between two reﬂection principles for justiﬁed belief, one which he deems plausible and one which he deems implausible. The plausible reﬂection principle has it that if it is justiﬁably believed at t 1 that it is not justiﬁably believed at t 1 that p, then it is not justiﬁably believed at t 1 that p. This is, of course, the reﬂection principle—(RP)—that we formulated above. In contrast, according to the implausible reﬂection principle, if someone at some time justiﬁably believes that no one ever justiﬁably believes that p, then no one ever justiﬁably believes that p. This principle can be formalized in the following way: (RP∗ )

(∀P ) ( (∃s, t ) (JBs, t¬(∃sl, t 1 ) (JBs 1, t 1 P ) ) → ¬(∃sl, t 1 ) (JBs 1, t 1 P ) ) )

Mackie claims, correctly and for the right reasons, that (RP∗ ) is false. He goes on to claim, again correctly, that the conclusion of Fitch’s argument can be avoided if the epistemic constraint is construed in terms of justiﬁed believability at some time—i.e., what we have called the justiﬁed believability principle. Unfortunately, however, this last claim, while correct, is made for the wrong reasons. For, as we are about to show, Fitch’s conclusion can be derived from (RP), which Mackie deems plausible, and the justiﬁed believability principle.

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To begin with, we start with the relevant assumption for reductio—someone at some time justiﬁably believes that p and that no one ever justiﬁably believes that p: (3 ) (∃s 2, t 2 ) (JBs 2, t 2 (P 1 &¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) )) If (3 ) is true, then so is an instance of it. Or, in other words, if someone at some time justiﬁably believes that p and that no one ever justiﬁably believes that p, then there must be a particular epistemic subject who believes this conjunction at a particular time. Let the epistemic subject and time be s 3 and t 3, respectively. We then get: (4 ) JBs 3, t 3 (P 1 &¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) ) ) Since justiﬁed belief distributes across conjunctions, we get: (5 ) JBs 3, t 3 P 1 &JBs 3, t 3 ¬(∃sl, t 1 ) (JBs 1, t 1 P 1 ) Now, if one justiﬁably believes that there is no one at any time who justiﬁably believes that P 1 , then one also justiﬁably believes that, currently, one does not justiﬁably believe P 1 oneself.¹⁷ Accordingly, from (5 ) we can derive: (6 ) JBs 3, t 3 P 1 &JBs 3, t 3 (¬JBs 3, t 3 P 1 ) Given (RP), however, the second conjunct of (6 ) entails that s 3 does not justiﬁably believe P 1 at t 3 : (7 ) JBs 3, t 3 P 1 & ¬JBs 3, t 3 P 1 From here the Fitch-style argument proceeds as rehearsed. So we can argue to its conclusion without having to appeal to the implausible reﬂection principle, (RP∗ ). All that we need is (RP) which Mackie deems a plausible reﬂection principle. So Mackie’s distinction between the two reﬂection principles will not help the semantic anti-realist. If the semantic anti-realist is to get any mileage out of rejecting the principle of knowability and replacing it by the weaker principle of justiﬁed believability, then she must have some other way of resisting the Fitch-style conclusion. Fortunately for the semantic anti-realist, however, there is excellent reason to believe that (RP) does not hold. Consider the following case due to Saul Kripke: Pierre is a Frenchman who has lived most of his life in France. Having just ¹⁷ Some may object that the step from (5 ) to (6 ) will not go through because justiﬁed belief is not closed under known entailment (never mind under entailment simpliciter). However, the argument does not depend on such closure principles. It is plausible that if (RP) is valid—that is, if one justiﬁably believes at t 1 that one does not justiﬁably believe that p at t 1, then one does not justiﬁably believe that p at t 1—then so is the following reﬂection principle: if one justiﬁably believes at a given time that no one ever justiﬁably believes that p, then at that time one does not justiﬁably believe that p. Stated formally: (RP∗∗ ) (RP∗∗ )

(∀P )(∀s)(∀t )(JBs, t¬(∃sl, t 1 )(JBs 1, t 1 P ) → ¬JBs, tP ) will validate the inference to (7 ) without relying on any further closure principles.

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returned from a trip to London, one of Pierre’s best friends asserts ‘‘Londres est jolie.’’ Since Pierre knows his friend to be a man of exceptional taste he believes what his friend asserted and hence comes to believe that London is pretty. Now suppose that, by some unfortunate circumstance, Pierre ﬁnds himself stuck in a particularly unattractive part of London. Pierre is forced to take on a badly paid job that will just pay him enough to buy food and accommodation. At this time he learns English ‘directly’—that is, by direct interactions with other English speakers rather than referring to, say, translation manuals. Pierre uses the term ‘London’ as his neighbours do and learns everything his neighbours know about it which, let us suppose for the sake of argument, is not very much. On the basis of his experiences in the city he comes to believe that London is not pretty. At the same time, Pierre is still sometimes thinking about his nice life in France, and sometimes even of his friend who told him about the pretty city of London. In such moments Pierre thinks to himself: ‘‘Si seulement je serais en Londres . . .’’ Obviously, Pierre still believes that London is pretty and hence he has inconsistent beliefs. Moreover, his inconsistent beliefs are both justiﬁed. The testimony from a person with exceptional taste justiﬁes his belief that London is pretty while his direct experiences justify his belief that London isn’t pretty. It is plausible that whilst having inconsistent beliefs that are both justiﬁed, Pierre may also believe, justiﬁably, that he does not believe that London is pretty. Perhaps some psychologist analyses him and tells him that the source of his recent unhappiness is simply that he no longer believes himself to be living in a pretty city. Pierre thus comes to believe, and justiﬁably so (since on the basis of the reliable testimony from the psychologist), that he does not believe that London is pretty. But if one justiﬁably believes that one does not believe a proposition, then one also justiﬁably believes that one does not justiﬁably believe that proposition. Accordingly, Pierre also justiﬁably believes that he does not justiﬁably believe that London is a pretty city. Pierre’s case thus indicates that one can simultaneously justiﬁably believe all of the following: (a) a proposition, P; (b) its negation, not-P; and (c) the proposition that one does not justiﬁably believe P. Given that this is so, however, it can easily be seen that the reﬂection principle (RP) must fail. For, if (RP) held, it would follow that Pierre both does and does not justiﬁably believe that London is pretty. (RP) turns an inconsistency in Pierre’s belief-system (in conjunction with a second-order belief), into an inconsistency in the world. So it must be false.¹⁸ If (RP) is false, however, then the relevant Fitch-style argument no longer goes through. The semantic anti-realist is off the hook. There is, however, a further difﬁculty for the semantic anti-realist who endorses the justiﬁed believability principle. It remains true that since there are some statements that are true but will never be justiﬁably believed, it must, by ¹⁸ If one wants to run the argument by appeal to (RP∗∗ ) instead of (RP)—see the above footnote—one will need a slightly different case to make the present point. We provide such a case below.

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the justiﬁed believability principle, also be possible for someone at some time to justiﬁably believe an instance of this. Among other things that means that it must be possible for someone at some time to justiﬁably believe statements of the form ‘‘P but no one ever justiﬁably believes P’’ and, similarly, ‘‘P but I don’t justiﬁably believe P’’. And, as Dorothy Edgington (1985: 558) has pointed out, one might think that this is already bad enough for the semantic anti-realist. After all, it would seem that one just couldn’t have any evidence for statements of either form. If so, then it would seem that one also cannot justiﬁably believe such statements. Moreover, recall that the semantic anti-realist introduces the justiﬁed believability principle in order to ensure that meaning, construed truth-conditionally, can always be manifested in use. If there are truths of the form ‘‘P and no one ever justiﬁably believes P’’ and ‘‘P and I don’t justiﬁably believe P’’, then one must be able to manifest the meaning of those statements in understanding. Since justiﬁed believability is supposed to secure manifestability, it must be possible to justiﬁably believe statements of the form ‘‘P and no one ever justiﬁably believes P’’ and ‘‘P and I don’t justiﬁably believe P’’. But if it is impossible to have evidence that would support statements of this form, then one cannot justiﬁably believe such statements. So even if the semantic anti-realist can deny the reﬂection principle, (RP), Fitch’s argument shows that things are already bad enough for the semantic anti-realist even before the problematic principle comes into play. It would seem, however, that there are ways for the semantic anti-realist to respond to this difﬁculty. Let us begin with statements of the form ‘‘P and I don’t justiﬁably believe that P’’. In order to argue that statements of this form can be justiﬁably believed, the semantic anti-realist can simply point to Pierre’s case again and claim that Pierre might well come to believe that London is pretty (by believing that the proposition expressed by ‘‘Londres est jolie’’ is true) and that he does not believe that London is pretty (by believing that the proposition expressed by ‘‘I don’t believe that London is pretty’’ is also true). Since both of his beliefs are justiﬁed and since we typically acquire a justiﬁed belief in a conjunction by conjoining the justiﬁcation we have for the beliefs in the conjuncts,¹⁹ it would follow that the semantic anti-realist can comfortably allow that Pierre justiﬁably believes that London is pretty and that he does not justiﬁably believe that London is pretty.²⁰ Things are a bit more difﬁcult when it comes to statements of the form ‘‘P and no one ever justiﬁably believes that P’’. However, the situation is not hopeless for ¹⁹ Cf. Kvanvig (2006: 21). ²⁰ Of course, another consequence is that Pierre can come to justiﬁably believe a contradiction. This may initially seem an unwelcome consequence of the view. However, it is not clear why one could not justiﬁably believe a contradiction (at least so long as it is not an obvious one). A clever logician could easily tell me that what, in effect, is a complicated contradiction is true and I could come to believe this contradiction on that basis. Since the logician is an otherwise reliable informant on such issues, and since the testimony of reliable informants furnishes us with justiﬁed beliefs, my belief in the contradiction is surely justiﬁed.

