Mistakes of Reason: Essays in Honour of John Woods 9781442677296

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MISTAKES OF REASON: ESSAYS IN HONOUR OF JOHN WOODS

John Woods

MISTAKES OF REASON Essays in Honour of John Woods

Edited by Kent A. Peacock and Andrew D. Irvine

UNIVERSITY OF TORONTO PRESS Toronto Buffalo London

www.utppublishing.com © University of Toronto Press Incorporated 2005 Toronto Buffalo London Printed in Canada ISBN 0-8020-3866-2

Printed on acid-free paper Toronto Studies in Philosophy Editors: Donald Ainslie and Amy Mullin

Library and Archives Canada Cataloguing in Publication Mistakes of reason : essays in honour of John Woods / edited by Kent Peacock and Andrew Irvine. (Toronto studies in philosophy) Includes bibliographical references and index. ISBN 0-8020-3866-2 1. Philosophy. I. Peacock, Kent Alan, 1952– . II. Irvine, A.D. III. Woods, John, 1937– . IV. Series. B29.M58 2005

191

C2005-902066-0

Frontispiece photograph of John Woods courtesy of Robert Cooney, Office of University Advancement, University of Lethbridge. University of Toronto Press acknowledges the financial assistance to its publishing program of the Canada Council and the Ontario Arts Council. University of Toronto Press acknowledges the financial support for its publishing activities of the Government of Canada through the Book Publishing Industry Development Program (BPIDP).

Contents

Preface

ix

Acknowledgements

xi

Introduction: John Woods in Profile d. irvine 3

kent a. peacock and andrew

I Reality 1 Through the Woods to Meinong’s Jungle 2 The Epsilon Logic of Fictions

nicholas griffin

b.h. slater

15

33

3 Animadversions on the Logic of Fiction and Reform of Modal Logic dale jacquette 49 4 Resolving the Skolem Paradox lisa lehrer dive 64 5 Are Platonism and Pragmatism Compatible?

victor rodych 78

6 A Neo-Hintikkan Solution to Kripke’s Puzzle

peter alward

Part One: Respondeo john woods

103

II Knowledge 7 The Day of the Dolphins: Puzzling over Epistemic Partnership bas c. van fraassen 111 8 Cognitive Yearning and Fugitive Truth

john woods

9 The de Finetti Lottery and Equiprobability

134

paul bartha

158

93

vi

Contents

10 The Lottery Paradox

jarett weintraub

173

11 Reliabilism and Inference to the Best Explanation samuel ruhmkorff 183 Part Two: Respondeo

john woods

197

III Logic and Language 12 Aristotle and Modern Logic

d.a. cutler

207

13 The Peculiarities of Stoic Propositional Logic david hitchcock 224 14 On the Substitutional Approach to Logical Consequence matthew mckeon 243 15 The Fallacy of Transitivity for Necessary Counterfactuals: On Behalf of (Certain) Non-Transitive Entailment Relations jonathan strand 264 16 Vagueness and Intuitionistic Logic: On the Wright Track david devidi 279 17 The Semantic Illusion Part Three: Respondeo

r.e. jennings john woods

296 321

IV Reasoning 18 Arguing from Authority

leslie burkholder

19 Premiss Acceptability and Truth

331

james b. freeman

348

20 Emotion, Relevance, and Consolation Arguments trudy govier 364 21 Temporal Agents

jim cunningham

380

22 Filtration Structures and the Cut Down Problem for Abduction dov m. gabbay and john woods 398 23 Mistakes in Reasoning about Argumentation Part Four: Respondeo

john woods

george boger

418

442

V Values 24 Engineered Death and the (Il)logic of Social Change stingl 453

michael

Contents vii

25 Incorrect English

michael wreen

474

26 Ameliorating Computational Exhaustion in Artificial Prudence paul viminitz 491 Part Five: Respondeo Contributors

511

Books by John Woods Index

521

517

john woods

504

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Preface

This book has its beginnings in a conference held in honour of John Woods, one of Canada’s most eminent logicians and philosophers. Most of the papers in this volume were presented (some in a different form) at the University of Lethbridge, 19–21 April 2002, while a few were added later. All are inspired (perhaps in some cases provoked or incited) by the themes that have animated Woods’ wide philosophical opus, which has ranged over the history and philosophy of logic, deviant logics, inductive and abductive reasoning, informal reasoning, fallacy theory, the logic of fiction, and the intense debates over ‘engineered death’ (abortion and euthanasia). In our Call for Papers, we invited authors to explore the following question: ‘What can reason accomplish in an often unreasonable world?’ This was the best way we could think of to capture the essence of the concern that has animated Woods’ teaching and his many books and papers. In this spirit, the contributors to this volume explore in various ways the nature and limits of human rationality, and the prospects for its improvement. The book is divided into five parts – Reality, Knowledge, Logic and Language, Reasoning, and Values – reflecting the editors’ best attempt to subdivide Woods’ wide philosophical interests. Each part is capped by a brief rejoinder by Woods himself.

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Acknowledgements

Neither the conference on which this book is based, nor the book itself, could have been possible without funding from the Social Sciences and Humanities Research Council of Canada through their Aid to Occasional Scholarly Conferences Programme. Generous support – financial, material, and moral – also came from Dr Bhagwan Dua, former Dean of the Faculty of Arts and Science at the University of Lethbridge, Dr Chris Nicol, present Dean of that Faculty, Dr William Cade, President of the University of Lethbridge, and the family of John Woods. We also wish to thank the conference participants, most of whom themselves bore some or all of the cost of their own attendance. It is tautological but hardly trite to observe that without conferees there can be no conference. Here is a complete list of those who attended and gave talks or papers: Peter Alward (University of Lethbridge), Rani Lill Anjum (University of Tromsø), Paul Bartha (University of British Columbia), George Boger (Canisius College), Jim Cunningham (Imperial College, London), Darcy Cutler (University of British Columbia), David DeVidi (University of Waterloo), Lisa Lehrer Dive (University of Sydney), James B. Freeman (Hunter College of City University of New York), Dov M. Gabbay (King’s College, London), Carlos E. Garcia (University of Florida), Trudy Govier (Independent Scholar), Nicholas Griffin (McMaster University), David Hitchcock (McMaster University), Sarah Hoffman (University of Saskatchewan), Dale Jacquette (Pennsylvania State University), Ray Jennings (Simon Fraser University), Tyrone Lai (Memorial University of Newfoundland), Matthew McKeon (Michigan State University), Mark Migotti (University of Calgary), Victor Rodych (University of Lethbridge), Timothy Rosenkoetter (University of Chicago), Samuel Ruhmkorff (Simon’s Rock College of

xii Acknowledgements

Bard), Timothy Schroeder (University of Manitoba), Jonathan Seldin (University of Lethbridge), Robert Sinclair (Simon Fraser University), Hartley Slater (University of Western Australia), Michael Stingl (University of Lethbridge), Jonathan Strand (Concordia University College of Alberta), Mariam Thalos (University of Utah), Bas C. van Fraassen (Princeton University), Paul Viminitz (University of Lethbridge), Jarrett Weintraub (University of California at Riverside), John Woods (University of Lethbridge), and Michael Wreen (Marquette University). In organizing and running the conference we received indispensable on-the-ground help from Peter Alward, Dawn Collins, Bob Cooney, Quincy Geiger, Rachel Harvey, Ryan Jade, Randa and Alexa Stone, Tina Strasbourg, Jillain Tuininga, Carol Woods, and Paul Viminitz. At the University of Toronto Press, Ron Schoeffel gave the project a leg up, and Len Husband and Frances Mundy have provided steady guidance throughout. There is no telling how long the manuscript could have taken to complete without Dawn Collins’ expert editorial assistance. Victor Rodych of the University of Lethbridge suggested ‘Mistakes of Reason’ as the title for both the conference and the book; and John Woods himself, indefatigable trouper that he is, contributed substantially to the conference and to this book in the way of good advice, moral support – and many pages of copy. Good reasoners, all!

MISTAKES OF REASON

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Introduction: John Woods in Profile KENT A. PEACOCK AND ANDREW D. IRVINE

Reason is one of the human species’ most important survival tools, and the capacity that philosophers once believed separates humanity from the lesser beasts. But it is notoriously fallible. ‘Mistakes of Reason,’ the title of this book, draws attention to the uncomfortable fact that our faculty of reason is beset with characteristic trompes l’oeil intellectuels, optical illusions of the mind. Many of these so-called fallacies are so typical of the human animal and so recurrent that we give them special names and teach them to undergraduates; but the flaws in reason are deep and systemic. They are not readily capturable in their full complexity in a neat taxonomy of fallacies, and we still do not fully understand how best to cope with them, despite all the progress that logic, philosophy, and mathematics have made since the days when Aristotle and his students ambled through the gardens of the Lyceum. Few Canadian philosophers have wrestled so long or so productively with the diagnosis and treatment of reason itself as has John Woods. In recent work, Woods, in collaboration with Dov Gabbay1 have dared to suggest that our systematic tendency to reason fallaciously in certain familiar ways is not merely a sort of failure of our neural hardware, like a computer chip failing to sum two numbers correctly. Rather, it may be a set of compromises adopted by the mind as a way of making the best of a bad job, a set of adaptations that often gets us by when we are in a hurry despite their tendency sometimes to lead us from true premises to false conclusions. Understood as a set of heuristics evolved ‘on the fly,’ under ceaseless pressure of the limitations of information and time, the characteristic failings of reason may not, in the end, be all that unreasonable; but they must, at the very least, be understood to be transcended.

4 Kent A. Peacock and Andrew D. Irvine

John Woods’ recent studies of the workings of practical reason are a natural product of the long development of his thought. We suppose one would have to call Woods an analytic philosopher, but his work (though displaying great technical adroitness when necessary) is in no way narrow or reductionistic. He has always been concerned, above all else, with the difference that how we think and how well we think make to the whole human project. Woods’ character and experience suit him to large intellectual investigations. Those of us who have been privileged to work with him have been constantly beguiled by his erudition and sheer intellectual energy. It seemed that at the drop of a hat he could go home and draft a 10,000word review of the latest work on quantum logic, abductive inference, or natural religion. Woods has never shied away from controversy but at the same time is a dogged and effective conciliator (a virtue that must have been honed by his many years of experience as a senior academic administrator). At the University of Lethbridge, his old-fashioned courtly manner and formal bow ties were sometimes jarring to students in these days of backwards baseball caps and shirt-tails out, but his office door was always open. He treated his students with unfailing courtesy and friendly respect, while challenging them mercilessly in his courses. He is always able to take a joke, even when we irreverently refer to him as ‘Canada’s leading fallacious thinker.’ In fact, Woods was certainly one of the most respected faculty members at the University of Lethbridge, and this is reflected in the fact that he was the first person at that university to win both its awards for outstanding researcher and outstanding teacher.2 A notable virtue possessed by Woods is his willingness, indeed eagerness, to foster a new approach or promote talent even if it is unconventional. (One of the editors of this book, in particular, is a beneficiary of this trait.) Quite apart from his own impressive intellectual productivity, Woods is a great facilitator and instigator, one of those people who somehow make things happen. This volume is a modest payment of interest on Canadian philosophy’s large debt to John Woods. Woods was born in Barrie, Ontario, in 1937. He describes himself as a fifth-generation Canadian of mainly Irish Catholic stock – an Irish Catholicism oddly tempered, Woods says, by a zealous Anglophilia and loyalty to Empire. Woods describes some early highlights of his education: An early philosophical moment occurred when, at age four, I was saying my bed-time prayers. ‘God bless Mummy and Daddy and [sisters] Barbie

Introduction 5 and Joanie,’ I bade the Almighty, ‘and make me a good boy after a while.’ Years later I would discover my Augustinian sensibility in the entreaty by the author of the Confessions that God render him chaste, but not now. Two years later, our Grade One teacher asked us what we thought enabled Our Lord to perform the miracle of the loaves and fishes. Up snapped my hand: ‘Because He is magic!’ ‘No, no,’ admonished Sister Anne, ‘it is because He is God.’ Thus I was introduced to the dissatisfaction of vacuous truths. A third episode involved my art teacher, Mrs Harvey, who offered instruction on Saturday mornings. She would give me, in effect, a lesson in the appearance-reality distinction. ‘You can’t get snow right with white paint,’ she insisted. ‘How could this be?’ I wondered. ‘Isn’t white the colour of snow?’ ‘Because,’ Mrs. Harvey replied, ‘when you see snow you always see more than its colour. You also see shadow. So you don’t paint what the colour of snow is; you paint what snow looks like.’3

Woods somehow survived the ministrations of Sister Anne, and enrolled, at the relatively tender age of seventeen, in the first year of Social and Philosophical Studies at the University of Toronto. He found himself drawn to philosophy partly because a high school history teacher named Mr Fisher, perhaps himself a thwarted philosopher, devoted most of a course that was supposed to be on North American history to philosophy. After a slightly shaky start marked by what Woods calls ‘a thirsty but undisciplined eclecticism and a fondness for late nights,’ Woods excelled in his studies and graduated in 1957 with high standing. He recalls two of his professors, David Gallop and Douglas Dryer, with special affection and respect. His fellow graduates included future luminaries Barry Stroud and Ted Honderich. He followed his bachelor’s degree with a master’s at Toronto, by which time he realized that he had been bitten incurably by the philosophical bug, and then went on to the University of Michigan at Ann Arbor for his doctorate. At Ann Arbor, Woods worked in the company of William Frankena, Paul Henle, Richard Cartwright, Julius Moravcsik, William Alston, Edmund Gettier, Terence Penelhum, J.O. Urmson, John Searle, and Alvin Plantinga. In 1962 he accepted a job offer from the University of Toronto and defended his thesis – Entailment and the Paradoxes of Strict Implication – in 1965. As Woods tells us, My time in Ann Arbor had equipped me with a nascent conception of how philosophy should be done. It emphasized the critical importance of

6 Kent A. Peacock and Andrew D. Irvine the counterexample, but it left me untutored further about how to manage the distinction between counterexamples that are nuisances and those that do genuine damage ... Also prominent ... was the disposition to privilege strong antecedent conviction with judgements in the form ‘We have it from the very concept of X, that P.’ In so thinking I was drawn to what, years later, I would call the Heuristic Fallacy. This is the mistake of inferring that beliefs that were necessary in thinking a theory up in the first place must be formally preserved in the theory itself. It took me many years to see that any philosophical method that privileges our ‘intuitions’ in epistemically strong ways is at risk for the Heuristic Fallacy; even so, I was not long back in Toronto before I started losing confidence in ordinary language philosophy.

Woods remained at the University of Toronto until 1971, although there were visiting appointments at Ann Arbor and Stanford University. During this time he developed (and published) his views on modal logic, counterfactuality, semantic kinds, and the logic of fiction. In 1971 he joined the Department of Philosophy at the newly created University of Victoria: My time at Victoria was hugely consequential for me. Not only had I seen The Logic of Fiction and Proof and Truth through the presses, but I had gradually come to the beginnings of a realization that, for me, the dominant philosophical question was how philosophy was to be done. This was not by any means an original thought, but in time I became convinced that it was an idea that most philosophers didn’t know how to exploit ... Knowing how philosophy should be done is therefore knowing, largely, what makes for successful argument in philosophy.