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the semantic anti-realist. Consider, for instance, a case in which I am stranded on a lonely island. The only thing I have with me is a book about psychology which is written in English. I read about a brain lesion, named ‘X’, and learn that the only symptom of X, which occurs in 99 per cent of the cases, is a continuous strong belief on the part of the patient that she has X. Through introspection, I ﬁnd that I don’t believe that I have X. I therefore come to believe that I don’t believe that I have X. My belief, since based on reliable introspective capacities, is justiﬁed. I justiﬁably believe that I don’t believe that I have X. Since if one justiﬁably believes that one does not believe a proposition, P, then one also justiﬁably believes that one does not justiﬁably believe P, I justiﬁably believe that I don’t justiﬁably believe that I have X. Moreover, since I am on a lonely island, without drinkable water, and since I have every reason to believe that no one will come to my rescue and that I will die fairly soon, I also justiﬁably believe that no one will ever justiﬁably believe that I have X. To ﬁnish the story off, suppose that just before I left for my holiday, I called my doctor to get the results for some brain tests that they had done on me. My doctor told me that everything was ﬁne except that I have a brain condition called ‘Y’, which is completely harmless, and that I could go on the trip without any problem. On the basis of the testimony from my doctor I come to believe, justiﬁably, that I have Y. Now, since I am German and have talked to my German doctor our conversation was naturally in German. What I don’t know is that the German expression ‘Y’ and the English expression ‘X’ are names for the same brain lesion and that I am among the lucky 1 per cent of patients who don’t suffer from the symptoms. I am now in a condition in which I justiﬁably believe that I have X (on the basis of testimony from my doctor: ‘Sie haben Y’) and I also justiﬁably believe that no one ever justiﬁably believes that I have X (on the basis of introspection, what I have read about X in the psychology book and my unfortunate predicament of being on a lonely island about to die). If I were to conjoin my two beliefs, I would justiﬁably believe that I have X and that no one ever justiﬁably believes that I have X. There is thus a way for the semantic anti-realist to respond to Edgington’s worry. The semantic anti-realist may point out that, contrary to what Edgington claimed, one can justiﬁability believe statements of the form ‘‘P and I don’t justiﬁably believe P’’ as well as ‘‘P and no one ever justiﬁably believes P’’. The semantic anti-realist is, again, off the hook. In general, there are good grounds for holding that the deﬂationary strategy of replacing the knowability principle with the justiﬁed believing principle may well offer the semantic anti-realist a way of avoiding Fitch-style reasoning.

21 Not Every Truth Can Be Known (at least, not all at once) Greg Restall

Ab s t r a c t According to the ‘‘knowability thesis,’’ every truth is knowable. Fitch’s paradox refutes the knowability thesis by showing that if we are not omniscient, then not only are some truths not known, but there are some truths that are not knowable. In this paper, I propose a weakening of the knowability thesis (which I call the ‘‘conjunctive knowability thesis’’) to the effect that for every truth p there is a collection of truths such that (i) each of them is knowable and (ii) their conjunction is equivalent to p. I show that the conjunctive knowability thesis avoids triviality arguments against it, and that it fares very differently depending on another thesis connecting knowledge and possibility. If there are two propositions, inconsistent with one another, but both knowable, then the conjunctive knowability thesis is trivially true. On the other hand, if knowability entails truth, the conjunctive knowability thesis is coherent, but only if the logic of possibility is weak.

1 There are many things that we don’t know to be true. Ignorance is a fact of life. However, it is tempting to think that of the things that are true but not known to be true, each of them could be known. If the signiﬁcance of a proposition is to be explained in terms of its veriﬁcation conditions, for example, then if it is See http://consequently.org/writing/notevery/ for the latest version of the paper, to post comments and to read comments left by others. ¶ Thanks to Conrad Asmus, Allen Hazen, Lloyd Humberstone, Nick Smith and Timothy Williamson, to audiences at Monash University, the University of Melbourne, and Oxford University, and to commentators at http://consequently.org/writing/notevery/ for helpful discussions. Feedback from anonymous reviewers for this volume was useful in cleaning up the presentation.

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true, there must be some veriﬁcation conditions, and it is tempting to say that we could (at least potentially, in theory) have access to them. So, it is tempting to endorse the claim Every truth is knowable.

(♦)

which has come to be known as the knowability thesis, and its formalization (∀p) (p ⊃ ♦Kp)

(1)

for any truth, it is possible (♦) that it be known (K). This position is tempting to many, but Fitch has shown that the temptation comes at a very high cost. Using only inference principles that are very tough to reject, we can show that, given the knowability thesis, every truth is, indeed, known.¹

1.2 Here is Fitch’s proof that the knowability thesis fails if we are not omniscient. Suppose, for a reductio, that we are ignorant of some truth, so suppose that p is true but not known to be true. Then p ∧ ¬Kp is true. So, by the knowability thesis, this is possibly known: ♦K(p ∧ ¬Kp). Now, this is very hard to take. How could we know that p ∧ ¬Kp? If knowing a conjunction entails knowing the conjuncts, then K(p ∧ ¬Kp) entails Kp and K¬Kp.² Now knowledge entails truth, so K¬Kp entails ¬Kp, a contradiction. So, by a reductio, it is not possible that K(p ∧ ¬Kp), and we have (using the knowability thesis) refuted the hypothesis of ignorance. If the knowability thesis holds, a much stronger thesis holds too: every truth is not merely knowable, but known.

1.3 This phenomenon has come to be called Fitch’s Paradox, after F. B. Fitch, who ﬁrst formulated it (Fitch 1963). This paradox has generated a vast literature, including on the one hand ‘‘search and rescue’’ missions designed to ﬁnd the true principle underlying knowability thesis, and to save them from similar paradoxical fate, ¹ I indicated before that Kp should be read as ‘‘it is known that p.’’ But known by whom? Kp can be read either as ‘‘α knows that p’’ for some ﬁxed agent α, or ‘‘someone knows that p’’ without too much strain in what follows. The existential reading, which requires that p merely be known by someone, ensures that K is nothing like a normal modal operator. We do not have Kp, Kq K(p ∧ q), since it may well be that someone knows that p and someone (else) knows that q without anyone knowing that p ∧ q. In what follows, however, we soon move from reading Kp as the straightforward ‘‘p is known’’ to the idealization ‘‘p is a logical consequence of what is known,’’ and this does satisfy the principle that distributes knowledge over conjunctions: if p and q are consequences of what is known (by someone or other) then so is their conjunction. For any of these readings, the knowability thesis has some bite. It seems like a substantial claim that any truth is knowable by someone or other. It seems like a more substantial claim that any truth is knowable by you. ² Maybe we could know a conjunction without knowing the conjuncts. No problem: Just interpret Kp as ‘‘p is a logical consequence of what I know.’’ If the knowability thesis works for knowledge, it works for this K too. So, from now on, K will allow for deductive closure: if Kp and p q then Kq.

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and ‘‘seek and destroy’’ campaigns aimed at hammering more nails into the cofﬁn of any so-called principle of knowability that shows signs of life.³ This paper contains elements of both kinds of discussion. I shall present and motivate yet another revision of the knowability thesis, and then show that this revision is consistent and not subject to Fitch-paradoxical refutation—that is the search and rescue part of the story—and then I will show that this revision is not only consistent but either it is also almost trivially true and therefore, it is not likely to do the work that a veriﬁcationist or anti-realist might require of a principle of knowability, or it’s an interesting, controversial thesis about knowledge, which is coherent under certain conditions.

1.4 The knowability thesis, cast as the statement (1), is dead. Fitch’s paradox is a conclusive refutation, and even though many interesting moves are possible with the logic in which these principles are couched, defeating the inference from (1) to omniscience (Beall 2000; Wansing 2002), these answers do not address the question I take to be asked by Fitch’s paradox. I say this because upon reﬂection, the principles motivating a knowability thesis in fact undercut its application in a case such as p ∧ ¬Kp. Consider any truth p, of which we are ignorant. Given the knowability thesis we can, indeed, imagine coming to know that p. This is all well and good, but any way we can go about knowing that p makes it no longer the case that ¬Kp. But, ¬Kp was true, and so, maybe it too could be known. If we do not enquire as to the status of p (so we don’t come to know that p is true) but rather take ourselves to consider whether or not Kp, it seems plausible to suppose that we could conﬁrm that ¬Kp. In other words, it’s quite coherent to suppose that there is nothing that we can see that makes it impossible for us to know that ¬Kp. But the conjunction p ∧ ¬Kp is true, and none of the ways we have considered, of coming to know p, or coming to know ¬Kp will provide a way to know both p and ¬Kp. The conjunction p ∧ ¬Kp cannot be known ‘‘all at once.’’ Fitch’s proof, it seems, is not a trick to be avoided or to be explained away but a result to be understood.

2 This reasoning points the way to a possible answer: the Fitch-paradoxical conjunction p ∧ ¬Kp cannot be known ‘‘all at once’’ but it can be known ‘‘in pieces.’’ In particular, the ﬁrst conjunct can be known (or rather, there seems to be nothing preventing us knowing it), but it cannot be known if the second conjunct is known. Similarly, the second conjunct can be known, but ³ Brogaard and Salerno’s ‘‘Fitch’s Paradox of Knowability’’ (2004) is a ﬁne guide to this literature.

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it cannot be known if the ﬁrst conjunct is known. They cannot be known together. (1) is refuted, but it begs for a reformulation. Instead of saying that any truth could be known, let’s attempt to maintain instead that every truth can be known ‘‘in pieces.’’ That is, for any truth p, there is some collection of truths, each of which could be known, and when taken together, entail the original truth p. In other words, p can be factored into components, each of which is knowable.

2.2 If we could defend the knowability thesis in this weaker form, according to which unknowable truths could be factored into knowable pieces, then we may be able to provide some comfort to the anti-realist who takes meaningfulness to be a matter of knowability. For the fact that p ∧ ¬Kp is unknowable is no counterexample to its meaningfulness any more than the unknowability of p ∧ ¬p renders it meaningless. No, p ∧ ¬p is meaningful when p is meaningful, because we can understand p and its negation and its conjunction, even if to understand this is to come to see that it can never be known for it can never be true. The same kind of process can be seen in p ∧ ¬Kp, though now we have a conjunction which we can see that we will never know even though it may be true. It is meaningful because it is a conjunction of meaningful claims.

2.3 In fact, one could say that in the original naïve formulation (♦) didn’t mean what is expressed by (1), at least in its application to the statement p ∧ ¬Kp. For the p ∧ ¬Kp is not, in itself, one truth that is knowable, but two. There are two knowable truths here, not one. (This is altogether too tendentious a reading to take seriously, however. Nothing in this paper hangs on the idea that conjunctive knowability is what we really wanted in the ﬁrst place.)

2.4 Now consider what it is for a sentence to be a conjunction of knowable sentences. (In what follows, I will call these ‘knowables’ for short.) From the perspective of pure logic it matters not whether the original sentence is complex or atomic. For whatever may be expressed by a complex sentence may be expressed by an atomic sentence too. In whatever model theory we like, if we have a model in which a complex sentence is interpreted in some way, then as far as logic is concerned, any simple sentence may be interpreted in just that way too.⁴ But suppose that our original sentence was a complex sentence like p ∧ ¬Kp. This sentence is ⁴ This phenomenon underlies one of the substitutional properties of formal logics. If φ is a tautology containing the atomic sentence p, then φ , found by replacing p everywhere by another formula B is also a tautology.

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unknowable. If this is because it expresses an unknowable sentence, then if we interpret the atomic sentence q as ‘‘meaning the same thing’’ as p ∧ ¬Kp, then it seems that q will (relative to this interpretation, of course) be unknowable as well. But q has no conjuncts at all: it is a simple sentence. Have we sunk the ‘‘factoring’’ analysis before it could set sail?

2.5 This factoring analysis may survive if we are prepared to agree that while the sentence q from our example has no explicit conjuncts, it may have conjuncts implicitly. The sentence q is equivalent (relative to this model, again) to the conjunction p ∧ ¬Kp. As far as logic is concerned this will sufﬁce for a factorization. We will say that q is conjunctively knowable (relative to a model) if it is equivalent (relative to that model) to a conjunction, each of whose conjuncts are knowable (relative to that model).

2.6 This is my proposed revision of the knowability principle: Every truth is conjunctively knowable.

(♥)

In the rest of this paper, we will examine the fate of this principle.⁵

3 For the proposal to be formally evaluated, it must be stated more sharply. This thesis assumes a number of exotic elements of logical vocabulary, such as propositional equivalence, propositional quantiﬁcation, epistemic and modal operators. To properly state this thesis will require a great deal of machinery. The syntax of the claim is straightforward enough. We may formalize one version of this claim as follows: (∀p) (p ⊃ (∃q, r ) ( (p = q ∧ r ) ∧ ♦Kq ∧ ♦Kr ) )

(2)

This version posits a very strong version of conjunctive knowability: every proposition may be factored into a conjunction of two propositions, each of which is knowable.⁶ ⁵ After writing a draft of this paper, Joe Salerno brought to my attention Risto Hilpinen’s paper ‘‘On a Pragmatic Theory of Meaning and Knowledge’’ (2004), in which he argues that a Peircean pragmatism motivates a conjunctive knowability principle just like this. I must leave it to the reader to determine whether or not the results of the investigation below are congenial to the pragmatist project. ⁶ It could be that something could be factored into three knowable conjuncts but not two. As far as I can see, there is no natural upper limit to the number of conjuncts one could require in a formalization of (♥), so a perfectly general formulation would perforce be quite complex indeed, as

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3.2 The work comes in when we are required to characterize the logical properties of propositional quantiﬁcation, propositional identity and the modal and epistemic operators. Thankfully, for our purposes, we need not attempt to pin down the right principles governing ♦, K, (∀p) and =. Qua logician my job is to investigate the consistency of (♥) and its formalization (2). Fitch showed that (1) is inconsistent with deeply plausible modal and epistemic principles. I will show that (2) does not suffer that fate. (2) is consistent, and compatible with the strong principles of modal and epistemic reasoning. To do this, we need not ﬁnd the right principles of such reasoning. In doing this, it is acceptable to overshoot and require too much. I will provide a class of models that show that the revised knowability thesis (♥) and its formalization (2) can be absolutely unrestrictedly true at no cost to ignorance or to many other epistemic or modal principles. (There will, however, be an important caveat to be discussed in Section 5.)

3.3 Our logic will be the incredibly strong modal epistemic logic in which ♦ and K are both governed by the principles of the logic s5. This is unrealistic in the extreme, for it commits us to wild epistemic principles such as the claim that if p is true then we must know that we don’t know that ¬p (if p then K¬K¬p) and even that if we don’t know something we know that we don’t know it (if ¬Kp then K¬Kp). Neither of these principles is particularly plausible (even if we take Kp to mean that p is a consequence of what we know) but we will use such a strong logic nevertheless, since nearly every epistemic or modal principle endorsed by someone or other is valid in this logic: s5 ♦ ⊕ s5 K .

3.4 This logic has models of the usual kind for modal logics. Here a model is a quadruple W , R♦ , RK , [[ · ]] where W is a non-empty set of worlds, R♦ and R K are accessibility relations on W and [[ · ]] is a function assigning to each atomic sentence (for example, p) an interpretation—a set of worlds (in this case, [[p]]). In this modal logic we place no restrictions on which sets can be used to interpret sentences. All sets may be propositions in our model. The set [[p]] is the set of worlds in which p is true. In the usual way, the interpretation function is it would have to quantify over collections of propositions (there are some propositions which factorize p). This seems to be the right condition: (∀p)(p ⊃ (∃P)(( P = p) ∧ (∀q)(Pq ⊃ ♦Kq))) (2 ) where P is the second-order propositional variable used in the second-order propositional existential quantiﬁer ∃P ranging over classes of propositions (which is ‘‘really’’ a third-order quantiﬁer over objects, since propositions are ‘‘really’’ zero-place properties) and the ‘‘connective’’ sends a class of propositions to its conjunction.

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extended to assign sets to arbitrary sentences in the language. For conjunction, disjunction, the material conditional and negation we use the standard Boolean operations. A conjunction p ∧ q is true at the worlds where both p and q are true: [[p ∧ q]] = [[p]] ∩ [[q]], and similarly for the other Boolean connectives. The relations R♦ and R K are used to model the operators ♦ and K respectively. In our case R♦ and R K are both equivalence relations. R♦ is the equivalence relation governing ♦: #

♦φ is true at w iff φ is true at a world in w’s R♦ equivalence class.

We will call the worlds in w’s R♦ equivalence class the modal alternatives of w.⁷ Another way to understand the interpretation is as a function from propositions to propositions. [[♦φ]] is the union of all modal equivalence classes overlapping [[φ]]. Think of approximating the proposition [[φ]] by equivalence classes, counting in our approximation any equivalence class that at least partly overlaps the original proposition: [[φ]] is approximated by its closure. Similarly, an equivalence relation R K governs K: #

Kφ is true at w iff φ is true at all worlds in w’s R K equivalence class.