At Victoria, too, he ‘thrilled to the prospect of helping to build a new university,’ and he rose to the position of Associate Dean of the Faculty of Arts and Science. Almost immediately he was tempted with the offer of the position of Dean of the Faculty of Humanities at the University of Calgary. By this time he had become fascinated with the complex and difficult task of running universities, and in 1976 ‘four broken-hearted Woodses and one rather guilty one made the move to Calgary,’ giving up the perpetual springtime of Victoria for the ‘wintery climes of a raucous oil town.’ In 1977, Woods completed his controversial book Engineered Death:4 I had originally approached the project thinking that liberal assumptions

Introduction 7 implied the permissibility of abortion and the impermissibility of euthanasia. On thinking it over, I became convinced that it was the other way around. This made me probably the only liberal in the country who saw his liberalism as precluding abortion (certainly abortion on demand). When I exposed this view to public scrutiny ... it was met with considerable disbelief and no little hostility.

(For more on this question, see the papers and response by Woods in Part V of this volume.) From 1979 to 1986 Woods was President and Vice-Chancellor of the University of Lethbridge: Lethbridge provided a large challenge of its own. Chartered in 1967, it was a slender undergraduate university with enrollments that were both small and declining, a huge budget crisis and a rather substantial general demoralization.

Woods turned the University of Lethbridge around. Although Lethbridge is still a relatively small university, it is growing vigorously and many of its departments have earned substantial research reputations. Woods insisted on keeping up his research and teaching despite the demands of his presidency, and after that period concluded in 1986 he returned to the philosophical fray with renewed enthusiasm. This is hardly to suggest that his administrative work ceased: he was Chair of the Department of Philosophy from 1991 to his retirement in 2002 and had numerous internal and external appointments including President of the Academy of Humanities and Social Science of the Royal Society of Canada (1996–8). He also served or continues to serve on several editorial boards and departmental review committees. Since he tends to work across the grain of traditional divisions, Woods’ thought is not easy to pigeon-hole. The categories that most readily capture his output (and which are reflected in the structure of this book) are Reality, Knowledge, Logic and Language, Reasoning, and Values; although it should be emphasized that, more than is customary perhaps, Woods’ writings subdivide in these ways less by design than by indirection; they also exhibit uncustomary levels of categorial spillage. A case in point is his pioneering work on fiction, which is a contribution as much to logic as to the metaphysics of the Reality/Unreality distinction. Similarly, although Woods’ earliest work on abortion and euthanasia was substantivally about values, it also was an attempt to elucidate the logical structure of intractable disagreement, a theme that

8 Kent A. Peacock and Andrew D. Irvine

has matured into a dominant emphasis in Woods’ philosophy. A further example is Woods’ highly influential work on the fallacies. From the earliest days of his twelve-year collaboration with Douglas Walton, Woods has seen fallacy theory as a natural part of logic and has regretted its neglect by the post-Fregean mainstream. We see in this an ambiguity in Woods’ conception of logic itself. If taken in the way of the modern mainstream, logic is an investigation of properties such as logical consequence of linguistic structures and relations such as satisfaction between linguistic structures and set-theoretic entities. But, historically, logic is also about reasoning, and specifically about inference. The difference is reflected most economically in a negative answer to the question ‘Is it always reasonable to infer a logical consequence of anything one now believes?’ Since 1972, Woods has been part of the No campaign, and in the ensuing years has been much reinforced in this opinion by work done in computer science and artificial intelligence. By 2000, it had become apparent to Woods that something important had happened to logic. If, in the last third of the nineteenth century, logic had taken a mathematical turn, a hundred years later it had taken a turn towards the practical. In a burst of memetic-ubiquity, work that was independent and largely concurrent in computer science, artificial intelligence, linguistics, psychology, forensic science, argumentation theory, and informal logic converged on the theme of the practical. Thus, by Woods’ lights, much of what is rightly classified as work on reasoning also makes a claim to be logic in this broader sense. In all these traditional categories of philosophy, Woods’ work has had a significant evolution in the past forty years. In the beginning, Woods was an uncritical realist. He simply took for granted the objectivity of all that exists and of what can be known about it. In logic, he was a classicist about logical consequence and invested some early effort in discrediting developments in relevant logic, wherein, he thought, the important difference between entailment and inference was heeded insufficiently. In epistemology, he scorned even moderate skepticism as either sophomoric cleverness or catastrophic confusedness. In philosophy itself, Woods was an a priorist-foundationalist, in the manner of G.E. Moore’s original conception of philosophical analysis and, later, of its Ordinary Language adaptation. So oriented, Woods was more than ready to make (as he now believes) two characteristic mistakes common to that position. One was to epistemically privilege one’s pre-theoretic intuitions. The other, relatedly, was to conflate nor-

Introduction 9

mativity with ideality. These same preconceptions extended to Woods’ work on values. In Engineered Death he assumed that there was a fact of the matter about the rightness or wrongness of abortion and that that fact was transparent to good reasoning. A turning point in Woods’ philosophical orientation came when he took over Lethbridge’s class in Ancient Philosophy, following the retirement of Peter Preuss in 1995. The great Greek accomplishment was to have assigned to logos the centrally important task of disciplining the appearance-reality distinction in a principled way. What appeared to have been learned from the Presocratics was that the early philosophers had catastrophically lost control of this distinction. If philosophy was to recover from this disaster, a correct theory of reasoning would have to be developed. Woods was much drawn to Aristotle’s audacious claim that the theoretical core of any such theory would be the logic of syllogisms. But, as Aristotle himself points out in the early parts of the Organon, logic itself was subject to this same lack of control; for some arguments appear to be syllogisms which are not in fact. These are the fallacies, and Aristotle saw them as viruses that threatened the very integrity of logic itself. (We will not dwell on the question of the extent to which this virus was expelled by the later perfectibility proofs of the Prior Analytics.) What matters in Greek philosophy for the development of Woods’ thought is his readiness to take seriously the corrosive skepticisms and the apparent lunacies of the Presocratics. In so doing, he shifted his focus from an interest in whether these doctrines were actually true (and the easy answer that ‘obviously’ they were not) to the procedural question of whether these doctrines could be made impervious to effective confutation short of begging the question. In pressing this point, Woods had made some collateral adjustments to his earlier philosophical presuppositions. He abandoned entirely the idea that what the theorist cannot help believing must be epistemically privileged in some way, a theme developed in Paradox and Paraconsistency,5 and he rejected all the standard ways of establishing normative authority in matters of correct reasoning, a theme also sounded in Paradox and Paraconsistency and in a paper given at the Dagstuhl Zeminar6 in 2002. He adopted a variation of Henry Johnstone’s notorious claim that the only tenable way to overcome deadlocks in philosophy is by way of arguments ad hominem. (The notoriety attached to Johnstone was not entirely deserved, since he used the term ‘ad hominem’ in the sense in which Locke used it, not in the modern sense in which it is perceived as an irrelevant attack on a

10 Kent A. Peacock and Andrew D. Irvine

disputant’s person or circumstance. For Locke and Johnstone, the main strategy of argumentation is to show your opponent that his position commits him to conclusions that he cannot accept.) Thus, for Woods, philosophical inquiry had become dominantly a dialectical matter (in the modern rather than ancient sense of that word). He found himself drawn to the idea that, whether the issue is abortion or disjunctive syllogism, conflict resolution in the non-empirical sciences is inherently economic in character – which, after all, was Locke’s insight in that long ago of 1690. A further part of this reorientation was a willingness to take skepticism seriously, whether about knowledge, freedom of the will, induction, realism, and so on. In some cases, induction, for example, he thought that there was a reasonable though defeasible answer to the skeptic (best set out in ‘The Problem of Abduction’).7 In the other cases, however, he had come to the view that the skeptic could not be answered with non-question-begging effect. Woods no longer thinks that the central questions are whether knowledge exists or whether we are free, and the like. The central task is to chart the course of intelligent reasonableness given that knowledge might not exist or that we probably are not free or that there is not an ascertainable fact of the matter about abortion or even that the world is absurd.8 The challenge is to answer the question of reason’s rightful role in a general context of justificatory failure. Fundamental to this repositioning of Woods’ conception of philosophy has been an approach to practical reasoning set out in his joint work with Dov Gabbay.9 On this view the reasonableness of a piece of reasoning, both about what is the case and about what to do, is a function of two factors. One is the nature and extent of the cognitive resources to which the agent has access (including the time in which the task must be performed); the other is the appropriateness of the evaluative target, given the nature of the task at hand and the resources available for its transaction. Out of this comes a radical proposal for the analysis of the fallacies, in which a piece of reasoning or an argument is fallacious, if at all, only relative to the task at hand, the available resources, and the evaluative target that is appropriate to them. According to Woods, not only is hasty generalization not intrinsically fallacious for beings like ourselves (although institutional agents, such as NASA, are a different story),10 but evaluative targets, such as validity and inductive strength, are rarely appropriate for agents of this type. Details of this orientation are being worked out in a multi-volume work with Gabbay under the generic title, A Practical Logic of Cognitive Systems, of which the first two

Introduction 11

volumes (Agenda Relevance: A Study in Formal Pragmatics and The Reach of Abduction: Insight and Trial) appeared in 2003 and 2004.11 Woods retired from the University of Lethbridge in 2002, but he continues to hold four adjunct or visiting appointments at universities in Canada and Europe, and carries on a research and publication program that would daunt many younger academics. Despite his intensely demanding career, Woods has somehow managed to keep family first – no easy feat. It is not inappropriate to conclude with a few of his words on a more personal level: Carol Arnold and I married virtually as children and had our own children early. We have seen nearly all of life’s bounties and vicissitudes and, as we have passed together through several of Shakespeare’s stages of man, we have been suffused and enriched by our love for each other and for our children, Catherine, Kelly, and Michael. I have enjoyed much good fortune in my academic life, but it is a second thing entirely to these four indispensable gifts.

notes 1 John Woods and Dov M. Gabbay, The Reach of Abduction: Insight and Trial, vol. 2 of The Practical Logic of Cognitive Systems (Amsterdam: North Holland, 2005). 2 Ingrid Speaker Medal for Distinguished Research, 1997; Distinguished Teaching Award, 1996. 3 All quotations in this introduction are from John Woods, ‘Sketches of a Philosophical Education,’ unpublished manuscript. 4 John Woods, Engineered Death: Abortion, Suicide, Euthanasia and Senecide (Ottawa: University of Ottawa Press/Éditions de l’Université d’Ottawa, 1978). 5 John Woods, Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences (Cambridge: Cambridge University Press, 2003). 6 German Research Institute, Dagstühl, Germany. 7 John Woods, ‘The Problem of Abduction,’ Algemeen Tijdschrift Wijsbegeerte 93 (2001): 265–72 8 See ‘The Dialectical Unassailability of Heraclitean Logic,’ Logic Journal of IGPL, to appear in 2005. 9 See, e.g., ‘Logic and the Practical Turn,’ in Handbook of the Logic of Argument and Inference: The Turn toward the Practical, ed. Dov M. Gabbay, Ralph H. Johnson, Hans Jürgen Ohlbach and John Woods (Amsterdam: Elsevier Sci-

12 Kent A. Peacock and Andrew D. Irvine ence, 2002); and John Woods and Dov M. Gabbay, ‘The New Logic,’ Logic Journal of the IGPL 9 (2001): 157–90. 10 See also Dov M. Gabbay and John Woods, ‘Filtration Structures and the Cut Down Problem for Abduction,’ in this volume. 11 John Woods and Dov M. Gabbay, Agenda Relevance: An Essay in Formal Pragmatics, vol. 1 of The Practical Logic of Cognitive Systems (Amsterdam: North Holland, 2003); John Woods and Dov M. Gabbay, The Reach of Abduction: Insight and Trial, vol. 2 of The Practical Logic of Cognitive Systems (Amsterdam: North-Holland, 2005).

Part I Reality

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1 Through the Woods to Meinong’s Jungle NICHOLAS GRIFFIN

John Woods’ The Logic of Fiction1 was a pioneering treatment of the semantics of fictional discourse. As Woods notes in his introduction, philosophers of language had previously not paid much attention to the treatment of fiction – and this for entirely feeble reasons. Moreover, Woods made it clear that the two then-standard approaches – treating fiction by means of Russell’s theory of descriptions and thereby rendering all primary fictional discourse false, or treating it by means of free logic, and thereby rendering most of it truth-valueless – were both totally inadequate. It does no justice to fictional discourse to treat both ‘Sherlock Holmes was a carpenter’ and ‘Sherlock Holmes was a detective’ as false, nor is it much of an improvement to suggest that both lack a truth-value. An essential task for the semantics of fiction is to make it possible to explain how it is that people can understand fictional writings, how they can reason (correctly and incorrectly) about fictional situations, and how they can have expectations (justified and unjustified) about what, for example, will happen next in a novel. None of this is provided by crudely applying either of the standard approaches to fiction. What is needed, as Woods clearly recognized, is a theory of fictional objects, objects which, despite the fact that they do not exist, nonetheless have properties and about which truths (and falsehoods) can be spoken. In 1974, when the rehabilitation of Meinong’s theory of objects was still in its infancy, this was a very radical suggestion. Meinong’s theory of objects had been taken to be decisively refuted sixty-nine years earlier by Russell,2 whose theory of descriptions, moreover, was for long taken to supply all that was necessary for a correct treatment of the issues that Meinong’s Gegenstandstheorie had addressed. The crit-

16 Nicholas Griffin

icisms of Russell’s theory which had emerged since Strawson’s ‘On Referring’ appeared in Mind in 1950 had not significantly changed the situation with regard to Meinong. Out of Strawson (malgré lui) grew free logic, which was not very much less hostile to Meinong and did not give a markedly better treatment of fiction than Russell. 1. Nonesuches vs. Fictional (and Other) Objects Not that Woods sought to rehabilitate Meinong. His is a theory of fictional objects, not of Meinongian objects. For Meinong, every singular referring expression refers to an object; for Woods, descriptions like ‘the present King of France’ do not refer at all. Woods is not entirely clear about how the distinction is to be drawn. He concedes that ‘the present King of France’ ‘might be an intentional object,’ but not in the way that Sherlock Holmes is. He notes that, even though Holmes does not exist ‘we know who he is. He is a non-entity who is a somebody’ (29). By contrast, the present King of France is what he calls a ‘nonesuch.’ The one clear demarcation between the two is that bound variables do not range over nonesuches, though they do over items like Sherlock Holmes (29).3 It is not altogether easy to see how this hangs together. For example, it is difficult to see how the present King of France could be an intentional object in any sense at all, if it does not fall within the range of the bound variables. For if a genuinely is the object of an intentional state 4, it would seem essential to be able to infer that something was the object of 4. Moreover, Holmes is rare among fictional objects in that we do know who he is. Many fictional objects are provided with little, if anything, by way of an identity. Who is the servant who enters in act V, scene II of Richard II and exits a minute later without saying a word? Though we don’t know who he is, he is certainly the value of a bound variable, for the Duke of York surely tells somebody to saddle his horse. In reply, it can be argued that the mere fact that Shakespeare explicitly introduces him ensures that there is such a fictional person and provides him with enough of an identity to save him from being a nonesuch, however little we know about him.4 What makes the King of France a nonesuch and Sherlock Holmes a genuine object cannot be merely the poverty of our knowledge of the King of France compared with our knowledge of Sherlock Holmes, for we know equally little about York’s servant, who yet remains a genuine fictional object. About the King of France we know only what the definite description tells us; we do not know whether he is bald or