We will call the worlds in w’s R K equivalence class the epistemic alternatives of w. [[Kφ]] is the set containing all epistemic equivalence classes totally included in [[φ]]. Here, [[φ]] is approximated by its epistemic interior. The modal alternatives of w need not be the same worlds as the epistemic alternatives: a modal alternative need not be an epistemic alternative (we can know things that are not necessary) and an epistemic alternative need not be a modal alternative (we can be ignorant of some necessary truths). Now for the propositional quantiﬁer: # (∃p)φ is true at a world w if and only if for some set X of worlds, φ is true at w when we take the formula p occurring unbound in φ to be true at exactly the worlds in X. # For identity, we will say that φ = ψ is true at a world just when [[φ]] = [[ψ]]. This sufﬁces to ensure that we may, for example, infer from φ = ψ that θ(φ ) = θ(ψ). Any formula containing φ (for example, ♦Kφ) is true in the same worlds as the formula found by replacing those φs by ψs (in this case, ♦Kψ). ∃p,= An argument is s5 ♦ ⊕ s5 K valid if for every model, if the premises are true in a world in that model, the conclusion is true in that world too. A sentence is ∃p,= an s5 ♦ ⊕ s5 K tautology if and only if it is true in every world in every model.

3.5 Let me reiterate: This model theory is not to be endorsed as giving us the ‘‘true picture’’ of knowledge, possibility, propositional quantiﬁcation and propositional identity. It is intended as a grab-bag sizeable enough to catch all principles thought ⁷ This leads to a slight infelicity: w counts as a modal alternative of itself.

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to govern epistemic modal logic with propositional quantiﬁers. If we can ﬁnd models that both validate the conjunctive knowability principle (♥) and allow for ignorance, then they show that no principle true in these models collapses conjunctive knowability into omniscience.

3.6 Figure 21.1 shows a simple model in which (♥) holds. There are four worlds {a 1 , a 2 , b 1 , b 2 }. a1

a2

b1

b2

∃p,=

Figure 21.1. A simple s5 ♦ ⊕ s5 K

frame

The modal accessibility relation R♦ relates a 1 to b 1 and a 2 to b 2 ; the epistemic accessibility relation RK is orthogonal to the modal relation: it relates a 1 to a 2 and b 1 to b 2 . So, a world’s modal alternatives are those worlds sharing a number, and its epistemic alternatives are those sharing a letter. In Figure 21.1 (and in all other diagrams) solid lines join epistemic alternatives, and dashed lines join modal alternatives.

3.7 (♥) says that any proposition true at the world of evaluation is a conjunction of two propositions which, at the world of evaluation, are knowable. What are the propositions in our model that are knowable at any world? Any proposition true at both a 1 and a 2 is knowable at all worlds, since it is known at a 1 and a 2 (and hence, it is possibly known there), and at b 1 , the world a 1 is possible, and at b 2 , a 2 is possible, so at b 1 or at b 2 , this proposition is also possibly known. So, if {a 1 , a 2 } ⊆ [[φ]], then φ is knowable at any point in the model. Similarly, any proposition true at both b 1 and b 2 is knowable at every world. And these propositions are the only propositions knowable at any world. The propositions which can not be known are ∅, each singleton proposition {a 1 }, {b 1 }, etc., and the two diagonal propositions {a 1 , b 2 } and {a 2 , b 1 } and the modal alternative propositions {a 1 , b 1 } and {a 2 , b 2 }. All other propositions are knowable, from the point of view of every world.

3.8 It will be helpful to consider why for any interpretation of p, the proposition denoted by p ∧ ¬Kp is not knowable at any world. If [[p]] = X ⊆ W , then [[Kp]]

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consists of the interior epistemic approximation of X, and [[¬Kp]], then, is the union of all equivalence classes not totally inside X. So, its intersection with X (the set [[p ∧ ¬Kp]]) consists of the union of all X-overlapping parts of epistemic equivalence classes that overlap X but do not fall completely inside X.⁸ In the case where [[p]] = {a 1 , a 2 , b 1 }, [[Kp]] = {a 1 , a 2 } and so [[¬Kp]] = {b 1 , b 2 }, and [[p ∧ ¬Kp]] = {b 1 }. This proposition is not knowable, because it contains no epistemic equivalence classes as a subset.

3.9 In this model, every true proposition is a conjunction of two knowable propositions. The singleton {a 1 } is the conjunction of {a 1 , a 2 } and {a 1 , b 1 , b 2 }. The same goes for each other singleton. The pair {a 1 , b 1 } is the conjunction of {a 1 , a 2 , b 1 } and {a 1 , b 1 , b 2 }. The same goes for each other pair. It follows that (2) is true in our model (Figure 21.2). a1

a2

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b2

Figure 21.2. {a 1 } as a conjunction of knowables

3.10 So, we have shown that (2) survives consistently and coherently. In this model there is much ignorance (a proposition true at a 1 alone is true there but not known to be true), yet every proposition is a conjunction of two knowable propositions. The principle (♥) of conjunctive knowability is secure. Truth and knowability can be intimately connected, even if not every truth is knowable. What the friend of knowability cannot have whole she is allowed to have if she will accept it in two pieces.

3.11 At this point, the story takes a different turn. Conjunctive knowability is secure, but it either is almost certainly not what the veriﬁcationist wants, or it is too high a price for the veriﬁcationist to pay. In the rest of this paper I shall show that if knowability does not entail truth (so some falsehoods are knowable while not being known), then conjunctive knowability, in the form of (2) is ⁸ In topological terms it is the part of the (epistemic) boundary of X that is also inside X.

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not only true, but it’s very hard to refute in an epistemic modal logic. It puts precious few constraints on knowledge or necessity, and so, is not useful as criterion for favouring one theory over another. If principles acceptable to the realist lead them to accept (2) while maintaining their realism, then (2) will do no good as a principle designed to favour the anti-realist. On the other hand, if knowability entails truth, then any non-trivial account of conjunctive knowability is inconsistent with plausible modal principles. (In particular, with the modal principle of transitivity: ♦♦p ♦p.) 4 ∃p,=

So, consider what we have done so far. We have a model of s5♦ ⊕ s5 K in which conjunctive knowability is satisﬁed. It turns out that this is not a one-off affair. ∃p,= In an epistemic modal logic like s5♦ ⊕ s5 K , and its much weaker cousins in which the modal and epistemic accessibility relations satisfy fewer constraints, (2) turns out to be very easy to validate. Not only are there many models in which (2) is true, it turns out that (2) is a consequence of other, unproblematic modal and epistemic principles. In particular, (2) follows from the following thesis about possible knowledge, satisﬁed in the models we have seen: (∃q) (♦Kq ∧ ♦K¬q)

(3)

This is relatively uncontroversial, given one understanding of how possibility and knowledge (or the consequences of what is known) are connected. Provided that, for some q, both q and ¬q are possibly true (and this is not too difﬁcult to imagine) then it is not much more difﬁcult to conclude that for some q, both q and ¬q are possibly known. Of course, a circumstance in which one knows q is, perforce, one in which ¬q is not known, and vice versa, for there to be a q such that ♦Kq and ♦K¬q, there must be at least two distinct modal alternatives, one at which q is known, and the other at which ¬q is known. All that requires is that we have two modal alternatives whose epistemic closures do not intersect, like so (Figure 21.3).

Figure 21.3. Two inconsistent knowables

Given a q such that both it and its negation are knowable, we can prove that for any true p, there are knowable p 1 and p 2 where p = p 1 ∧ p 2 and

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both p 1 and p 2 are knowable. If q is such that (♦Kq ∧ ♦K¬q), then we may choose p 1 to be p ∨ q and p 2 to be p ∨ ¬q. Then by simple Boolean reasoning, p 1 ∧ p 2 = (p ∨ q) ∧ (p ∨ ¬q) = p. However, since ♦Kq, we have ♦K(p ∨ q).⁹ Similarly, since ♦K¬q, we have ♦K(p ∨ ¬q). Both p ∨ q and p ∨ ¬q are knowables, regardless of how unknowable p might be! (2) is a trivial consequence of the trivial truth (∃q) (♦Kq ∧ ♦K¬q). It looks as if (2) tells us little about the connection between truth and knowability.