Through the Woods to Meinong’s Jungle 17

wise or neither. Then again, we do not know these things about York’s servant either. Indeed, there is also a great deal we don’t know about Sherlock Holmes: we don’t know whether he took a size eleven shoe, or whether he had a mole on his back. As already indicated, the epistemic idiom is not really appropriate here. It is not that there is some fact of the matter about Holmes’ shoe size that we are ignorant of. There is nothing to be known about Holmes’ shoe size simply because Conan Doyle was silent on the matter. The basis of a semantics of fiction is the author’s say-so (35ff). Conan Doyle could have given Holmes shoes of any size simply by saying so. Specifying the truth-conditions for fiction via the say-so semantics is not as simple a task as it might seem and will not be attempted with any rigour here.5 In general, fictional objects are said to be incomplete with respect to properties about which the author gives no clue. And, since no author can give an exhaustive description of any object, every fictional object will be incomplete. By contrast, there is always much more to know about someone who actually exists. We might not know Julius Caesar’s shoe size but, barring general anti-realist concerns, there is a fact of the matter which may (or may not) be amenable to investigation, and thus there is a bet to be won or lost.6 Incompleteness, thus, comes in grades, but it would be a mistake to suppose that nonesuches occupy the higher grades and fictional objects the lower. It is certainly possible, by means of the absurd elaboration of a fanciful example (there is no theoretical limit to the number of adjectives that can be crammed into a definite description), to say much more about a nonesuch than Shakespeare says about York’s servant. It is not, therefore, the incompleteness of the present King of France that makes him a nonesuch, nor even his relative incompleteness compared to Sherlock Holmes. It seems, rather, that what makes him a nonesuch is that there is no work of fiction in which the definite description ‘the present King of France’ is used to refer to an object. It is hard to imagine that this is quite correct as it stands, for there are no doubt dozens of literary works in which ‘the present King of France’ is used to refer to a fictional object. But, leaving that issue to one side, the account is still too wide, for there are many genuine fictional objects, including some known only by their description, in works where the description by which they are known does not occur. For example, I don’t think Conan Doyle ever uses the description ‘the youngest of the Baker Street irregulars’ or any synonym of it.7 We have no idea who this person was, but it seems clear that he is a genuine fictional person and not a nonesuch. At all events, he is clearly the value of a bound

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variable, for when Holmes summons all the Baker Street irregulars, the youngest of them is surely included. Evidently, the say-so semantics of fiction must be sufficiently liberal to admit items which are not explicitly referred to but which can reasonably be inferred from what the author says.8 In the case of the irregulars, Conan Doyle refers to a small group of people, from which we can infer that one of them will be younger than all the others. It is not absolutely guaranteed – two of them may have been born at the very same instant and after all the others – but at the very least it is bet-sensitive. This inference condition can be extended further. For example, I don’t think that Conan Doyle makes any mention of Holmes’ grandfather, but we can surely infer that Holmes had a grandfather even though the stories don’t mention him. For we can infer from the stories that Holmes was not the product of an immaculate conception nor of parthenogenesis.9 So it seems that ‘the grandfather of Sherlock Holmes’ refers to a fictional object, rather than a nonesuch, even though there is no work of fiction in which that object appears. Woods explicitly draws the line at ‘Mrs Sherlock Holmes’ (27). She, for Woods, is definitely a nonesuch, and this because Conan Doyle makes it clear that Holmes was never married. It is certainly true that there is no such object in the Conan Doyle stories. This could also be said, as we’ve seen, of Holmes’ grandfather, but Mrs Holmes is different, for she is explicitly excluded by the stories, whereas Granddad Holmes is there waiting to be inferred. As far as the Conan Doyle stories are concerned, Mrs Holmes is not the value of a bound variable. But can we conclude from this that she is not the value of a bound variable at all? It is obviously open to some future author of a noncanonical Holmes story (already a substantial genre) to have Holmes give up detecting, get married, and fall into suburban domesticity. Such a story will have a stock of fictional objects – including Mrs Holmes and their suburban villa – substantially different from that of the canonical Holmes stories. This, however, is merely to create a new fiction with a new set of fictional items. Until this is done, Woods will maintain, there is no such item as Mrs Holmes. There is much more to be said about this kind of contextualization of fictional objects, which confines such objects to domains associated with particular works of fiction, for it seems likely that the solution to the chief problems of the semantics of fiction are to be found there. The issues are complex and take us well beyond the scope of this paper, though I shall return to them briefly at the end. However, the suggestion which naturally arises from this, and

Through the Woods to Meinong’s Jungle 19

I think it is a correct one, is that there is no one division between fictional objects and nonesuches, but that (if one wishes to keep the nonesuch terminology) certain items are nonesuches (and others fictional) with respect to a particular work of fiction. Other considerations tell against a single nonesuch/fictional object distinction. To begin with, there are other types of non-existent objects apart from fictional objects which do not seem to fall into the nonesuch category, for example, ideal theoretical entities in science such as the ideal gas and the frictionless plane. It seems clear that these are not nonesuches and that, at least in a non-reductive account of the sciences in which they occur, they fall within the theory’s domain of quantification. The objects of myth and legend are another category, though perhaps one that can be subsumed under fiction. An intermediate type of case would be that of the golden mountain, which might seem to be a nonesuch but which explorers actually sought and which had a rich, intentional history – unlike many nonesuches. A natural suggestion for all these would be for Woods to liberalize his account by treating myths, legends, and theories as works of fiction, or at least as analogous to works of fiction. Thus rational mechanics produces the frictionless plane, and eighteenth-century heat theory produces caloric, just as the Conan Doyle stories produce Holmes. From our present perspective, we could well treat eighteenth century heat theory as a literal fiction – albeit one that was at one time believed. While caloric is a nonesuch in contemporary heat theory, it was a genuine item in the eighteenth-century theory. But if this approach is adopted, what do we make of ‘the greatest prime’? Here the item is not produced by a theory but explicitly denied by it. On the analogy with fiction, the greatest prime should have the same status as Mrs Sherlock Holmes. Intuitions may differ – as Woods notes explicitly in connection with Mrs Holmes (81n) – but I find it strongly counter-intuitive to treat the greatest prime as a mere nonesuch when its properties (including its non-existence) are precisely and rather fully known and admit of exact mathematical proof. It is fruitless to speculate on how Woods would handle these cases, for the semantics of theoretical and mathematical terms was not part of the task he set himself. On the other hand, he is explicit about his treatment of the present King of France: he will not grant it the same status as his fictional items, nor will he allow it to be the value of a bound variable. One disadvantage of this approach is that inferences such as the following fail:

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(1)

The present King of France is bald

? (å x)(x is bald)

It seems to me that whether ‘The present King of France is bald’ is true, false, or truth-valueless, (1) ought to be valid. Rejecting (1) while accepting a similar argument in which ‘Professor Moriarty’ replaces ‘the present King of France’ requires complex restrictions on particularization which embrangle logic not just with ontology but with the messier exigencies of fiction. Such a policy will appeal neither to those who think that logic should not be a branch of metaphysics nor to those who share Russell’s robust sense of reality. Another problem concerns identity conditions for nonesuches. If the present King of France is not an object at all, it would seem absurd to inquire how he differs from the present King of Spain, and yet an adequate semantics has to provide an account of how ‘the present King of France’ differs semantically from ‘the present King of Spain.’ This is not to say an account cannot be provided, but any satisfactory account is likely to be substantially more complex than the one offered by Meinong’s theory of objects. It is not altogether clear that Woods can avoid this problem by claiming that it is no business of a theory of fiction to account for the semantic behaviour of nonesuches, for the present King of France may intrude into fiction. Obviously one could write a story about the present King of France, though this is not the difficult case. Woods can handle it by claiming that the fiction created an entirely different object from the nonesuch that is introduced in ‘On Denoting,’ which we might call Russell’s present King of France. The natural account of a story about the present King of France is that it introduces a new fictional object with no connection at all with Russell’s present King of France – just as Mrs Holmes might be introduced by a new Holmes story and have no relation as a result with the nonesuch wife of Sherlock Holmes of the existing stories. But a more difficult scenario can be imagined, one in which the author of the fiction makes it clear that the hero of the story is intended to be identical to Russell’s King of France. It could begin one night, late in 1905, with a knocking on the door of Russell’s house in Bagley Wood. Russell opens the door to find a diminutive, hatted figure who says reproachfully: ‘I am the present King of France whose bald and crownless head you have exposed to the ridicule of logicians everywhere. Little wonder I’ve been forced to embrace Hegelianism and wear a toupee.’ Russell might have written the tale himself and included it as ‘Lord Russell’s Nightmare’ in Night-

Through the Woods to Meinong’s Jungle 21

mares of Eminent Persons. Of course we need not conclude that this character is Russell’s King of France merely on the character’s say-so: for the character could be a prankster. On the other hand, the story could envisage much more elaborate links between the two, including causal ones. It could be part of the story, for example, that Russell’s visitor comes from a realm in which all logical examples are produced when they are thought of. It is the hero’s misfortune to have been created there in 1905 when Russell wrote ‘On Denoting.’ It would be hard to understand this story correctly if one insists that Russell’s King of France is a nonesuch. 2. Objections to the Theory of Objects Though Woods is more grudging in his admission of non-existent objects than either Meinong or the neo-Meinongians, there was one important respect in which Woods’ radicalism outdid even that of many neo-Meinongians. For many years, the new Meinongians sought to tame the realm of non-existent objects, to show how Meinong’s original theory could be modified to escape the apparently intolerable consequences Russell had drawn from it. One obvious requirement of a theory of non-existent objects is to give some account of the properties they have – an item without properties would be a genuine nonesuch. If non-existent objects are to play the role Meinong intended for them in an account of intentionality, for example, it is plain that they must have properties. Explorers have sought both the golden mountain and the fountain of youth, but these were two separate quests, for two different objects, neither of which existed. If non-existent objects lacked properties, it would be impossible to distinguish these two and the role of non-existent objects in an account of intentionality would fail. Faced with the problem of ascribing properties to non-existent objects, Meinong makes the initially reasonable suggestion that they have the properties ascribed to them. In the case of fictional objects, this conforms well to the say-so semantics: Sherlock Holmes has all the properties Conan Doyle ascribes to him. It also works well – at least to a first approximation – with nonesuches: the golden mountain is both golden and a mountain. It seems plain that if we think of an object which lacks either of these properties then we have not thought of the golden mountain. So Meinong’s principle is this: a non-existent object has those properties which are used to characterize it.

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Against this principle Russell raised two apparently devastating objections, the objections mainly responsible for the long eclipse of Meinongian semantics. First, the round square is characterized as being both round and square. But being square entails being not round and thus the round square is both round and not round. It thus violates the law of non-contradiction. Second, just as the golden mountain is both golden and mountainous, the existing golden mountain is golden, mountainous, and exists. But no golden mountain exists: thus the theory of objects entails claims which are factually false. Meinongians resolved the first of these problems, as had Meinong himself, by distinguishing between predicate and sentence negation, maintaining that while, for sentence negation, the law of non-contradiction held unrestrictedly, for predicate negation the law failed in the case of impossible objects. They resolved the second by distinguishing properties like existence from other properties and denying that non-existent objects could be characterized by means of them. Both Meinong and Parsons offer weakened (‘watered-down’) companion predicates to replace the problematic ones.10 Neither policy works well for fiction. Fictional works often contain objects which not only don’t exist in the real world but don’t exist in the imaginary world of the fiction either. In the movie Harvey, the title character, a six-foot invisible rabbit, arguably does not exist. There is, it is true, some suggestion to the contrary in the final sequence, but this will hardly give solace to either a Russellian or a free logician.11 The point is that Harvey’s existence is intelligibly discussed both within and outside the fiction, making it difficult plausibly to maintain that fictional items cannot be characterized as existing. In order to understand the movie, we have to be able to use ‘exists’ as, at least, a predicate-like expression which distinguishes some of the items spoken about within the fiction from others. Similarly, it is entirely possible for there to be a fiction in which some character squares the circle, and not just in some weakened predicational sense but in a full-blooded sentential sense.12 Woods does not discuss these issues in detail. He was not, after all, attempting to defend Meinong, and many of the devices adopted by the neo-Meinongians were only being explored as Woods was writing, Meinong’s original exploration of them being largely forgotten. Nonetheless, Woods’ focus on fiction encourages a recognition of the range of phenomena to be accounted for. It has not been uncommon for those working in semantics to dismiss impossible objects as literally unthinkable and thus not in need of semantic treatment. Woods points out that

Through the Woods to Meinong’s Jungle 23

one can imagine a fiction in which one of the characters squares the circle and thus, by the say-so semantics, it will be true in that fiction that the circle was squared. By the same token, we can imagine fictions in which Euclid’s parallel postulate is proved, the greatest prime found, or the law of identity violated. Rather than try to minimize the affront that these extravagances pose to our normal way of thinking by the sorts of weakening devices traditionally favoured by Meinongians, the proper way to understand the fiction is to take the affront at face value and treat it as every bit as outrageous as it purports to be. Writers who have gone to the trouble of envisaging a world in which the circle can be squared will not think they have been understood if they are told that all they’ve done is envisage some watered-down version of circlesquaring which is not mathematically offensive. One supposes that they intended to be mathematically offensive. Logically, the realm of fiction is wild. Postmodernism did not teach us much, but at least it taught us the ways in which fiction might play with logical inconsistency, nest stories within each other, and play with the boundaries between them. Not that these things were unknown before, merely that they weren’t much talked about. Woods, to his credit, makes allowance for all this. It means that Meinong’s ideal of a single overarching object-theory whose principles are universally applicable to the existent and non-existent alike will not be able to do justice to the vagaries of fiction. In particular, it means that we can’t create a logic for fiction by simply rescinding particular laws of logic (as Meinong rescinds the law of predicate non-contradiction for impossibilia) to meet the needs of particular cases, and making do with what’s left. For the needs of particular cases are infinitely variable, and no law of logic is immune to rescission in a sufficiently bizarre fiction. Contrary to Frege’s claim that when this happens all thought becomes impossible,13 thought proceeds by different means. What this suggests is that while the vast majority of fictions will have perfectly ordinary logics and arithmetics (as well as ordinary geometry and physics), fictions can certainly be envisaged in which the usual logic and arithmetic break down – just as science fiction regularly envisages a breakdown in the usual laws of geometry and physics. What is needed is not to contrast ordinary physics with a single fictional physics but to note that the (usually assumed) background physical theory may vary from fiction to fiction. The physics of Star Trek is not the physics of Dr Who, yet, by the say-so semantics, each is correct within

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its own fiction. Fictions in which logics and arithmetics vary are much rarer, but the range of fiction is incredibly broad and surrealist and dadaist stories need to be allowed for, as well as jokes and puzzles. There is, after all, a story about a village barber who shaves everyone in the village who doesn’t shave himself. The lesson of these two problems, therefore, seems to me to be essentially the same as the lessons of the problem of demarcating nonesuches, namely, that the notions of logical and physical possibility vary from fiction to fiction, just as the class of nonesuches does. This lesson, I believe, will be reinforced by attempts to deal with the third great problem facing the theory of objects – the problem of how to understand relational characterization. The third problem is by far the most difficult; it arises from the simple fact that non-existent items are often characterized by their relations and often by their relations to items that exist. This is particularly the case in fiction, where the action very typically takes place in some actually existing country, or at least on an actually existing planet, the Earth. Thus, to use an example that Woods has made famous, the Conan Doyle stories characterize Holmes as living in London. By the say-so semantics it follows that Holmes has the property of living in London. But then, by the elementary logic of relations and their converses, London has the property of counting Holmes among its inhabitants. But this is false: Holmes was never among the inhabitants of London. This problem, which is discussed at length by Routley and Parsons,14 seems to have originated with Woods.15 It is a bit surprising that Russell himself did not think of it, given his heavy interest in relations. I suspect, though I have no hard evidence, that he may have thought that relations were still so controversial that a counter-example based on them would be taken to be an objection to relations rather than to non-existent objects. At all events, it is the most difficult of the three, for, while there is some hope that the other two might be avoided by watering-down or restrictions of some kind (even though the mechanism for doing this is far from clear), there is little hope that the same devices will work with relations. Nonetheless, this is the approach that Meinongians have favoured. (Meinong himself seems to have been blissfully unaware of the difficulty.) Routley favours a distinction between entire and reduced relations (a version of wateringdown), and Parsons a doctrine of plugging-up relations (essentially a form of restriction on characterization). Neither approach works very well: Routley’s messes with the fundamental logic of relations in

Through the Woods to Meinong’s Jungle 25

essentially ad hoc and unpredictable ways; Parsons’ is cleaner but seems equally ad hoc and seems also to impugn the genuineness of relational characterization. Woods deals with the problem by distinguishing between an item’s history-constitutive properties and fictionalizations about it. To abbreviate his account somewhat, both fictional and real items can have both history-constitutive and fictionalization properties. In the case of real items, fictionalizations are true of the item simply by the author’s say-so, history-constitutive predications are not. In the case of fictional items, history-constitutive predications are true of them by the author’s say-so. Thus it is a history-constitutive truth about Holmes that he lived in London, but it is a fictionalization about London that it included Holmes among its inhabitants. It is also a history-constitutive truth about London that it never had Holmes among its inhabitants, but this is not inconsistent with the previous claim: there can be no contradiction between a history-constitutive truth and its fictionalized negation (or vice versa). Nor are there entailments from fictionalizations about real items to history-constitutive claims about them;16 though there are entailments from history-constitutive claims about fictional items to fictionalizations about real items, for from the history-constitutive fact about Holmes that he lived in London, we can infer the fictionalization about London that it numbered Holmes among its inhabitants. The formal representation of this is likely to be somewhat messy because we cannot characterize the claims themselves as either history-constitutive or fictionalized; they are one or the other in respect of a certain argument place (rather in the manner of Parsons’ plugged-up relations). Moreover, we need to know, of each argument place, whether it is occupied by a fictional or a real item. Things get more complicated yet, for fictional items can themselves be fictionalized. The Sherlock Holmes stories not written by Conan Doyle do just this. Inevitably, they characterize Holmes in ways inconsistent with the way Conan Doyle characterizes him. In the Conan Doyle stories Holmes never collaborated with Sigmund Freud, as he does in Nicholas Meyer’s The Seven Per-Cent Solution. Yet it is crucial to Meyer’s story that it is Sherlock Holmes himself who leads the investigation and not some look-alike with the same name and habits. Invoking counterparts here just gets things wrong. Woods’ account avoids them. On it, Meyer fictionalizes both Freud and Holmes. A fictionalization is true of a fictional object when it is true of the object by the author’s say-so, but the author in question is not the creator of the

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character. Thus Conan Doyle has the monopoly on history-constitutive discourse about Holmes – for which, I am sure, his literary estate is grateful. 3. Problems with the Woods Solution Woods says little explicitly about our first two problems, perhaps because he thinks they arise mainly in connection with nonesuches. Indeed, it is hard to apply Woods’ account to nonesuches. One cannot treat ‘Lord Russell’s Nightmare’ as a fictionalization of a nonesuch, for there are no nonesuches to be fictionalized. On the other hand, to treat it as a history-constitutive account of a fictional item would seem to miss the point. In most of the other cases we have considered it would seem possible to treat the fictions as history-constitutive of fictional items, even in the cases where (absent the fiction) the item in question would be considered a nonesuch. Along these lines, a fiction which included an existing golden mountain17 would be one in which ‘The golden mountain exists’ would be a history-constitutive fact about a fictional object. This would not be at odds with the history-constitutive fact about the real world, that no golden mountain exists, provided at least that we are allowed the principle that no history-constitutive fact about fictions can be inconsistent with anything claimed about a nonesuch. On the other hand, if we treat the original golden mountain that was sought long ago in South America as a fictional item rather than a nonesuch (as was proposed in §1), then we need a rule to ensure that what is history-constitutive of one fiction cannot conflict with what is history-constitutive of another. But finally, if the fiction in which the golden mountain exists is to be taken as a story about the original golden mountain itself, a story which considers what would have happened if that golden mountain actually existed, then the second story is a fictionalization of the original story, and we need further rules to prevent one fictionalization of an item from conflicting either with another fictionalization of the same item or with a history-constitutive account of a fictional item. All this suggests a very considerable complexity, and yet the case is really relatively simple (compared with others from literary history). The need is to be able to keep some track of fictionalizations of fictionalizations back to the original, history-constitutive story – what I shall call the ‘genealogy’ of the item. In many cases, however, it is impossible even to trace this genealogy,

Through the Woods to Meinong’s Jungle 27

let alone to keep track of it in explicating the story. Who was the creator of the original golden mountain, considered now as a fictional item? And what properties did its creator ascribe to it? We have here a kind of ‘folk fiction’ for which no original text can be identified, and the problems we face as a result point out the difficulties of treating such cases analogously to fiction. For all we know the golden mountain began life as a nonesuch. But then, how was a nonesuch ever fictionalized? In this the golden mountain is different from other, apparently similar, cases, such as Atlantis, where we do have texts to appeal to – Plato’s Timaeus and Critias, of which all subsequent accounts will be fictionalizations. Or at least, we think we do. If Plato got the story from some other source now completely lost, his account, too, will be a fictionalization but of a history-constitutive text which is completely inaccessible. The Holy Grail gives us a good idea of the sort of difficulties that can beset us. We might readily agree that most modern versions – including Tennyson’s – are fictionalizations of some prior fiction. But which prior fiction? It is natural to suggest (at least for modern English versions) that it was Malory’s Le Morte d’Arthur, since that is the major English source of the story. But the Holy Grail was created long before Malory, and ten major versions of the legend appeared between 1180 and 1230. Were these fictionalizations of one another? If so, which fictionalized which? Were they all fictionalizations of even earlier stories? At the very least it would be a work of massive literary scholarship to figure out the Grail’s genealogy, and at worst quite impossible. (We may be grateful nowadays to copyright lawyers for preventing the recurrence of such muddles.) Yet none of this really affects our ability to understand the later texts, and there seems little reason why any of it should be relevant to the semantics of fiction. It seems to me that one ill consequence of Woods’ account is that it writes too much literary history into the semantics of a work of fiction. In some cases, as when a new Holmes story is written, in order to understand a fiction it is essential to know that it is a fictionalization of a previous fiction. Someone who is entirely ignorant of Holmes (or of Freud, for that matter) will miss the point of The Seven Per-Cent Solution. But when one writer simply retells a story originally written by someone else, the new story can be understood perfectly well without knowing of the earlier one. It seems simply unnecessary to bring the entire previous – and in some cases astoundingly complex – history of the item into the semantics of the later version. Nor does it seem plausible to maintain that the semantics of Tennyson’s Morte d’Arthur would be

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changed were scholarship to reveal that version A of the Grail legend was a fictionalization of version B, instead of vice versa as formerly supposed. Indeed, the semantics – as distinct from our literary appreciation of the work – would seem to remain essentially untouched even if all the previously believed history of the tale turned out to be spurious. This occasionally happens, when an author provides a fake genealogy for their work. The poems of Thomas Chatterton are an elaborate example. Of course, one reads the poems differently when one knows of Chatterton’s fabrications, but this seems a literary rather than a semantic matter. Yet keeping the genealogy of a fictional item clear is important for Woods’ theory, since different authors will ascribe different properties to their version of the item, and unless the different versions are kept distinct the fictional item will end up with inconsistent properties. Woods discusses this in the case of Faust, but there he is able, very conveniently, to point out that there was in fact a historical Faust, so that all the competing fictional versions were fictionalizations of a real person. But, again, the question of whether there was a historical Faust and of who, in subsequent tellings of his tale, was influenced by whom seem a matter for literary scholarship rather than semantics. For that matter, does it matter to the semantics of Conan Doyle’s Holmes stories that the author is said to have based Holmes upon one of his teachers? Does this make Holmes a fictionalization of a real person? And, if not, what would? And if biographical research into Conan Doyle reveals that the teacher was not a model for Holmes, would that make a semantic difference to the stories? Conan Doyle did not, of course, intend the Holmes stories to be about his teacher – Holmes in the stories is not Conan Doyle’s teacher under an alias. But, by the same token, it is not clear that the authors of the various Faust stories intended their stories to be about the historical Faust, and, even if they did, I doubt it ought to make much difference to the semantic treatment of the stories. Authorial intentions are notoriously obscure. It is difficult enough, especially with old texts, to know whether one author could have even known of the work or another, let alone whether they intended to write about the same character. At all events, Woods is quite firm on the matter: all accounts of a fictional item after the first are fictionalizations of it. Thus Hamlet is not a character created by Shakespeare, and what Shakespeare says about Hamlet is not history-constitutive of him. He is a fictionalization of a

Through the Woods to Meinong’s Jungle 29

character who appeared in an earlier play, now lost, by an author we do not know. This does not strike me as plausible. Moreover, Woods’ insistence that a fictionalization of a fictional character be by an author other than the character’s creator prevents an author from fictionalizing his own creation. Of course, Conan Doyle did not fictionalize Holmes when he wrote a new Holmes story (he added more historyconstitutive detail). But it would surely have been possible for him to have done so, writing, for his own amusement perhaps, a spoof Holmes story or a satire, for example. The genealogy of a fictional character is important for Woods because it eliminates the inconsistencies which, in the say-so semantics, would otherwise arise from a succession of authors heaping their own accounts upon a single fictional item. To avoid this, on Woods’ theory, one has to establish what is history-constitutive of the character and what is subsequent fictionalization. Conflicts between the original account and a later one are avoided because, as already noted, a history-constitutive truth about an item is not inconsistent with its fictionalized negation. Since a fictional item may receive many conflicting fictionalizations, these also have to be distinguished and the consistency principle extended to the claim that one fictionalization of an item is not inconsistent with its fictionalized negation, provided the fictionalized negation comes from a different fictionalization. We do not therefore have a single realm of actual entities to be contrasted with a single fictional realm which contains fictionalizations of the actual entities as well as free-standing fictional creations. We have, rather, the realm of actual entities, and up against that a panoply of different fictional realms, each of which has to be kept distinct from all the others as well as from the realm of actual entities. Thus, in addition to the historical Faust, we have Marlowe’s version, Goethe’s version, and goodness knows how many others. This way of putting it may suggest that the different fictional accounts of Faust are all of different fictional items, that there are a multitude of Fausts. But this is plainly wrong: they are all accounts of the same fictional item. A story which creates Faust II, a counterpart Faust in a parallel universe, would be quite different. Even radically different versions of the story – versions in which, for example, Faust refuses Mephistopheles’ contract – would still be stories about the same character. The character is Faust, if the author says he is – that much is required by the say-so semantics. But the say-so semantics also requires that he have the properties ascribed to him in the story.

30 Nicholas Griffin

How then is inconsistency to be avoided? The answer, it seems evident, is by indexing the ascription of truth-values to the work itself. Thus, for example, Marlowe does not give us the truth simpliciter about Faust but the truth with respect to Marlowe’s play. In this it is difficult (but not impossible) for an author to go wrong. The occurrence of a claim in a fiction generally guarantees the truth of that claim with respect to that fiction – that is the effect of the say-so semantics – but it does not guarantee the truth of the claim with respect to any other fiction or with respect to the real world. There is, moreover, no inconsistency in assigning the value true to a proposition in one context and assigning the value false to the same proposition in a different context. The advantage of this account over Woods’ is that, on it, all that is required in order to ascribe contextually relativized truth-values to fictional claims is a knowledge of what is said about an item in a particular work. It thus preserves the say-so semantics without invoking the long (and often unknowable) genealogy of a fictional object. My proposal, of course, requires much elaboration before it is in a position to be compared with Woods’. We are, I think, still a long way from an adequate account of the semantics of fiction, and Woods’ important contribution in 1974 still has a great deal to teach us. notes 1 John Woods, The Logic of Fiction (The Hague: Mouton, 1974). All page references are to this work unless otherwise indicated. 2 Russell: ‘On Denoting’ (1905), Collected Papers of Bertrand Russell, vol. 4: Foundations of Logic, 1903–05, Alasdair Urquhart, ed. (London: Routledge, 1994), 418; Review of Meinong et al., Untersuchungen zur Gegenstandstheorie und Psychologie (1905), ibid., 596–604; Review of Meinong, Über die Stellung der Gegenstandstheorie im System der Wissenschaften, Mind 16 (1907): 436–9. Meinong’s replies have pretty much been ignored. See his Über Möglichkeit und Wahrscheinlichkeit (Leipzig: Barth, 1915), 171–4, 278ff; Über die Stellung der Gegenstandstheorie im System der Wissenschaften (Leipzig: Voitlander, 1907), 16–17, 62. 3 Woods (27) introduces an ontologically neutral particular quantifier, ‘åx’, read ‘there is an object x such that ...’ Its universal mate, ‘3x’, is defined in the usual way: (3x)A = df ™(åx) ™A. 4 It would be better here to say: ‘however little there is to be known about him.’ For it is not to be supposed that there are facts about him other than those revealed by (or inferable from) the play.