4.2 Where can the fan of conjunctive knowability resist this analysis? It might be thought that a friend of relevance would quail at the identiﬁcation of p with (p ∨ q) ∧ (p ∨ ¬q), as well they should. The inference from p to (p ∨ q) ∧ (p ∨ ¬q) is valid in almost every logic you care to mention, as it is found by composing the inferences from p to p ∨ q, and from p to p ∨ ¬q and from these to their conjunction. All are simple lattice moves. The problem with relevance is in the other direction. To get from (p ∨ q) ∧ (p ∨ ¬q) we need q ∧ ¬q p, and this is relevantly invalid. Nonetheless, rejecting this identity¹⁰ is not going to stop this argument from getting off the ground. The crucial premise in the argument was that q and its negation were both knowable, and could be used in the factorization of p. There is no requirement that q and its negation be used for this purpose. Provided that we are given two incompatible propositions (say, q 1 and q 2 ) that are knowable—so q 1 , q 2 ⊥ for the trivial proposition ⊥, and ♦Kq 1 and ♦Kq 2 —then even in relevant logics, the sentences p and (p ∧ q 1 ) ∨ (p ∧ q 2 ) are true in exactly the same situations. Blocking the inference from (p ∧ q 1 ) ∨ (p ∧ q 2 ) (with the rule q 1 , q 2 ⊥) to p requires blocking the distribution of conjunction over disjunction, not any odd behaviour about negation or relevance. So, pleading relevance or paraconsistency will not give the fan of conjunctive knowability (or its enemy, for that matter) a straightforward way out of the problem.¹¹

4.3 Denying that one can infer K(p ∨ q) from Kp is not going to help, either, for as we have seen, we can replace talk of what is known by talk of what is a consequence of ⁹ By distribution of both K and ♦ over logical consequence: since q p ∨ q, then Kq K(p ∨ q) (remember, we read K(p ∨ q) as ‘‘p ∨ q is a consequence of what is known’’) and so, ♦Kq ♦K(p ∨ q): all are reasonable principles. ¹⁰ Which, it must be said, is not the same identity as that between p and (p ∧ q) ∨ (p ∧ ¬q), a factorization seen again and again in different kinds of reasoning. ¹¹ A not-quite-straightforward way out of the problem is to deny that there are any incompatible pairs of propositions. To be sure, in many relevant logics, there is no way to construct formulas φ and ψ such that φ, ψ ⊥. Nonetheless, in most models for such logics there are ways to interpret φ and ψ such that their conjunction is absolutely inconsistent. Merely take [[φ]] and [[ψ]] to have empty intersection, so their conjunction is true nowhere.

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what is known (at least in decidable logics), and clearly, if p is a consequence of what is known, so is p ∨ q. Furthermore, possibly being a logical consequence of what is known is not very far removed from being possibly known, so reading Kp throughout as ‘‘p is a consequence of what is known’’ does little violence to the principles in question, and it validates the inferences used in our deduction. So requiring high standards for knowledge, so high that logical consequence can lead you from what is known to what is not, is also not a way out of the problem.

4.4 So, conjunctive knowability is not only consistent, but it is trivially so, if possibility and knowledge are connected as given by (3), that is, if possible knowledge can sometimes outrun truth, just as Fitch’s paradox has shown us that truth can sometimes outrun possible knowledge.

5 Nonetheless, (3) is by no means uncontroversial.¹² What if we reject (3) and hold, instead, that only truths may be possibly known? So, let us embrace (4): ♦Kp p

(4)

5.2 Before proceeding, I wish to do away with a bad argument for (4). No-one should argue as follows: ‘‘What is possibly known must be true, because of necessity, what is known is true. It is, therefore, impossible for what is known to be false. It follows that if something is possibly known, it is true.’’ This contains a modal fallacy. We have attempted to infer from the innocuous ‘‘it is impossible for what is known to be false’’ (¬♦(Kp ∧ ¬p) ) to the much stronger ‘‘if something is possibly known, it is true’’ (which as a material implication is ¬♦Kp ∨ p). In the ﬁrst, the truth of p (or its not being false) is under the scope of the possibility operator, and in the second it is not. That is a bad argument for (4). If you contemplate (4), do not do so for that reason.

5.3 In the case of an epistemic modal logic modelled with an accessibility relation R♦ for possibility and R K for knowledge, (4) is straightforward to guarantee: we need simply that (∀x) (∀y) (xR♦ y ⊃ yR K x) ¹² Thanks to Nick Smith for pressing me on this point.

( 4 )

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for then, when we are at x and we have some modally accessible world (call it y) in which every epistemically accessible world has p true, p is true at x, since x is epistemically accessible from y. Conversely, if we have some x and y where xR♦ y but not yR K x, then if p is true at everywhere other than x (or, if you like, everywhere epistemically accessible from y, if everywhere other than x seems like overkill) then at y, Kp is true, and hence at x, ♦Kp is true. However, at x, p is false. So, if we are allowed to assign the extension of a proposition at whim in our models (and it is hard to see why not) then condition ( 4 ) corresponds precisely to the validity of (4).

5.4 Similarly, we can say, precisely, what condition on R K and R♦ corresponds to conjunctive knowability in its weakest possible form. First note that, in a given model, if p is conjuctively knowable when [[p]] is a singleton set (so p is true at one world only) then every proposition is conjunctively knowable. (If φ is true at x, and if p is true at x alone, then consider the propositions, each knowable, which jointly entail p. These jointly entail φ —relative to that model—too, which shows that φ is also conjunctively knowable.) So, what does it take for a proposition true at x alone to be conjunctively knowable? Well, we must ﬁnd for any world y distinct from x, a proposition which is knowable but not true at y. If that is not found, the conjunction of all knowable propositions will not entail p, since it will also be true at y, where p is not true. So, we require the following condition (∀x) (∀y) (x = y ⊃ ∃z(xR♦ z ∧ ¬zR K y) )

(5)

for if (5) does not hold, then any z modally accessible from x will include y as epistemically accessible, so no proposition false at y will be knowable from x, as it will not be known at any modally accessible worlds.

5.5 It follows that normal epistemic modal models for conjunctive knowability satisfy (5). Alas if ( 4 ) and (5) both hold, then if R♦ is transitive, it is trivial in the sense that xR♦ y only if x = y. Here is why: if ( 4 ) holds, then ¬zR K y means that ¬yR♦ z, which when substituted in ( 4 ) gives (∀x) (∀y) (x = y ⊃ ∃z(xR♦ z ∧ ¬yR♦ z) ) but if x and y are non-identical and yR♦ x, then whenever xR♦ z by transitivity yR♦ x, which contradicts what we have assumed.

5.6 We have a syntactic proof of this modal collapse as well. We can show that (4), ♦♦p ♦p and conjunctive knowability (in the most general form ( 2 )), ensure that ♦p p, in the presence of propositional quantiﬁcation.

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5.7 Here is the proof: Suppose ♦p. So there is some possible circumstance in which p is true. Consider one. In this circumstance p is true, so there are propositions r 1 , r 2 , . . ., which together entail p, and each of which are possibly known. So, we have ♦Kr 1 , ♦Kr 2 , etc., and r 1 , r 2 , . . . p. Now consider the actual circumstance in which ♦p is true. In this circumstance, each ♦Kri is possible: that is, ♦♦Kri . But possible possibility is (we assume) possibility, so we have ♦Kri for each i. But by (4), ♦Kri ri , so each ri is true. But r 1 , r 2 , . . . p, so p is true too. In other words, we have inferred p from ♦p.

5.8 So, we cannot have (4), (5), ♦♦p ♦p and the non-triviality of ♦. One, at least, must go. Which one is to go? I am tempted to do away with (4), but we have already seen what can be done without (4): it makes conjunctive knowability all too easy. Making R♦ trivial is unacceptable, for then the only possibilities will be truths, so if every proposition is a conjunction of knowables, it will be a conjunction of knowns, and hence, every truth will be a consequence of what is known, making all ignorance vanish. We avoid Fitch’s paradox and its heirs by denying the premise that we are not omniscient. To do away with (5) is to give up the task of exploring the consequences of conjunctive knowability. The only remaining option here (given the machinery of normal epistemic modal logics and their possible worlds models) is to explore the rejection of the transitivity of R♦ . As a result, we will examine what follows if we deny the inference from ♦♦p to ♦p.

5.9 Denying transitivity of R♦ is a severe price to pay to save conjunctive knowability. It turns out that it is enough. In the remaining paragraphs of this section I will show that we may maintain (4), making every knowable a truth, and (5), making every truth conjunctively knowable, without concluding that every truth is known. A model showing this is relatively simple (Figure 21.4). The worlds are the (positive and negative) integers Z. We have xR♦ y iff y = x or y = x + 1. (Notice that this is not transitive, since 0R♦ 1 and 1R♦ 2, but we don’t have 0R♦ 2. Nonetheless, it is reﬂexive, so at the very least, p ♦p.) We have xR K y iff y = x or y = x − 1. (Notice that this is not transitive either, so we do not have Kp KKp, but it is reﬂexive, so Kp p, as one would hope.)

x−2

Figure 21.4. The Model

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5.10 That is the model. Let’s see how it manages to satisfy (4 ) and (5). We have satisﬁed (4 ) by ﬁat: if xR♦ y, then y = x or y = x + 1, in which case either x = y or x = y − 1, ensuring that yR K x. So, (4 ) is satisﬁed, ensuring that ♦Kp p. Conjunctive knowability, in the form of (5), is satisﬁed too. If x = y, then there is always some z where xR♦ z but not zR K y. If we don’t have xR K y, then choosing x for z will sufﬁce (since xR♦ x always). If we do have xR K y, then since x = y, we have y = x − 1. Then choose x + 1 for z. We have xR♦ z (z is one step up from x) but we don’t have yR K z (y is two steps down from z, which is just too far).