Through the Woods to Meinong’s Jungle 31 5 Those who hold a substantive theory of truth might, for one reason or another, balk at the notion of fictional truth. These concerns might be avoided by means of Woods’ concept of ‘bet-insensitivity.’ If you bet that Holmes is a detective while I bet that he is a carpenter, you win and I lose. By contrast, there is no bet to be made on whether the King of France is bald or not, or on whether Holmes takes size 11 shoes – these matters are bet-insensitive. Of course it would be easy (and natural) to reduce betsensitivity to truth: you win the bet about Holmes because ‘Holmes is a detective’ is true; I lose because ‘Holmes is a carpenter’ is false; no bet is to be made on whether the King of France is bald because ‘the King of France is bald’ is neither true nor false. On the other hand, it may be possible to develop the notion of bet-sensitivity in a way that did not require ascribing truth-values to fictional statements: though the difficulty here is somehow to draw the distinction between correct and incorrect statements about fiction without making it look like the distinction between true and false statements under a new label. At all events, Woods does not develop this approach but instead casts the say-so semantics in terms of truth. 6 Some authors, including Meinong and Routley, have been led to claim that existent objects are complete with respect to every property, i.e., E(a) o (I)(I(a) › ™I(a)), where ‘E’ is the predicate ‘exists.’ This may or may not be the case – objections from vagueness or quantum mechanics need to be taken seriously – but it does seem that we should avoid taking this condition as a definition of ‘exists.’ 7 If he did, we can readily change the example to a description he did not use, e.g., ‘the Baker Street irregular whose birthday occurs first in the calendar.’ 8 The inference will not, in general, be a deductive one. 9 Woods (64–5) rightly includes as part of his semantics the assumption that the author will signal – directly or indirectly – any such significant deviation from our usual expectations regarding fictional objects. The say-so semantics will (and ought) to allow characters to be born by immaculate conception, but our ability to understand the stories depends upon the author and the audience sharing a set of assumptions about the characters, and this requires that the author signal deviations from the norm or, more accurately, deviations from what the author thinks the audience will expect. 10 Cf. Meinong, Über die Stellung, 17; Terence Parsons, Nonexistent Objects (New Haven: Yale University Press, 1980), 44, 65, 73. 11 I owe my knowledge of this James Stewart classic to Carolyn Swanson. 12 For a charming example, deliberately constructed to make this very point,

32 Nicholas Griffin

13 14

15

16 17

see Graham Priest, ‘Sylvan’s Box: A Short Story and Ten Morals,’ Notre Dame Journal of Formal Logic 38 (1997): 573–82 Gottlob Frege, The Foundations of Arithmetic, trans. J.L. Austin (Oxford: Blackwell, 1959), §14, 21. Cf. Richard Routley, Exploring Meinong’s Jungle and Beyond (Canberra: Department of Philosophy, Australian National University, 1980), 267–9, 577–90, 718–20; Parsons, Nonexistent Objects, 26–7, 59–60, 64–9, 75–7, 156– 60, 234–40. It was certainly from Woods that Routley got it. I happen to own Routley’s copy of Woods’ Logic of Fiction, and he’s written ‘major problem’ in the margin where Woods introduces the problem (p. 42). Woods devotes two long discussions to it: 42ff, 135ff. Beyond, that is, the history-constitutive claim that they have been thus fictionalized. One thinks of a story in which explorers search for a fabled golden mountain – and eventually find it.

2 The Epsilon Logic of Fictions B.H. SLATER

1 John Woods considered a whole panoply of ways of treating fictions in his 1974 book The Logic of Fiction.1 Notably, he considered the many forms of free logic which were then prevalent, and also several manyvalued logics. A number of specific problems ran through the discussion, concerning such examples as ‘Sherlock Holmes had tea with Gladstone,’ ‘Kingsley Amis admires James Bond,’ ‘Freud psychoanalysed Gradiva,’ Meinong’s notorious cases (‘the round square’ and ‘the gold mountain’), and the difference between, say, ‘Sherlock Holmes lived in Baker Street’ and ‘The present King of France is bald.’ I will return to some of the specific problems arising with such examples in the course of, but mainly at the end of, this paper. In the body of it I shall explain a general approach to fictions which John Woods did not consider. It is an approach I developed using the epsilon calculus.2 Epsilon calculi are conservative extensions of the predicate calculus which incorporate epsilon terms. Epsilon terms are individual terms of the form ‘HxFx’, being defined for all predicates in the language. The epsilon term ‘HxFx’ denotes a chosen F, if there are any F’s, and has an arbitrary reference otherwise. It is the latter case which primarily enables epsilon terms to handle fictions, as when there is reference to The Gold Mountain even when there is no gold mountain. Epsilon calculi were originally developed to study certain forms of arithmetic and set theory, and also to prove some important meta-theorems about the predicate calculus. Later formal developments have included a variety of intensional epsilon calculi, of use in the study of necessity and of more general intensional notions like belief. It is in the latter context that an epsilon logic of fictions comes to be appropriate.

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Epsilon terms were introduced by the German mathematician David Hilbert in 1923 to provide explicit definitions of the existential and universal quantifiers and to resolve some problems in infinitistic mathematics.3 But the extended presentation of an epsilon calculus, as a formal logic of interest in its own right, in fact only first appeared in Bourbaki’s Éléments de Mathématique. Bourbaki’s epsilon calculus with identity4 is axiomatic, with modus ponens as the only primitive inference or derivation rule. Thus, in effect, we get (X › X) Š X, X Š (X › Y), (X › Y) Š (Y › X), (X Š Y) Š ((Z › X) Š (Z › Y)), Fy Š FHxFx, x = y Š Fx { Fy, (x)(Fx { Gx) Š HxFx = HxGx. This adds to a basis for the propositional calculus an epsilon axiom schema, then Leibniz’ Law, and a second epsilon axiom schema, which is a further law of identity. Bourbaki used the Greek letter tau rather than epsilon to form what are now called ‘epsilon terms’; nevertheless, they defined the quantifiers in terms of their tau symbol in the manner of Hilbert, namely (x)Fx { FHxFx, (x)Fx { FHx ™Fx. An epsilon term such as ‘HxFx’ Hilbert read as ‘the first F’ – in arithmetical contexts ‘the least F.’ More generally it can be read as the demonstrative description ‘that F,’ when arising either deictically – i.e., in a pragmatic context where something is being pointed at – or in linguistic cross-reference situations, as with, for example, ‘There is a red-haired man in the room. That red-haired man is Caucasian.’ The appropriate epsilon term then symbolizes the demonstrative description ‘that redhaired man,’ and does so even if there is no red-haired man in the room, since the cross-reference is just a matter of grammar and does not depend on whether the antecedent is true. Likewise an epsilon term might symbolize ‘the man with a martini’ in some pragmatic context, even if no man there has a martini. The application of epsilon terms to natural language thus shares some features with the use of iota terms in

The Epsilon Logic of Fictions 35

the theory of descriptions given by Bertrand Russell, but it differs in not assuming uniqueness and, more important, in formalizing aspects of the alternative theory of reference given by Keith Donnellan.5 More recently, epsilon terms have been used by a number of writers to formalize cross-sentential anaphora, which would arise if ‘that red-haired man’ in the linguistic case above was replaced with a pronoun such as ‘he.’6 There is also the similar application in intensional cases like ‘There is a red-haired man in the room. Celia believed he was a woman.’7 Certain specific theorems in the epsilon calculus relate closely to these matters. One theorem demonstrates directly the relation between Russell’s attributive and some of Donnellan’s referential ideas. For (x)(Fx.(y)(Fy Š y = x).Gx) is logically equivalent to (x)(Fx.(y)(Fy Š y = x)).Ga, where a = Hx(Fx.(y)(Fy Š y = x)). This arises because the latter is equivalent to Fa.(y)(Fy Š y = a).Ga, which entails the former. But the former is Fb.(y)(Fy Š y = b).Gb, with b = Hx(Fx.(y)(Fy Š y = x).Gx), and so entails (x)(Fx.(y)(Fy Š y = x)), and Fa.(y)(Fy Š y = a). But that means that, from the uniqueness clause, a = b, and so Ga. Hence the former entails the latter, and therefore the former is equivalent to the latter. The former, of course, gives Russell’s ‘Theory of Descriptions,’ in the case of ‘The F is G’; it explicitly asserts the first two clauses to do with

36 B.H. Slater

the existence and uniqueness of an F. The presuppositional theory of Strawson, formalized in van Fraassen’s theory of super-valuations, would not explicitly assert these two clauses: on such an account they are a precondition before the term ‘the F’ can be introduced. But neither of these theories accommodates improper definite descriptions. Since Donnellan it is more common to allow that we can always use ‘the F’ referringly: if the description is improper then the referent of this term is simply found in the term’s practical use. Such a sense of ‘the F’ is captured in the epsilon term ‘a’ above, and specifically we can then say that the one and only F is G – i.e., Ga – even if there is not a single F. The term ‘a,’ therefore, still has a referent in this case, which might not be how one ordinarily thinks of a fiction. But this is quite in tune with how we take fictions in real life, as I have explained at length,8 and it is the basis for the theory of fictions I shall present in the remainder of this paper. Woods reminds us that Faust was in fact a real person, even while the various stories about him were fictional,9 but the point has a much wider application. Thus, for instance, the same point can be made about the personages in Swift’s Gulliver’s Travels, since in that book it is many of Swift’s political contemporaries who are the (disguised) targets of his satire. With regard to ‘the (present) king of France’ one has to be more imaginative, since there are no known stories about this character. Let us see ... An Englishman might call his next-door neighbour ‘the king of France’ and tell all sorts of extravagant stories about him if, for instance, that next-door neighbour exhibited enough puffed-up majesty and was difficult to get along with. But a spy might call even some non-person, say a safe house, ‘the king of France,’ if he wanted to be mysterious and lead people off the scent. A comparable collection of considerations might be involved in finding a reference for Meinong’s ‘the round square.’ Is this some corpulent conservative? Maybe it is just a sound. Ironically, given the theoretical opposition by Fregeans to ‘Millian names,’ pragmatic considerations have found a non-attributive – that is, Millian – reference for Frege’s ‘The Morning Star,’ since this now conventionally names Venus, even though Venus is not a star. Clearly neither The Round Square nor The Morning Star can live up to its name, since in neither case is there any such thing. But how can something be the one and only F ‘if there is no such thing’? In part this is just a matter of how the word ‘such’ operates: it brings in things with the spoken-of character rather than things with

The Epsilon Logic of Fictions 37

the spoken-of name. It is here where another theorem provable in the epsilon calculus is illuminating: (Fc.(y)(Fy Š y = c)) Š c = Hx(Fx.(y)(Fy Š y = x)). The important thing is that there is a difference between the left-hand side and the right-hand side, that is, between something being alone F, and that thing being the one and only F. For the left-right implication cannot be reversed. We get from the left to the right when we see that the left as a whole entails (x)(Fx.(y)(Fy Š y = x)), and so also its epsilon equivalent FHx(Fx.(y)(Fy Š y = x)).(z)(Fz Š z = Hx(Fx.(y)(Fy Š y = x))). Given Fc, then from the second clause here we get the right-hand side of our original implication. But if we substitute ‘Hx(Fx.(y)(Fy Š y = x))’ for ‘c’ in that implication then on the right we have something which is necessarily true. But the left-hand side is then the same as (x)(Fx.(y)(Fy Š y = x)), and that is in general contingent. Hence the implication cannot generally be reversed. Having the property of being alone F can be contingent, but possessing the identity of the one and only F is necessary. The distinction is not made in Russell’s logic, since possession of the property is the only thing which can be formally expressed there. In Russell’s theory of descriptions, c’s possession of the property of being alone a king of France is expressed as a quasi-identity c = LxKx, and that has the consequence that such identities are contingent. Indeed, in counterpart theories of objects in other possible worlds the idea is pervasive that an entity may be defined in terms of its contingent properties in a given world. It was in this context that free logics were born, with the presence of an entity in, or its absence from, a world being taken to be a matter of whether certain contingent proper-

38 B.H. Slater

ties are instantiated there. Using the epsilon calculus, however, we come to realize that contingent properties are not necessary to identify objects, and that fact allows that it is no mere counterpart of an entity which appears in another world, but the very entity itself. Fictions, while they may have their supposed character in some other possible world, do not have it in this world – that is just what makes them fictions – but from this it must not be deduced that they do not exist in this world. Pegasus, The Winged Horse, simply is not winged, or is not a horse in this world, the world of fact. Russell’s identity above is maybe a ‘contingent identity,’ but it must be crucially distinguished from necessary identities, as became very apparent when transworld identities came to be studied in modal and general intensional contexts in the 1960s. 2 Hughes and Cresswell discussed Russellian and Donnellan-like theories of descriptions in their chapter on identity in modal logic. They differentiated between contingent identities and necessary identities in the following way: Now it is contingent that the man who is in fact the man who lives next door is the man who lives next door, for he might have lived somewhere else; that is living next door is a property which belongs contingently, not necessarily, to the man to whom it does belong. And similarly, it is contingent that the man who is in fact the mayor is the mayor; for someone else might have been elected instead. But if we understand [The man who lives next door is the mayor] to mean that the object which (as a matter of contingent fact) possesses the property of being the man who lives next door is identical with the object which (as a matter of contingent fact) possesses the property of being the mayor, then we are understanding it to assert that a certain object (variously described) is identical with itself, and this we need have no qualms about regarding as a necessary truth. This would give us a way of construing identity statements which makes [(x = y) Š L(x = y) – where ‘L’ means ‘necessarily’] perfectly acceptable: for whenever x = y is true we can take it as expressing the necessary truth that a certain object is identical with itself.10

There are more consequences of this matter, however, than Hughes and Cresswell drew out. For once we have proper referring terms for

The Epsilon Logic of Fictions 39

individuals to go into such expressions as ‘x = y,’ we first see better where the contingency of the properties of such individuals comes from – simply the linguistic facility of using improper definite descriptions. But we also see, because identities between such terms are necessary, that proper referring terms must be rigid, that is, have the same reference in all possible worlds.11 This is not how Thomason and Stalnaker saw the matter in their treatment of directly referring terms. They12 said that there were two kinds of individual constants: ones like ‘Socrates,’ which can take the place of individual variables, and others like ‘Miss America,’ which cannot. The latter, as a result, they took to be non-rigid. But by being non-rigid, they are also non-constant; indeed, it is strictly ‘Miss America in year t’ which is meant in the second case, and that is a functional expression, even though such functions can take the place of individual variables. It was Routley, Meyer, and Goddard who most seriously considered the resultant formal possibility that all properly constant individual terms are rigid. At least, they worked out many of the implications of this position, even though Routley was not entirely content with it. Routley described several systems of rigid intensional semantics.13 One of these, for instance, just took the first epsilon axiom to hold in any interpretation and made the value of an epsilon term itself. On such a basis Routley, Meyer, and Goddard derived what I have called ‘Routley’s Formula’: L(x)Fx Š (x)LFx. In fact, on their understanding, this formula holds for any operator and any predicate, but they had in mind principally the case of necessity illustrated here, with ‘Fx’ taken as ‘x numbers the planets,’ making ‘HxFx’ ‘the number of the planets.’ The formula is derived quite simply in the following way: From L(x)Fx, we can get LFHxFx, by the epsilon definition of the existential quantifier, and so

40 B.H. Slater

(x)LFx, by existential generalization over the rigid term.14 Routley, however, was still inclined to think that a rigid semantics was philosophically objectionable: Rigid semantics tend to clutter up the semantics for enriched systems with ad hoc modelling conditions. More important, rigid semantics, whether substitutional or objectual, are philosophically objectionable. For one thing, they make Vulcan and Hephaestus everywhere indistinguishable though there are intensional claims that hold of one but not of the other. The standard escape from this sort of problem, that of taking proper names like ‘Vulcan’ as disguised descriptions, we have already found wanting ... Flexible semantics, which satisfactorily avoid these objections, impose a more objectual interpretation, since, even if [the domain] is construed as the domain of terms, [the value of a term in a world] has to be permitted, in some cases at least, to vary from world to world.15