5.11 How does this model work? At every point, x, knowledge is a little limited because x is epistemically indistinguishable from x − 1. Only propositions true at both x and x − 1 may be known at x. Nonetheless, the world x + 1 is modally accessible from x, and at this world, x − 1 is not epistemically accessible but x is. This means that any proposition true at x is a conjunction of two knowable propositions (Figure 21.5). If p is true at the set X (including x) then consider two propositions q 1 and q 2 , true at X ∪ {x − 1} and X ∪ {x + 1} respectively. q 1 , true at X ∪ {x − 1}, is known at x (and so is possibly known at x) and q 2 , true at X ∪ {x + 1}, is known at x + 1 (and so is also possibly known at x). In this case, as in our other models, every proposition true at a point is a conjunction of two knowable propositions. Nonetheless, not every proposition is known: at every point there is ignorance.

x−2

x−1

x

x+1

x+2

Figure 21.5. {x} as a conjunction of two propositions knowable at x

5.12 So, if knowability entails truth then we can maintain the conjunctive knowability thesis in the form of (2), but only at the cost of rejecting the s4 principle for possibility: ♦♦p ♦p.

6 Fitch’s paradox shows us that not every proposition is knowable: at least not all at once. Fitch’s paradoxical sentence is an example of a proposition that cannot be

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known, but which can nonetheless be split into pieces, each conjunct of which can be known. It turns out that this modest fallback position is coherent. We may coherently hold that every proposition can be factored into a conjunction, each of which are knowable. Thinking of this in terms of possible worlds, it comes quite close to one original consideration in motivating of knowability. Propositions divide possible worlds into those that are in and those that are out. Conjunctive knowability tells us that for any world that a proposition takes to be out, we can know something that would rule out that world. Think of the discriminations that a proposition makes as constituted by all of the worlds inconsistent with it. According to conjunctive knowability, no proposition makes a discriminiation essentially beyond our grasp. This is coherent. If two inconsistent propositions are knowable, then conjunctive knowability is coherent but trivial. If knowability entails truth, then conjunctive knowability is both coherent and substantial.

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Index adjunction 107 agnostic ﬁctionalism 273–7 algebra 25 n. 7 aliases, problem of 156, 157 Analyticity 285 Anderson, C. A. 170 n. 13 Anderson, A. R. 21 n. 2, 26 n. 7, 119 anonymity, problem of 24 n. 5, 156, 157 anti-realism 5–6, 65–6, 74, 183–204 and equivalence thesis for truth (ET) 62, 63 and equivalence thesis for warranted assertibility (EA) 63–4 global 298, 302, 303 intuitionism and 78 knowability and 209–11 about meaning 289–92, 297–8 and recognition-transcendence 289–92, 293–4 semantic 53–5, 77, 250–1, 325–8, 332–8 and truth 184–8, 289–92 Armour-Garb, B. 207 n. 6 Armstrong, D. 307 n. 6 Artemov, S. 131 assertibility of disjunctions 192–3 warranted 63–4 asserting 217–18, 273 assertions 285–6 self-defeating 286–7, 294 self-fulﬁlling 285 Ayer, A. J. 293 Azzouni, J. 276 Balaguer, M. 254, 255, 258 Baltag, A. 139 Barcan Marcus, R. 21 n. 3 Beall, Jc 7–8, 102 n. 9, 119 n. 23, 125 n. 32, 303 n. 1, 319, 341 belief 3–4, 39, 40, 164–6, 169–75, 176 disbelief principle 166, 171 false common 145 type theory and 169–70 believing 22, 31 Belnap, N. 26 n. 7, 119 van Benthem, J. 8, 33 n. 4, 130–46 Bigelow, J. 5 n. 4 Binkley, R. 166 bivalence 66–7

Boolos, G. 321 n. 1 Brock, Stuart 243 Brogaard, B. 5 n. 5, 9, 10, 47 n. 22, 130, 147, 158, 187 n. 5, 207 n. 3, 227 n. 4, 230 n. 5, 231 n. 6, 242 n. 3, 247, 249 n. 20, 250 n. 21 & n. 22, 252 Bueno, O. 9, 261 n. 12 & n. 13, 265 n. 17, 277 Burali-Forti paradox 80 Burge, Tyler 322 Burgess, J. 8, 35 n. 10 Burks, A. W., 23, 33 n. 5, 46 Burm´udez, J. 7 Cantorian sets 254 Cantor’s proof 80 Carnap, Rudolf 21 n. 3,173 causal necessity 23 causal possibility 23 causation, partial 25–6 Church, Alonzo 8, 67, 168, 179, 302, 322 ﬁrst referee report on Fitch’s ‘‘Deﬁnition of Value’’ 13–15, 36–40, 163 n. 1 identiﬁcation as anonymous referee 35–7 second referee report on Fitch’s ‘‘Deﬁnition of Value’’ 15–18, 41–5, 169 n. 11 theory of types 168 Clark, P. 82 closure 107, 111, 118 Cogburn, J. 5 n. 4, 8, 20 Colyvan, M. 265 n. 16 coming to know 131–3, 139 common knowledge 134, 135–6, 140, 145 and multi-agent learning 143–4 and public announcements 138, 141–3 compositionality 298 conditional fallacy 38, 46, 48, 250 n. 22 conjunctive knowability thesis 341–53 worlds theory and 344–6, 351, 352, 353, 354 consistency 117, 171, 254, 256–61, 267, 269, 271–3 contradictions 109–10, 112, 117 contraposition 107 in Fitch’s paradox 97–9 da Costa, n. C. A. 261 n. 12, 277 Costa-Leite, A. 9–10 Cozzo, Cesare 189 n. 9, 207, 292 n. 11, 305–6, 308

368

Index

de dicto knowledge 155–7 de re knowledge 155–7 decidability 66–7, 190–3, 195–6, 290 decidable sentences 59–61, 66–7, 209 deontic necessity 23 deontic possibility 23 desire 23, 27 knowledge and 40, 41–3 Devidi, D. 6 n. 7, 8, 189 n. 9, 207 n. 3, 249 dialetheism 97, 100–1, 102 disbelief axiom 173 disbelief principle 166, 171 discovery principle 147–57 de dicto knowledge 155–7 de re knowledge 155–7 formalization of 151–5 objections to 149–51 temporal paradox 147–9 distribution 106, 111, 117 distribution axiom 165 van Ditmarsch, H. 138, 139 Divers, John 244, 245, 246, 249, 250 doing 22, 26 Douven, I. 309 Dummett, Michael 54, 63, 64, 66, 129, 183, 192–3, 207, 237, 248 n. 18, 250, 289, 290, 292 n. 11 anti-realism 1, 5–7, 53–4 basic/atomic sentences 73 indeﬁnite extensibility 78, 79–81, 82–4 inductive speciﬁcation 89–90 intuitionistic logic 186–7 justiﬁcationism 51–2 knowability principle, restriction of 6–7, 187 n. 5 manifestation argument 250–1, 326 n. 4, 326–7 meaning theory 297, 327 natural numbers 80–1 negation 189 ordinal numbers 79–80, 83–4 quantiﬁcation 82–4 response to Fitch’s paradox 89–90 restricted knowability principle 6–7, 73, 305, 297–300, 304 n. 2, 305 Dunn, J. M. 114 n. 18, 119 n. 24 Edgington, Dorothy 9, 140, 158, 189 n. 10, 204 n. 28, 207, 248, 304 n. 2, 306, 337, 338 Egr´e, P. 146 Enderton, H. B. 35 n. 10, 80 ephemeral truths 150–1 epistemic logic dynamic 134–6, 138, 145 event updates 136–7, 139–40

and evidence 130–1 static 133–4 epistemic realism 278 n. euclidean geometry 275 evidence-transcendent truths 326–8, 332 excluded middle 97–8, 108 factiveness 225 and normalization 233–5 operators 30–2, 33–4 restriction on 230–7 factivity principle: weakening of 329–31 Fagin, R. 141 false common belief 145 Fara, M. 9 ﬁctionalism 241–80 agnostic ﬁctionalism 275 intuitionistic strategy 247 mathematical 264–73 modal 242–5, 250 modal fallacies strategy 247–8 restriction strategy 248–50 rigidiﬁer strategy 248 ﬁctionalist epistemology 255–6 Field, Hartry 255, 264, 265–6, 267–8, 269, 270, 277, 279 Fitch, Frederic axioms 14, 20 Cartesian restriction 41–5, 46, 47–8 ‘Deﬁnition of Value’: withdrawal of 45 deﬁnitions 14–15, 16–17, 19, 33–4, 37, 41 empirical necessitation 37 n. generalized paradox 3–4 operators 4, 18, 30–1 theorems 1, 3, 20, 31–3, 36, 37 n., 45–6, 76–8 theory of value, collapse of 47–8 Fitch self-defeat 288, 289, 294 Fitch’s paradox 85, 320–3, 340–1 anonymous referee, identity of 34–7 basic rules 106–7 deﬂationary approach I: weakening the factivity principle 329–31 deﬂationary approach II: weakening the knowability principle 332–8 premises of 321 and proof 320–1 responses to 77–8, 89–90, 205–7 Florio, S. 229 van Fraassen, B. 279 Frege, Gottlob 272 Gabbay, D. 131 game theory 135, 139, 144, 145 Gerbrandy, J. 138

Index global anti-realism 298, 302, 303 G¨odel, Kurt 34–5, 212, 257, 262, 271–3, 277, 279 G¨odel-mapping (G¨odel-McKinsey-TarskiRasiowa-Sikorski mapping) 57, 58, 62, 66, 72–3, 74 G¨odel’s axiom 321 G¨odel sentences 257, 264 Greenough, P. 325 n. 3 Grim, P. 86 Guldmann, F. 62 n. 10 Gupta, Anil 322 Hagen, J. 244 n. 9, 246 n. 15 Hale, B. 243–4, 249 n. 19 Halpern, J. 141 Hand, Michael 8, 9, 197, 207, 214 n. 11 & n. 12, 219 n. 12, 249, 285 n. 3, 291 n. 10, 297 n. 14 Hart, W. D. 5, 9, 39, 45 n. 20, 54, 59, 95, 193 n. 16 Hazlett, Allan 331 n. Herzig, A. 139 Heyting, A. 66 Hilbert spaces 254 Hilpinen, R. 343 n. 5 Hintikka, Jaakko 130, 135, 146, 165, 172, 175 Hoshi, T. 139 Humberstone, L. 5 n. 5 idealism, problem of 300–1 ignorance 1, 29, 35: see also knowledge incompleteness 34–5, 123, 212, 257, 271–2 inconsistency 121–3: see also consistency indeﬁnite extensibility 78–81, 82–4, 85–6, 87 natural numbers 80–1, 82 ordinals 79–80 of propositions 86, 88, 90 real numbers 80 sets 80 indexical sentences 285, 289 indispensability argument 264–5 ineffable truths 150 inference 284 in Fitch’s paradox 95–7, 99–100 global restrictions on rules of 225–6 inscribing 218 intuition: Platonism and 264 intuitionism 6, 56–7, 65, 78, 186–8 intuitionistic logic 186–7 intuitionistic provability 57 irrationality 28 irrealism 321: see also anti-realism; realism

369

Jago, M. 146 James, P. 26 n. 7 Jehle, D. 245 n. 12 Jenkins, C. S. 9, 213 n. 9, 247, 306–9, 317 n. 17 justiﬁcationism 51–2 justiﬁed belief 332–8 reﬂection principles for 333–4, 335, 336 Kamp, Hans 322 Kelly, K. 145 Kelp, C. 9 Kenyon, T. 8, 207 n. 3, 249 knowability 107, 111–12, 117 and anti-realism 209–11 global restriction on 226–30 and inconsistency 123 and knowledge 121–2 and non-omniscience 105–6, 112, 113, 118 and paraconsistent veriﬁcationism 118–21 and theism 207–9, 218, 220–1 and trivial world 111, 112, 113 truth and 184–6 validity of 109, 111, 112, 113, 118–21 validity, non-normal semantics of 114–17 and veriﬁcationism 113–14, 118–21 knowability paradox 1–3, 32–3, 283–5, 302–3 classic puzzle 303–8, 318 and lost logical distinction 209–22 new puzzle 304, 308–16, 318 and omniscience 241 operators and 214–19 plausibility of 324 and syntactic generalization strategy 216–19 knowability principle 1–3, 5–6, 77–8 global restriction 223–38 normalization and 229–30 and omniscience 252–3 and Peirceanism 161–2 performance principle and 295–6 restrictions on 6–7, 187 n. 5, 223–38, 296–300 and temporal/modal analogy 157–9 and temporal/modal combination 159–60 weakening of 332–8 knowability theorems 1, 3, 20, 24–5, 31–3, 36, 37 n., 45–6, 76–8 Knower paradox 101–3, 108 knowing 22, 31 deﬁnition of in terms of believing 26–7 knowledge 39–40, 121–2, 164, 171–3, 178 and abnormal epistemic possibilities 124–5 and desire 40, 41–3 explicit temporal perspectives on 140–1

370

Index

knowledge (cont.) in Fitch’s paradox 95–6 interpretation of ‘know’ 194–6 limits of, and contradiction 102–3 see also ignorance Kooi, B. 138 Krause, D. 261 n. 12, 277 Kripke, S. 114 n. 18, 117 n. 22, 139, 209 n. 4, 335 K¨unne, Wolfgang 35 Kuppfer, M. 285 n. 3 Kvanvig, Jonathan L. 5 n. 5, 8, 9, 197, 204 n. 28, 207 n. 3, 247, 249, 297 n. 14, 304, 309–19, 311, 312, 318–19, 328 n. 12, 337 n. 19 learning (coming to know) theory 145 Lewis, David 242, 243–6, 270 Lewis/Langford: deﬁnitions and theorems 13, 18–19 liar paradox 197 n. 21, 323 de Lima, T. 139 Linsky, B. 3 n. 3, 5, 8, 39, 322 Linstr¨om, S. 9, 189 n. 10, 204 n. 28 Liu, F. 135 logical necessity 23 logical possibility 23, 256–7 McDowell, J. 70 n. 19 McGee, Vann 212 McGinn, C. 5, 45 n. 20, 54, 59, 95, 276 McGlynn, Aidan 315 n. MacIntosh, J. J. 5 n. 4 Mackie, John 5, 45 n. 20, 214, 287, 294, 311 n. 10, 316 n. 15, 334, 335 Makinson, D. C. 167 n. 8 manifestationism 250–1, 298 mapping objection 57–9, 63–4, 71, 73 negative part 58–9, 62, 63, 64, 74, 75 positive part 58, 59, 74 Martin-L¨of, P. 129 mathematical ﬁctionalism 264–73 mathematical propositions: indeﬁnite extensibility of 88 mathematics, philosophy of comprehension principles 261–4, 275–6, 277, 280 and full-blooded Platonism 254–61, 280 mathematical objects, existence of 254, 255–6, 261, 262, 263, 264, 265, 271, 273–4, 275–7, 276–7 and standard Platonism 261–4, 280 meaning theory 73, 300, 325–7 acquisition challenge 327 anti-realist 289–92, 297–8 manifestation challenge 327

realist theories of 325–6 and truth 297–8 Melia, J. 9, 204 n. 28 Meyer, Robert 34–5 modal ﬁctionalism 242–5 modal realism 243–4, 249 & n. 19 molecularism 298 Moses, Y. 141 Moore’s Paradox 36, 130, 132, 165 Moretti, L. 5 n. 4 Muddy Children puzzle 141–3 Murzi, J. 9, 36 Nagel, Ernest 13, 34 n. 7, 35–6, 40–1 natural numbers: indeﬁnite extensibility of 80–1, 82 NBG (von Neumann-Bernays-G¨odel) set theory 263, 267–8, 269, 271–2, 273 necessity 21–3 negation 170, 189 Nolan, D. 242 n. 4, 243 n. 6 & n. 7 nominalism, see mathematical ﬁctionalism non-assertibility 66 non-contradiction 107 non-omniscience: 105–6, 112, 113, 118: see also omniscience Nozick, R. 31 n. 2 object theory 261 n. 13 obligation 27, 213 omnipotence, paradox of 4 omniscience 328–9 full-blooded Platonism and 257–8 and knowability paradox 241 and knowability principle 295–6 see also non-omniscience omniscience principle 1–3, 78, 85 ontological argument 318 Orayen, Ra´ul 323 n. 6 ordinal numbers 79–80, 83–4 ordinary language: value concepts and 21–2 Osborne, M. 145 Pacuit, E. 140 Pagin, P. 189 n. 9 paracompleteness 108–9 and inconsistency 121–2 semantics 110–11 Strong Kleene framework 108–9 and trivial worlds 111, 112, 123 paraconsistency 7, 97–8 paraconsistent veriﬁcationism 118–23 and inconsistency 121–3 and knowability, validity of 118–21 Parry, William 25 n. 6, 36 Parsons, Charles 268, 322