As a result, while Routley, Meyer, and Goddard were still prepared to defend Routley’s Formula, and say, for instance, that there was a number which necessarily numbers the planets, namely the number of the planets (np), they thought that this was only in fact the same as 9, so that one still could not argue correctly that as L(np numbers the planets), so L(9 numbers the planets). ‘For extensional identity does not warrant intersubstitutivity in intensional frames.’16 They held, in other words, that the number of the planets was only contingently 9. Routley viewed the imposition of the first epsilon axiom in any model as ‘ad hoc.’ He therefore did not see it as universally true, determining, in part, what is a possible model, but instead thought of it as being something which itself is applicable in certain models rather than others. And while the move which Routley mentions – discriminating as disguised descriptions ‘Vulcan’ and ‘Hephaestos’ – is still available, there are other ways of making relevant discriminations while maintaining that these two terms are co-referential names, which Routley did not consider. Thus Vulcan/Hephaestos has a different place in Greek mythology than in Roman mythology, for instance on account of the set of Greek gods being different from the set of Roman ones. This means that while Routley, Meyer, and Goddard denied ‘(x = y) Š L(x = y)’, there are ways to hold onto this principle, that is, to

The Epsilon Logic of Fictions 41

maintain the invariable necessity of identity. To see how this can be done formally, we must consider some further work which has helped us to understand how reference in modal and general intensional contexts must be rigid. But it involves some different ideas in semantics and even starts in the semantics of propositional logic, which is outside our main area of interest, namely predicate logic. 3 When one thinks of ‘semantics,’ one may think of the valuation of formulas. Since the 1920s a meta-study of this kind was added to the previous logical interest in proof theory. Traditional proof theory is commonly associated with axiomatic procedures, but, from a modern perspective, its distinction is that it is to do with ‘object languages.’ Tarski’s theory of truth relies crucially on the distinction between object languages and meta-languages, and so semantics generally seems to be necessarily a meta-discipline. In fact, Tarski believed that such an elevation of our interest was forced upon us by the threat of semantic paradoxes like The Liar. If there was, by contrast, ‘semantic closure’ – that is, if truth and other semantic notions were definable by means of predicates at the object level – then there would be contradictions. But there is another way of looking at the matter which is explicitly non-Tarskian and which others have followed.17 This involves defining truth non-predicatively by means of the locution ‘it is true that.’ This expression is not a meta-linguistic predicate but an object-level operator, and when it is used the truth tabulations in Truth Tables, for instance, become just another form of proof procedure. Operators are intensional expressions, as in the often-discussed ‘it is necessary that’ and ‘it is believed that,’ and trying to see such forms of indirect discourse as meta-linguistic predicates was very common in the middle of the last century. It was pervasive, for instance, in Quine’s many discussions of modality and intensionality. Wouldn’t someone believe that The Morning Star is in the sky, but The Evening Star is not, if, respectively, they assented to the sentence ‘The Morning Star is in the sky’ and dissented from ‘The Evening Star is in the sky’? Anyone saying ‘yes’ is still following the Quinean tradition, but after Montague’s and Thomason’s work on operators18 many logicians are more persuaded that indirect discourse is not quotational. One particular application of this concerns the necessary status of the first epsilon axiom – something Routley was not convinced of, as

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above. Certainly the associated formula might be given a variety of unusual interpretations, on some of which it was false. But if one is concerned not with the formula, as in ‘“Fy Š FHxFx” is true,’ but the formula as standardly interpreted, as in ‘It is true that Fy Š FHxFx’, then that is necessarily true, since the first axiom is just a specification of the provable predicate calculus thesis, (x)(Fy Š Fx). So it cannot be false. Note that the second epsilon axiom is not similarly necessary, since with ‘F’ as a predicate like ‘is a god,’ and ‘G’ an associated set membership statement, there would need to be necessary co-extensionality for an identity to ensue.19 We must allow the set of gods to vary from one possible world to the next – which helps one to distinguish, as before, Vulcan/Hephaestos in one mythology from the same creature in another. On an operator reading of the attitudes, moreover, the identity of such objects of thought as Vulcan and Hephaestos becomes much clearer. That is because it involves seeing the words ‘The Morning Star is in the sky’ in such an indirect speech locution as ‘Quine believes that The Morning Star is in the sky’ as words merely used by the reporter, which need not directly reflect what the subject actually says. Hence the fact that the subject might not assent to ‘The Evening Star is in the sky’ is no bar to the implication that Quine believes that The Evening Star is in the sky – although one might then call this belief an unconscious one. It is indeed central to reported speech, putting something into the reporter’s own words rather than just parroting them from another source. Thus a reporter may say Celia believed that the man in the room was a woman, but clearly that does not mean that Celia would use ‘the man in the room’ for the person she was thinking about. So referential terms in the subordinate proposition are only certainly in the mouth of the reporter and as a result only certainly refer to what the reporter means by them. It is a short step from this thought to seeing There was a man in the room, but Celia believed that he was a woman as involving a transparent intensional locution. So it is here where

The Epsilon Logic of Fictions 43

rigid constant epsilon terms are needed, to symbolize the cross-sentential anaphor ‘he,’ as in (x)(Mx.Rx).BcWHx(Mx.Rx), where ‘Bc’ means ‘Celia believed.’ This is a further crucial way in which the epsilon treatment of fictions reduces objects of the imagination to things in this world. The Fregean theory of intensions would not allow any cross-reference, since on that theory the mental object of Celia’s belief is opaque, and not extensional. But here there is transparency, and what is on Celia’s mind is a straightforward physical object. To understand the matter fully, however, we must make the shift from meta- to object language we saw at the propositional level above with truth. Routley, Meyer, and Goddard realized that a rigid semantics required treating such expressions as ‘BcWx’ and ‘LFx’ as simple predicates, and we must now see what this implies. It is a matter of how one gets the ‘x’ in such a form as ‘BcWx’ and ‘LFx’ to be open for quantification. For, what one finds in traditional modal semantics20 are formulas in the meta-linguistic style, like V(Fx, i) = 1, which say that the valuation put on ‘Fx’ is 1, in world i. There should be quotation marks around the ‘Fx’ in such a formula to make it metalinguistic, but by convention they are generally omitted. To effect the change to the non-meta-linguistic point of view, we must simply read this formula as it literally is, so that the ‘Fx’ is in indirect speech rather than direct speech, and the whole becomes the operator form ‘it would be true in world i that Fx.’ In this way, the term ‘x’ gets into the language of the reporter, and the meta/object distinction is not relevant. Any variable inside the subordinate proposition can now be quantified over, just like a variable outside it, which means there is ‘quantifying in,’ and indeed all the normal predicate logic operations apply, since all individual terms are rigid. An example that illustrates this rigidity involves the actual top card in a pack and the cards which might have been top card in other circumstances.21 If the actual top card is the ace of spades, and it is supposed that the top card is the queen of hearts, then clearly what would have to be true for those circumstances to obtain would be for the ace of spades to be the queen of hearts. The ace of spades is not in fact the

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queen of hearts, but that does not mean they cannot be identical in other worlds.22 Certainly, if there were several cards people variously thought were on top, those cards in the various supposed circumstances would not provide a constant c such that Fc is true in all worlds. But that is because those cards are functions of the imagined worlds – the card a believes is top (HxBaFx) need not be the card b believes is top (HxBbFx), and so forth. It still remains that there is a constant, c, such that Fc is true in all worlds. Moreover, that c is not an ‘intensional object,’ for the given ace of spades is a plain and solid extensional object, the actual top card (HxFx). Routley, Meyer, and Goddard did not accept the latter point, wanting a rigid semantics in terms of ‘intensional objects.’23 Stalnaker and Thomason accepted that certain referential terms could be constant, and others functional, when discriminating ‘Socrates’ from ‘Miss America’ – although the functionality of ‘Miss America in year t’ is significantly different from that of ‘the top card in y’s belief.’ For if this year’s Miss America is last year’s Miss America, still it is only one thing which is identical with itself, unlike the case of the two cards. Also, there is nothing which can force this year’s Miss America to be last year’s different Miss America, in the way that the counterfactuality of the situation with the playing cards forces two non-identical things in the actual world to be the same thing in the other possible world. Other possible worlds are thus significantly different from other times, and as a result other possible worlds should not be seen from the realist perspective appropriate for other times – or other spaces. Other possible worlds, one must remember, are not real. 4 The foregoing throws considerable light on a great many aspects of Woods’ discussion of fiction. Thus the examination of the case of truth, for a start, shows that no departure from classical two-valued logic is necessary. Some statements, namely those relating to fictions, certainly have no determinate or ‘factual’ basis, but using the sort of choice formalized in the epsilon calculus, we can settle on a truth-value, even arbitrarily, and that is enough to show that no many-valued logic, or supervaluation, need come in. The shift from a sentential to a propositional account of indirect speech, as in the shift from a predicative to an operatorial theory of the attitudes, relates to Woods’ question about whether one takes into

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account translations of literary works.24 If Conan Doyle said that Sherlock Holmes lived in Baker Street, and one wants to win a bet on the question, does one have to speak English? On strict ‘say-so’ semantics this might be the case, but what Conan Doyle said, in the appropriate sense, is not a matter of direct quotation, since reported speech is involved. And in that connection, also, if one is to win the bet, does one just say that Sherlock Holmes lived in London, or is it more proper to mention that Conan Doyle, or some English writer, said this?25 We have a clear answer, since we now have unearthed the formal expression for statements of the latter kind, the operator expression: V(Fx, i) = 1. If one says merely ‘Fx’, by contrast, then one is just playing along with the storytelling – joining in the fiction or pretence. A different speech act, other than assertion, is then involved, even though an indicative sentence is uttered. But a more objective report is generally still available, in the above operatorial form, with ‘i’ being the author, or collection of authors, in question. That also settles the question of the difference between Sherlock Holmes and the present king of France. No one has written a story about the latter (or at least not a well-known one), which means that no report is available which would give the authority for the story. So in the latter case one is left, like the Englishman with his neighbour before, to make up some fiction spontaneously. The intrusion of further elements from real life into known stories, as in Woods’ case ‘Sherlock Holmes had tea with Gladstone,’ can be handled in the same way, since even if Conan Doyle did not say it, we could still make up such a story ourselves. But the interplay between real life and fiction is more complicated than this, as Woods appreciated, with some statements involving fictions being proper parts of real people’s histories.26 This would be the case with Woods’ ‘Kinglsey Amis admires James Bond,’ for instance, to which we might also add the hoary old philosophical chestnut ‘Ponce de Leon was looking for The Fountain of Youth.’ But the bare form, X was looking for Y, is invariably about someone’s relations to some thing in this world, whether or not there is, or is believed to be, anything of the kind looked for. If Ponce de Leon were to say ‘There is no such thing as The

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Fountain of Youth, but I am still looking for it,’ we would maybe judge him crazy – but formally it is just a matter that there is no point in making any effort to find HxFx if one knows that ™(x)Fx, since an arbitrary nomination of which thing ‘HxFx’ is to refer to is enough to locate it. In the reverse case, certainly, we have a more normal, rational situation, but then a fuller presentation of the case would be along the lines, ‘Ponce de Lion believed there was such a thing as The Fountain of Youth, and he was looking for it,’ and so clearly it is whether there is, or someone believes there is, such a thing as The Fountain of Youth which brings in other things than purely the relation of looking for. These points relate to the debate over modal realism, which has been shown to be inappropriate, given the epsilon calculus analysis. Certainly many readers and film viewers believe the fictions involved are real, or are similar enough to real events, or are quite likely true, and on that kind of basis understandable human relations might be developed with what would otherwise be just arbitrary entities. But some such further presumption must be added before we can have an emotional relation with fictions; otherwise the much-discussed paradox of fiction would arise,27 and it is not generally the case that such full emotional relations are experienced.28 One remaining case concerning our relations with fictions which Woods discussed brings in some further important matters – the case of Freud’s psychoanalysis of Gradiva.29 This case, of course, concerned not Freud’s psychoanalysis of the eponymous heroine of the book in question but instead his psychoanalysis of its author, Wilhelm Jensen, through the light Freud took the book to provide on Jensen’s early years. However, the general form of the conjecture Freud was thus enabled to make, which became a standard in psychoanalytic literary criticism thereafter, bears closely on the epsilon analysis of fictions. As in Freud’s theory of ‘dreamwork,’ the real events located in an author’s early years through psychoanalysis are said to be the ‘latent content’ behind the ‘manifest content’ available directly from the dreams the author put into books like Gradiva. So the fact that an epsilon term, when fictional, still has a real referent can be put by saying that that real referent is the ‘latent content’ behind the ‘manifest content’ given by the properties embodied in the epsilon term itself. The Gold Mountain in El Dorado was not strictly such, but it was still the source of all the stories, and comparable ‘Chinese whispers,’ as arose in that case, commonly transform other banal physical realities into weird, barely recognizable beasts. The Unicorn was in fact The Rhinoceros, as is clear from early

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medieval bestiaries, but its origins in Africa got lost to public consciousness somehow, leading to the much more romantic image we know. One difference from Freud, however, concerns the nature of the connection between the two kinds of content: Freud held that the connection is always rational, even causal, and he had a detailed theory of how the unconscious mind is supposed to work to mask reality. The epsilon analysis, however, allows for more absurd relations, since it can be just a matter of pure choice. The epsilon analysis may be said to be more ‘existential’ on account of this. notes 1 John Woods, The Logic of Fiction (The Hague: Mouton, 1974). 2 B.H. Slater, ‘Fictions,’ British Journal of Aesthetics 27 (1987): 145–55; B.H. Slater, ‘Hilbertian Reference,’ Nous 22 (1988): 283–97; B.H. Slater, ‘The Incoherence of the Aesthetic Response,’ British Journal of Aesthetics 33 (1993): 168–72. 3 A.C. Leisenring, Mathematical Logic and Hilbert’s H-symbol (London: Macdonald, 1969). 4 N. Bourbaki, Éléments de Mathématique, book 1 (Paris: Hermann, 1954). 5 K. Donnellan, ‘Reference and Definite Descriptions,’ Philosophical Review 75 (1966): 281–304. 6 See, for instance, W.P.M. Meyer Viol, Instantial Logic (The Hague: CIPGegevens Koninklijk Bibliotheek, 1995), chap. 6. 7 B.H. Slater, Intensional Logic (Aldershot: Avebury, 1994), passim. 8 B.H. Slater, ‘Fictions.’ 9 Woods, The Logic of Fiction, 46. 10 G.E. Hughes, and M.J. Cresswell, An Introduction to Modal Logic (London: Methuen, 1968), 191. 11 Compare with R. Barcan Marcus, Modalities (Oxford: Oxford University Press, 1993), esp. 11, 225. 12 R. Thomason, and R.C. Stalnaker, ‘Modality and Reference,’ Nous 2 (1968): 363. 13 R. Routley, ‘Choice and Descriptions in Enriched Intensional Languages II and III,’ in Problems in Logic and Ontology, E. Morscher, J. Czermak, and P. Weingartner, eds. (Graz: Akademische Druck- und Velagsanstalt, 1977), 185–6. 14 R. Routley, R. Meyer, and L. Goddard, ‘Choice and Descriptions in Enriched Intensional Languages I,’ Journal of Philosophical Logic 3 (1974): 308; see also Hughes and Cresswell, An Introduction to Modal Logic, 197, 204.