Index partial causation 25–6 Peano axioms 275 Peirceanism 161–2 Percival, P. 6, 9, 99 n. 3, 189 n. 9 & n. 10, 247 n. 17 performance principle: and knowability principle 293, 295–6 Plantinga, Alvin 176–7, 209 n. 5 Platonism 253, 274 comprehension principles in mathematics 261–4 full-blooded 254–61, 280 and intuition 264 standard (traditional) 261–4, 280 plurality of worlds theory 243–4, 246, 247, 249 Poincar´e, H. 322 n. 2 possibility 21–2, 93, 254 causal 23 deontic 23 in Fitch’s paradox 96–7 logical 23, 256–7 lost distinction between actuality and 209–13, 219 possible worlds theory 131, 220, 242–3, 249–50, 270, 306–7, 314 and conjunctive knowability thesis 344–6, 351, 352, 353, 354 possibly true contradictions 109–10, 112, 117 Potter, M. 82 pragmatic self-defeat 286–9 Prawitz, Dag 31 n. 3, 237, 292 n. 11 Preface Paradox 163, 166–8 Priest, G. 8, 100 n. 5, 103 n. 11, 104 n. 2 & n. 3, 110 n. 10, 111 n. 11, 119 n. 23, 125 n. 32 Prior, Arthur 151, 159–61, 167 n. 8 Pritchard, D. 9, 325 n. 3 propositions and conjunction elimination 22–3, 24–5, 27 and conjunction introduction 23, 27 compound 87, 89 indeﬁnite extensibility of 78–9, 86, 88, 90 mathematical 88 sets of 86, 87 simple 87, 88, 90 truth classes of 23–4, 27, 76–7, 164 proving 22 public announcements 132, 134–5, 136 and common knowledge 138, 141–3 Putnam, H. 1, 5, 265 n. 16, 327 n. 10 quantiﬁcation over indeﬁnitely extensible domains 78–9, 81, 82–7

371

and principles of transmissibility 84–5, 86, 87–9 and truth values 78–9, 82–3, 85 Quine, W. V. O. 169, 265 n. 16 Raatikainen, P. 292 n. 11 Rabinowicz, W. 9, 189 n. 10, 204 n. 28 Ramsey, F. 321–2 Rasmussen, S. 6, 7, 54–5, 62 Ravnkilde, Jens 6, 54, 55 Rea, M. 5 n. 4 real numbers 80 realism and decidable sentences 59–61 epistemic 278 n. meaning theories 325–6 modal 243–4, 249 n. 19 semantic 56 recognition-transcendence 289–92, 293–4, 300 relevance logic 349 reliabilism 272–3 Rescher, N. 33 n. 4, 35 n. 9 Resnik, M. 264 n. 15 Restall, G. 9, 114 n. 18, 119 n. 23 restricted realism 61, 62 restriction strategies 74 Dummett 6–7, 73, 305 on rules of inference 225–6 Tennant 8, 41, 55, 71–3 Robinson, A. 275 Rosen, Gideon 242 & n. 4, 243, 244 n. 9, 249, 250 Rosenkranz, S. 47 n. 22, 187 n. 5 Routley, Richard 5, 7, 8, 34, 35, 45 n. 20, 102 n. 9 Rubenstein, A. 145 R¨uckert, J. 9, 158 n. 5, 189 n. 10, 204 n. 28, 306 Russell, Bertrand 5, 79, 163 n. 1, 170 n. 13, 178 n. 17, 321 Russell set 277 Russell’s paradox 315 Russell’s theory of types 17, 321–3 Sack, J. 140 Salerno, J. 5, 6, 6 n. 5, 9, 10, 47 n. 22, 130, 147, 158, 187 n. 5, 207 n. 3, 227–8, 230 n. 5, 231 n. 6, 242 n. 3, 247, 249 n. 20, 250 n. 22, 252, 297 n. 14, 302, 304 n. 2, 322 n. 4, 329 n. 13, 341 n. 3, 343 n. 5 Salerno proof: global restriction 227–8 Segerberg, K. 9, 189 n. 10, 204 n. 28 self-defeat 294, 300 and indexicals 289 pragmatic 286–9

372 self-defeating assertions 286–7, 294 self-fulﬁlling assertions 138–9 self-reference 177 type theory and 167–9, 171–2 semantic anti-realism 53–5, 77, 325–8 and justiﬁed belief 332–8 timid 250–1 semantic realism 56 set theory 274–5, 321–2 NBG (von Neumann-Bernays-G¨odel) 267–8, 269, 271–2, 273 Zermelo-Fraenkel (ZFC) 323 n. 6 Shalkowski, S. 270 Shapiro, S. 264 n. 15, 267 Shope, Robert 178 simpliﬁcation 107 Smith, N. 350 n. 12 Solomon, G. 6 n. 7, 189 n. 9 Sorenson, Roy 36 n. 12, 101 n. 7, 165 n. 5, 168–9 Spicer, Finn 331 n. Stanley, J. 10 Stebel, K. 316 n. 16 Stjernberg, Frederik 166 n. 7 Stone, Jim 44 n.19 striving 22 strong modal ﬁctionalism 242, 247, 250 structuralism 264 n. superassertibility 56 n. Surprise Examination Paradox 138, 163,168–9, 170 n. 12 Szabo, Z. 10 Tarski, Alfred 265, 322, 323 n. 6 Taylor, Richard 221 n. Tennant, Neil 6, 47 n. 22, 54–5, 130, 183–204, 188 n. 7, 207, 223–38, 251 n. 23, 291 n. 10, 300, 327 n. 10 anti-realism 54, 305 anti-realist response to Fitch argument, soft 197–204 anti-realist treatment of Fitch argument, moderately hard 189–97 decidability 190–3, 195, 196, 197 factiveness 225, 230–7 interpretation of ‘know’ 194–5 intuitionistic relevance logic 198–200 restricted knowability principle 197–204, 297–300 restriction: Cartesian 5, 8–9, 41, 47, 55, 71–3, 197–204, 248, 249, 304 n. 2, 305 restriction: global 223–38 restriction: importance of normal form for 229–30, 233–5

Index theism 207–9, 218, 220–1 time-indexing 60, 61, 62, 64–5, 73, 74, 75 timid modal ﬁctionalism 242 n. 4, 250 timid semantic anti-realism 250–1 truth 23 anti-realism and 184–8, 298–92 equivalence thesis for 62, 63 and knowability 184–6 recognition-transcendence of 289–92, 293–4 veriﬁcation and 176–7, 178–9 truth classes of propositions 23–4, 76–7 theorems about 24–5 truthmaker necessitarianism 307 truths ephemeral 150–1 evidence-transcendent 326–8, 332 ineffable 150 Tsohatzidis, S. 285 n. 3 type theory 177–8, 321–3 and belief 169–70 and self-reference 167–9, 171–2 veriﬁcationism and 131 understanding 83, 291, 326–7 unrestricted claims problem 245–6 Usberti, G. 189 n. 9, 204 n. 28 uttering 217 value 15, 27–8 informed-desire theory of 33 and ordinary language 21–2 Vardi, M. 141 veridicality 106, 111, 117 veriﬁcation 193–4 canonical 283 n. and truth 176–7, 178–9 veriﬁcation principle 93–4, 96–7, 101 veriﬁcation procedures: performance of 292–3, 296 veriﬁcationally inconsistent sentences 68–70 veriﬁcationism 45, 77, 108–9, 133, 146, 163–4, 173–6 knowability and 113–14 and knowledge and abnormal epistemic possibilities 124–5 non-normal semantics of validity 114–17 paraconsistent 118–23 and paradoxes 129–30 and Peirceanism 161–2 strong 303, 304, 309, 310, 312, 313 and type theory 131 weak 303–7, 308–13, 318–19 veriﬁcationist thesis 129, 132, 133, 140, 145

Index Walton, D. 5 n. 4 Wansing, H. 7, 341 Warﬁeld, T. 39 n. 16 warranted assertibility 63–4 Wehmeir, K. 158 n. 5 Weiss, Bernhard 51, 55, 64 n. 14, 65, 74, 75 Whitehead, A. N. 78 n. 13, 163 n. 1, 170 n. 13, 178 n. 17 Williamson, Timothy 1, 2, 4, 6, 9, 29 n., 62, 47 n. 22, 53, 57, 62–4, 75, 101 n. 6, 158, 166 n. 7, 169 n. 10, 170 n. 12, 175, 183–204, 206–7, 211, 213 n. 9, 247, 249 n. 20, 273, 297, 302, 309 n. 9, 323 n. 6, 324 n. 1, 329 n. 13 Wittgenstein, Ludwig 323 n. 5, 326–7 Wright, Crispin 5 n. 4, 6, 9, 38 n. 14, 54, 63, 178, 189 n. 9 & n. 10, 193 n. 16,

373

247 n. 16, 288, 294 n. 12, 326 n. 4, n. 5, & n. 6, 327 n. 7, n. 8, & n. 10, 328 n. 11 anti-realism 54, 55 lost-opportunity cases 294 n. provisional biconditionals 173–4, 176–8 on realism 326 n. 4 superassertibility 56 n. veriﬁcationism 173–4 writing 218 Yap, A. 140 Zalta, Ed 261 n. 13 Zermelo, E. 263, 321 Zermelo-Frankel set theory 263, 264 n. 15, 315 n. 14 Zimmerman, T. 285, 286 n. 4