48 B.H. Slater 15 Routley, ‘Choice and Descriptions in Enriched Intensional Languages II and III,’ 186. 16 Routley, Meyer, and Goddard, ‘Choice and Descriptions in Enriched Intensional Languages I,’ 309. 17 For example, A.N. Prior, Objects of Thought (Oxford: Oxford University Press, 1971), chap. 7. 18 For example, R. Montague, ‘Syntactic Treatments of Modality,’ Acta Philosophica Fennica 16 (1963): 155–67; R. Thomason, ‘Indirect Discourse Is Not Quotational,’ Monist 60 (1977): 340–54; R. Thomason, ‘A Note on Syntactical Treatments of Modality,’ Synthese 44 (1980): 391–5. 19 Hughes and Cresswell, An Introduction to Modal Logic, 209–10. 20 Ibid., passim. 21 B.H. Slater, ‘Intensional Identities,’ Logique et Analyse 121–2 (1988): 93–107. 22 Hughes and Cresswell, An Introduction to Modal Logic, 190. 23 Routley, Meyer, and Goddard, ‘Choice and Descriptions in Enriched Intensional Languages I,’ 309; see also Hughes and Cresswell, An Introduction to Modal Logic, 197. 24 Woods, The Logic of Fiction, 25. 25 Ibid., 35. 26 Ibid., 42ff. 27 C. Radford, ‘How Can We Be Moved by the Fate of Anna Karenina?’ Aristotelian Society Proceedings Supplementary Volume 49 (1975): 62–80. 28 Slater, ‘The Incoherence of the Aesthetic Response,’ 168–72. 29 Woods, The Logic of Fiction, 25.

3 Animadversions on the Logic of Fiction and Reform of Modal Logic DALE JACQUETTE

1. Against the Grain The idea that a logically possible world is identical with or can be described as a maximally consistent proposition set is a fundamental assumption of conventional semantics for modal logic. Although the concept is formally unproblematic, philosophically there are serious difficulties in the received definition of a logically possible world. I want to raise conceptual objections to the standard analysis and then sketch a proposal for modal semantics that strikes at the root of the problem in order to avoid these limitations. The conflict to which I call attention has recently been discussed as a dispute between modal realism and modal actualism. It will quickly emerge that I favour modal actualism and oppose modal realism. The alternative by which I propose to avoid objections to conventional modal semantics is to interpret non-actual, merely logically possible worlds as fictional objects in a very specific sense, which I argue can best be understood in terms of John Woods’ theory of fictional objects in his groundbreaking 1974 study, The Logic of Fiction: A Sounding of Deviant Logic. 2. What’s Wrong with Standard Modal Semantics? The definition of a logically possible world as a maximally consistent proposition set is the heart and soul of conventional modal semantics. It is easy in retrospect to understand why. The first modal syntax and axiom systems developed by C.I. Lewis in 1918 were formally uninterpreted until Saul A. Kripke and Jaakko Hintikka independently worked out set-theoretical semantics for modal logics and quantified

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modal logics in the mid-1960s.1 Building on the firmly established mathematical foundation of Zermelo-Fraenkel set theory, Kripke and Hintikka provide an exact formal interpretation of these logical languages that has proven invaluable in the formal modelling of philosophical, scientific, and everyday discourse.2 The availability of a powerful mathematical method for interpreting modal logic made a deep impression on the analytic philosophical imagination in the second half of the twentieth century. It quickly became a favourite tool for symbolizing many difficult logical concepts. The idea that a logically possible world is a maximally consistent proposition set has been so integral to standard modal semantics that it has been accepted as part of the same remarkable package, without much philosophical objection and, indeed, without much philosophical question or scruple. The brilliance and usefulness of these semantic models and the unified interpretation of the variety of modal logics that they afford have made modal semantics a powerful paradigm of analytic philosophy, comparable in impact, and deservedly so, only to Russell’s theory of definite descriptions. The defects of standard modal semantics are less immediately appreciated, partly, no doubt, because proponents are thoroughly convinced of its usefulness and committed to its truth. The concept of a logically possible world as a maximally consistent proposition set is nevertheless philosophically problematic, once we look beyond the pragmatics of formal analysis. There is no way to sugarcoat the fact that if sets exist, as mathematical realism implies, then any logically possible world as a maximally consistent proposition set exists, even in the case of non-actual worlds consisting entirely of non-existent objects and non-existent states of affairs. Moreover, in the prevailing climate of extensionalism in philosophical semantics, it is unavoidable to make logically possible worlds into something existent even when they are nonactual. How can we refer to and truly predicate properties of nonactual, merely logically possible worlds, how can we say anything about them, and how can they stand as true predication subjects if they do not exist? The idea that non-actual, merely logically possible worlds are maximally consistent proposition sets combined with a default Platonistic or realist ontology of mathematical entities according to which proposition sets exist is nevertheless profoundly confused. Nor does it help to retreat to a redefinition whereby logically possible worlds are only described or represented rather than constituted by maximally consistent proposition sets. The problem in that case is

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just the opposite of the one that plagues the interpretation of logically possible worlds as maximally consistent proposition sets. If all logically possible worlds are described by maximally consistent proposition sets, then we face the equally difficult question of what it is, as far as the conventional semantics of modal logic is concerned, that is supposed to single out the actual world as having special ontic status from all other non-actual, merely logically possible worlds. To this pressing ontological query, there is no satisfactory answer within the conventional modal semantic framework. While modal logicians have formalized appropriate modal semantic relations among logically possible worlds, they have not looked into or tried in any meaningful way to characterize the metaphysics of being or to establish the principles of pure philosophical ontology for actually existent entities. The usual practice is for a modal semantics to define an enormous number of combinatorially generated logically possible worlds, typically by invoking the equivalent of a Lindenbaum maximal consistency recursion, each as a distinct, maximally consistent proposition set, and then simply to declare that one of these sets is to be ‘distinguished’ as the actual world. Notations differ, but it is common practice to adopt the mnemonic symbol alpha, ‘D,’ or the at-sign, ‘@,’ to designate the actual world in modal semantics as ‘w@.’3 The philosophical question that urgently remains is how a logically possible world, if universally defined as a maximally consistent proposition set, can be correctly identified as the actual world? 3. General Existence Conditions for Entities Where, then, can we start, if we agree that it is important to explain what makes the actual world actual? How can we think of the actual world as distinguished from non-actual, merely logically possible worlds? As a first, inadequate, approximation, we can say that the actual world is the world consisting of all and only existent states of affairs involving all and only existent objects, and conversely that an existent object or state of affairs is an object or state of affairs that belongs to the actual world, as opposed to a merely logically possible world. The next step in analysing the concept of being is to clarify what is meant by an actual world. Kripke, in his lectures on Naming and Necessity, reminds us that logically possible worlds are not viewed through high-powered telescopes to discover the objects they contain and to

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identify the same objects descriptively from world to world.4 We similarly need to understand that the actual world is not selected from a beauty pageant lineup of all logically possible world hopefuls, only one of which is to be ‘designated’ as actual. We need to break ourselves of misleading ways of imagining the semantics of modal logic and the existence requirements for logically possible objects, states of affairs, and the actual world. The actual world does nothing to deserve its actuality in competition with non-actual, merely logically possible worlds. It is the actual world as a matter of fact because it satisfies the requirements of being, whatever these turn out to be. Without further ado, I now want to propose what I consider to be the correct analysis of the concept of being. The definition is intensional, involving an object’s properties, in what I shall refer to as a property combination. A property combination is the set of properties nominalistically associated with an object, corresponding to the Fregean sense of an object’s logically proper name or definite description. Accordingly, I shall say that to be is to have a maximally consistent property combination. An entity is an existent object, state of affairs, or the actual world as a whole, which by the present account is one that has a maximally consistent property combination. A non-existent intended object, including fictional objects that belong to the semantic domain of a logic of fiction, is an object whose property combination is either inconsistent, containing both a property and its complement, or incomplete, failing to contain either a property or its complement. I shall refer to a theory of this kind as a combinatorial analysis of the concept of being. A defender of conventional modal semantics is committed to denying that maximal consistency is sufficient for an object to be actual. Modal logicians, recognizing the difficulty, and desperate to find a way to resolve conflicting intuitions, relativize existence to particular logically possible worlds. They say, for example, that the Statue of Liberty exists in the actual world but that in other logically possible worlds there exist objects like the Fountain of Youth or city of El Dorado that do not exist in ‘our’ world.5 Combinatorial modal semantics thus entails that there are objects and states of affairs that truly exist in worlds that truly do not exist. Such a position is logically incoherent if we believe that all and only the entities and states of affairs belonging to the actual world exist. By defining existence as the maximal consistency of an object’s property combination, we explain what it means for the actual world to exist, and we recover consistency in maintaining that only the actual world exists, along with the objects and states of affairs by which the actual

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world is constituted.6 We can then say that for the actual world to exist, to be designated or distinguished as actual from among all other logically possible worlds, means for it to be a maximally consistent statesof-affairs combination, represented by a particular maximally consistent proposition set. This in turn is equivalent to the actual world’s consisting only of objects and states of affairs whose property combinations are maximally consistent and to the property combination of the actual world in its entirety as an existent entity being maximally consistent. These are all purely logical concepts, by which the actual world is distinguished as a maximally consistent states-of-affairs combination to be represented linguistically as a uniquely maximally consistent proposition set, unlike, by the proposed definition of existence, all non-actual, merely logically possible worlds. By developing a combinatorial ontology along these lines, we can also answer the longstanding metaphysical problem of why there exists something rather than nothing and of why there exists at most only one logically contingent actual world.7 4. Maximal Consistency of the Actual World Of course, nothing prevents a conventional modal logician from devising a Lindenbaum-style recursive procedure whereby all distinct complete and consistent sets of propositions are projected. The method is to consider each proposition in turn and add it to a given set if and only if it is logically consistent with the propositions already collected in the set until there are none left, and otherwise adding its negation, following the process in the case of every logically distinct combination of propositions until every proposition or its negation is incorporated. Much the same recursion is followed in consistency and completeness proofs in standard logical meta-theory. This no doubt played an important role historically, along with the default realist or Platonistic ontology for mathematical entities, and extensionalist semantic presuppositions, in the conventional concept of a logically possible world as a maximally consistent proposition set. If we are Platonic realists in the applied scientific ontology of mathematics, and semantic extensionalists, then we may be strongly inclined if not irrevocably committed to regarding such sets as themselves existent mathematical entities to which we can appeal ad libitum in theory construction, especially in designing a formal semantics for modal logic. The problem is whether the resulting maximally consistent sets of propositions deserve to be called or considered as describing non-

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actual, logically possible worlds. As indicated, the identification or description of a logically possible world as a maximally consistent proposition set, or as logically derivative from a maximally consistent statesof-affairs combination, is warranted only in the unique case of the actual world. Non-actual, merely logically possible worlds, by comparison, are fictional creatures of combinatorial modal semantics and as such inherently predicationally incomplete. The sense in which non-actual, merely logically possible worlds and the objects and states of affairs that belong to them are fictional remains to be specified. The following arguments provide reasons for adopting the unconventional concept of a logically possible world, of the actual world as maximally consistent, and of non-actual, merely logically possible worlds as fictional, that has now been introduced for a reformed semantics of modal logic. Objection 1: Kripkean Transworld Identity Stipulations Are Inherently Submaximal The first objection to considering maximally consistent proposition sets as non-actual, merely logically possible worlds depends on Kripke’s answer to the transworld identity problem. Objects and states of affairs in the actual world might have been so different than they actually are that it appears impossible even in theory and certainly in practice to identify the same objects from world to world by positive correspondence with their descriptions in any given world. Kripke sidesteps the problem by arguing that transworld identity is not a matter of discovery, but of decision. We stipulate, in Kripke’s terminology, that there is a non-actual, logically possible world in which Richard Nixon’s chromosomes are so radically altered prior even to his development in the womb that at no time within that world is he recognizable as the Richard Nixon we know from experience of his appearance in the actual world, but instead exactly resembles Marilyn Monroe.8 Kripke’s response to the transworld identity problem has gained wide acceptance among modal logicians. Taken literally, although Kripke does not acknowledge the consequence, the Kripkean transworld identity stipulation implies a constitutional incompleteness in the proposition sets associated with non-actual, logically possible worlds, by which they can only be submaximal even if logically consistent. Stipulation involves real-time human decision making that is incompatible with the possibility of including all the items in a consistent proposition set in order to qualify as maximally consistent. We,

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finite creatures that we are, can only submaximally stipulate limited numbers of objects and facts in describing distinct logically possible worlds in the brief time we can devote to such theorizing, in which this or that is different from the actual world, and leave the rest unspecified. Objection 2: Submaximal Consistency Is Adequate for the Modal Semantics of Non-Actual Worlds Second, it is significant that submaximally consistent proposition sets are adequate for the formal semantics of modal logic in the case of all non-actual, logically possible worlds. It is good enough for the purposes of formalizing a general semantics of modal logic to recognize maximal consistency only in the case of the actual world. There is nothing we can practically do with maximally consistent sets in understanding the truth-conditions for sentences in modal logic that we cannot do with submaximally consistent sets or submaximally consistent states-of-affairs combinations. As a further theoretical advantage, submaximally consistent proposition sets do not incur the difficulties of conventional modal semantics. They encourage an answer to the question of being with respect to worlds, explaining the actual world as uniquely maximally consistent in its fully consistent complement of actually existent states of affairs as determined by the actually instantiated properties of actually existent entities. They further avoid the need to relativize existence to specific logically possible worlds, dispensing with the confusing assertion that the Fountain of Youth exists in a particular world, whatever this is supposed to mean, when we assume on the contrary without qualification that the Fountain of Youth does not exist or does not actually exist. The very idea of ‘existence in a (non-actual) world’ ought to be avoided if at all possible, because it saddles modal logic with a counter-intuitive way of distinguishing the actual world from alternative logically possible worlds. The actual world in that case cannot be identified as the world containing all and only existent states of affairs involving all and only existent entities. All worlds, paradoxically, then, each have their own world-indexed existent entities. We are further obliged in that case to index actually existent entities to the predesignated actual world, whose facts and objects are existent-w@, rather than existent-w1,-w2,-w3, and so on, where @ = 1, 2, 3, ... What, then, could possibly justify conventional modal logic’s demand that non-actual merely logically possible worlds be maximally consistent rather than submaximally

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consistent proposition sets? Why go maximal except in the unique case of the actual world? The only imaginable reason might have to do with an intuition about the meaning of the word ‘world,’ according to which submaximal consistency does not deserve to be called or associated with a world, or that a world, in the true sense of the word, even if it is non-actual and merely logically possible, must be maximal. Any such reasoning appears unfounded and any such counter-objection inconclusive. Objection 3: Non-Actual Worlds by Definition Are Submaximally Consistent A third justification for distinguishing the actual world as the only maximally consistent proposition set or states-of-affairs combination, by contrast with the submaximal consistency of non-actual, merely logically possible worlds, is based on a trilemma. The argument reveals a deeper reason why non-actual, merely logically possible worlds are submaximally consistent. A maximally consistent set of propositions, even on the weakest accessibility relations between logically possible worlds in conventional modal semantics, must include true or false propositions about the actual states of affairs obtaining in the actual world. Non-actual worlds need to look over their shoulders at what is happening in the actual world, so to speak, and include information about the situation there, in order to be maximal. Otherwise, there will be propositions that are entirely left out of their proposition sets, which by definition are thereby less than maximal. The proposition set of a non-actual, merely logically possible world as a result has a conflicting set of responsibilities if it is to be maximally consistent. It must pretend, in certain cases, that the Fountain of Youth ‘actually’ exists, or exists in or relative to its associated world, and must accordingly include a proposition to this effect, while at the same time declaring that the Fountain of Youth does not exist or does not actually exist in the actual world. We have already seen that indexing the truth of propositions to particular logically possible worlds within proposition sets associated with worlds is a philosophically questionable practice in modal semantics. Now we are prepared to see worse problems arise whether or not world-indexing of propositions is introduced. Consider a proposition set S for a non-actual, merely logically possible world, wi, that is striving for maximal consistency in the spirit of conventional modal semantics. Set S either includes or does not include the proposition that the Fountain of Youth does not exist, and

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does so either indexically, by making reference to the truth of the proposition in the actual world, or non-indexically. Thus, there are three possibilities: (1) If S does not include any proposition expressing the fact that the Fountain of Youth does not exist (in actual world w@), then S is submaximal, even if it is logically consistent. (2) If S includes a proposition expressing the fact that the Fountain of Youth does not exist indexically by referring to the proposition’s truth in or with reference to the actual world w@, ‘The Fountain of Youth does not exist (in w@),’ then it must also assert the existence of the Fountain of Youth in or with reference to wi, ‘The Fountain of Youth exists in wi’ (i = @). Then, problems of indexicality for an extensionalist semantics of modal logic aside (and they are considerable, including the danger of outright logical paradox), S contains propositions that acknowledge by their explicit indexing that the Fountain of Youth does not actually exist, in effect declaring its own falsehood, a false description of the world, and thereby rendering S unfit as a description of wi. (3) If S includes a proposition non-indexically expressing the fact that the Fountain of Youth does not exist (in w@) and non-indexically expressing the fact that the Fountain of Youth does exist (in wi, i = @, as before), then, without benefit of the indices indicated in parentheses, S is inconsistent, even if maximal. We want to know what it means for the actual world to be distinguished as actual, by comparison with all non-actual, merely logically possible worlds. We also want to be able to say that only the actual world exists, that all and only the objects and states of affairs in the actual world exist, that existence is not to be relativized to worlds, but that ‘existence’ means real existence or actuality. Thus, it seems we have no choice but to rethink the conventional wisdom of standard settheoretical semantics for modal logic. Set S is logically inconsistent if it recognizes the facts of the actual world but shuns indexicality, submaximal if it ignores the facts of the actual world, and inadequate as a description of wi if it embraces indexicality in order consistently to include facts about the actual world, such as the fact that the Fountain of Youth does not actually exist. It may appear as somewhat of a relief at this juncture to consider that even if logically possible worlds are defined combinatorially rather than

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set-theoretically, all the standard set-theoretical machinery of conventional modal semantics on which we have come to rely can remain in place, leaving the formal semantics of modal logic untouched. We can preserve set-theoretical relations among sets of worlds just as before, even if worlds are not themselves sets, and in particular even if they are not maximally consistent proposition sets. A logically necessary proposition, if there is any, is still one that is true in every logically possible world, involving the stipulation-exempt properties of abstract entities exclusively. A logically possible proposition is still one that is true in at least one logically possible world. We can continue to invoke differential accessibility relations between logically possible worlds to interpret iterated and especially quantified iterated alethic modalities. 5. Alethic Modality and Woods’ Logic of Fiction We are now in a position to appreciate the significance of the logic of fiction in a combinatorial semantics of alethic modal logic. The conventional approach is to develop a formal semantics for modal logic and then to apply modal logic in trying to understand the logic of fiction. If my objections are sound, then this otherwise reasonable strategy has things reversed. Logically possible worlds other than the actual world in that case are mere semantic fictions, so that we stand in need first of an adequate logic of fiction in order to formalize an exact interpretation of alethic modality. The fictions in which non-actual, merely logically possible worlds are presented might be stories, novels, poems, and other forms of entertainment literature, in scientific writings, including theories of natural phenomena that happen to be false, and in Kripke-style stipulative but intensionally combinatorially interpreted modal semantic constructions about non-actual, merely logically possible objects and states of affairs. This is as it should be if we assume that the actual world is uniquely existent and that non-actual, merely logical possible worlds do not exist even and especially as abstract mathematical or propositional structures. When we produce a formal semantics for alethic modal logic, on the present account, we refer to the uniquely existent, maximally consistent actual world, and we also engage in fiction, creating an imaginary order of non-actual, merely logically possible submaximally consistent worlds that are different in their nonexistent constituent facts and usually also in their non-existent constituent objects from the actual world.

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When we hear a philosopher say ‘It is logically possible that pigs fly means that there is at least one logically possible world inhabited by airborne swine,’ we are to understand this as equivalent to a logician’s tale, beginning with the preamble, ‘Once upon a time ...’ We further note that completeness is not a feature of logically possible worlds that we can freely stipulate. If completeness itself could be stipulated, if we could simply declare and it would then be true that there is a maximally consistent state-of-affairs combination in which pigs fly, then when A stipulates a complete world in which pigs fly and B stipulates a complete world in which pigs fly, then, other things being equal, A and B presumably stipulatively identify the same world. If that in turn were true, however, then there should be no further questions, as there obviously are, about whether or not the supposedly complete world A stipulates – as does the supposedly complete world B – that donkeys as well as pigs fly. Either possibility could hold true in a complete world in which pigs fly, so which is it? When A and B stipulate a complete, maximally consistent world in which pigs fly, are they stipulating a world in which donkeys fly or one in which donkeys do not?9 We need an intensional logic of fiction rather than an extensional mathematical theory of sets for the semantics of modal logic. The reason is that non-actual, merely logically possible worlds are the imaginative creatures of formal logical theoretical fictions. Logically possible worlds other than the actual world are not real things, but modal theoretical fictions. What Kripke and others do not seem to have fully appreciated, in the grip of the conventional set-theoretical apparatus for interpreting modal logic, is that to stipulate a logically possible world is to fictionalize, to tell an inevitably incomplete story about places and times that do not in any sense exist. If this is our new nonset-theoretical model of modality, what formal features does it have? There are numerous logics of fiction currently on the market. I shall conclude my diatribe against conventional modal semantics and pay homage to Woods’ original investigations into the logic of fiction by offering three reasons why I prefer a logic of fiction, specifically of the sort Woods has advanced, for purposes of interpreting the logic of possibility and necessity and forging a new Woods-like synthesis of modal logic and the logic of fiction.10 First, Woods’ logic of fiction recognizes fictional objects as incomplete or inconsistent in their properties, just as the combinatorial analysis of being which I have advanced requires. Second, Woods adopts Sukasiewicz’s three-valued propositional logic in order to accommo-

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date truth-value gaps in predications of properties to fictional objects for which those objects are indeterminate, or, in my terminology, for which those fictional objects have incomplete or submaximal property combinations. Thus, I agree that the truth-value of a proposition predicating a property of an object in a non-actual, merely logically possible world, the stipulation of whose properties does not include the relevant property predicated of it by the proposition, is neither true nor false but truth-functionally indeterminate. Third, Woods’ logic of fiction struggles, unnecessarily, in my view, in his chapter on ‘Many-Valued and Modal Logics,’ with problems about bringing propositions about fictional entities into a non-deviant framework of extensional satisfaction and conventional quantified modal semantics that might be satisfactorily resolved if he were to consider the advantages of a non-standard modal semantics in which non-actual, merely logically possible worlds are themselves treated as fictional objects.11 I do not suggest that Woods had these applications of his logic of fiction in mind. Nor do I claim that Woods anticipated my heterodox proposals for reforming the semantics of modal logic, turning the dependence relation between the logic of fiction and modal semantics upside down. Indeed, there are several respects in which Woods to my way of thinking does not go far enough along the route he opens in his philosophical sounding of deviant logic. What, on the other hand, I can unhesitatingly affirm is that reading Woods’ provocative exposition of a logic of fiction years ago powerfully shaped my understanding of logic. Woods’ analysis suggested to me new possibilities of formalizing everyday discourse more faithfully with respect to its intuitive meanings and the intentions of colloquial language users. As a direct result, my appreciation for the ways in which modal contexts might be more sensitively interpreted has been manifestly influenced by Woods’ pioneering explorations of the logic of fiction. notes I am grateful to the Alexander von Humboldt-Stiftung for supporting this and related research projects as Forschungsstipendiat during my sabbatical leave from the Pennsylvania State University, 2000, at the Franz Brentano Forschung, Bayerische-Julius-Maximilians-Universität, Würzburg, Germany. A much-condensed version of this paper was presented under the title ‘Nonstandard Modal Semantics and the Concept of a Logically Possible World’ at the International Symposium on Philosophical Insights into Logic and Mathe-

The Logic of Fiction and Reform of Modal Logic 61 matics, Nancy, France, 30 September 4 October 2002, and is scheduled to appear in a forthcoming edition of Philosophia Scientiae. 1 C.I. Lewis, A Survey of Symbolic Logic (Berkeley and Los Angeles: University of California Press, 1918). 2 Saul A. Kripke, ‘Semantical Considerations on Modal Logics,’ Acta Philosophica Fennica 16 (1963): 83–94; Kripke, ‘Semantical Analysis of Modal Logic I: Normal Modal Propositional Calculi,’ Zeitschrift für mathematische Logik und Grundlagen der Mathematik 9 (1963): 67–96; Jaakko Hintikka, ‘The Modes of Modality,’ Acta Philosophical Fennica 16 (1963): 65–81. Virtually all modal logicians agree in characterizing logically possible worlds as maximally consistent states of affairs or proposition sets. Hintikka and Kripke nevertheless replace all reference to logically possible worlds in their original formulations of model set-theoretical semantics for modal logics by syntactical structures consisting of ordered sets of sentences and operations on sets of sentences. A model in these formulations is not a world, nor is it suggested that a set of sentences provides an interpretation of modal formulas except as the description of a world. These distinctions understandably are sometimes blurred in expositions of modal logic, as in Alvin Plantinga’s The Nature of Necessity (Oxford: Clarendon Press, 1974), esp. 44– 8, and G.E. Hughes and M.J. Cresswell, An Introduction to Modal Logic (London: Methuen, 1972): 75–80. The problem is compounded by the fact that there is no generally agreed-upon distinction between sentences and propositions in philosophical logic and by unresolved disputes about what constitutes an adequate interpretation of the well-formed formulas in a formal system. There is accordingly a significant potential for disagreements about what should be considered the ‘standard’ or ‘conventional’ semantics of modal logic. For my purposes, I consider conventional modal semantics to involve, as I believe most modal logicians when pressed do also, maximally consistent sets of propositions corresponding to actual or non-actual states of affairs, and I regard syntactical model set-theoretical semantics in the original sense of the term as mediating analyses of modal sentences leading to a complete standard interpretation that needs to be further cashed out in terms of a satisfactory semantics of the syntax of sentences expressing propositions. 3 See David K. Lewis, Counterfactuals (Oxford: Blackwell Publishing, 1973). Some modal semantic systems do not refer to the actual world as such but make only passing reference to the fact that the actual world is to be included as one among all logically possible worlds, and consider only generalized accessibility relations relative to any arbitrary world D. A good

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4

5

6 7

8

9

example is Brian F. Chellas, Modal Logic: An Introduction (Cambridge: Cambridge University Press, 1980), who does not define the concept of a logically possible world but treats it as primitive or intuitive. Saul A. Kripke, Naming and Necessity (Cambridge: Harvard University Press, 1980), 44: ‘A possible world isn’t a distant country that we are coming across, or viewing through a telescope. Generally speaking, another possible world is too far away. Even if we travel faster than light, we won’t get to it. A possible world is given by the descriptive conditions we associate with it ... “Possible worlds” are stipulated, not discovered by powerful telescopes. There is no reason why we cannot stipulate that, in talking about what would have happened to Nixon in a certain counterfactual situation, we are talking about what would have happened to him.’ An example of this widespread practice is found in Graeme Forbes, The Metaphysics of Modality (Oxford: Clarendon Press, 1985), 28: ‘The discussion of the previous section should have imparted a general picture of what model theory for quantified S5 is going to look like. As in the sentential case, there will be a set of possible worlds, but in addition, each world will be assigned a set of objects, the things which exist at that world.’ I explore these topics in Dale Jacquette, Ontology (Acumen Publishing / McGill-Queen’s University Press, 2003), chaps. 2–5. The commitment to the idea of a logically possible world as maximal is clearly stated in Forbes, The Metaphysics of Modality, 8: ‘A possible world is a complete way things might have been – a total alternative history ... In terms of our model theory, the requirement that worlds be complete is reflected in the constraint that every sentence letter occurring in the argument in question be assigned one or other truth value at each world.’ This intuitive statement of the semantic concept of a logically possible world is equivalent to the conventional definition of a world as a maximally consistent proposition set. Forbes significantly adds: ‘We shall see in §4 of this chapter that we can get by without this sort of completeness, but that we pay a price in terms of simplicity’ (ibid.). See Hugues Leblanc, ‘On Dispensing with Things and Worlds,’ in Leblanc, Existence, Truth, and Provability (Albany: State University of New York Press, 1982), 103–19 and Forbes, The Metaphysics of Modality, esp. 70–89. Recent sources on the modal realism-actualism controversy include Charles Chihara, The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic (Oxford: Clarendon Press, 1998); Robert Stalnaker, ‘On Considering a Possible World as Actual I,’ Proceedings of the Aristotelian Society, Supplement 65 (2001): 141–56; Christopher Menzel, ‘Actualism, Ontological Commitment, and Possible Worlds Semantics,’ Synthese 85

The Logic of Fiction and Reform of Modal Logic 63 (1990): 355–89; Stephen Yablo, ‘How in the World?’ Philosophical Topics 24 (1996): 255–86. 10 Chihara in The Worlds of Possibility argues that a Cantorian cardinality paradox afflicts Plantinga’s set-theoretical principles of modal semantics in The Nature of Necessity (Oxford: Oxford University Press, 1974). The combinatorial analysis of alethic modality avoids Chihara’s Cantorian paradox by detaching the concept of a logically possible world from that of a maximally consistent proposition or states of affairs set. The relevance of Chihara’s conclusions as a result is limited to set theory in the abstract and to conventional set-theoretical models of logical possibility. 11 John Woods, The Logic of Fiction: A Philosophical Sounding of Deviant Logic (The Hague and Paris: Mouton, 1974), especially 109–44.

4 Resolving the Skolem Paradox LISA LEHRER DIVE

Cantor’s diagonal argument proves that the set of all sets of integers is uncountable. The Skolem-Löwenheim theorem proves that for any first-order theory there will always exist an enumerable model. How can these two results be reconciled? Putnam has argued that the tension between these two results refutes what he calls ‘moderate realism.’1 Either we are forced to accept traditional epistemology in order to justify the truth of claims about non-denumerable sets or we are forced to abandon classical truth theory. This is a version of Benacerraf’s famous dilemma,2 which requires philosophers of mathematics to choose between Platonism and a standard theory of truth and reference. In this paper I will defend a version of moderate realism, one in which a variety of mathematical models may all be said to correspond to a single mathematical reality. This is suggestive of a wider phenomenon, namely the inability of formal systems to capture all aspects of reality in their entirety. The first section of this paper briefly introduces one of the most fundamental problems in the philosophy of mathematics: the tension between epistemology and semantics. This will provide context and reveal the motivation for this paper. The next section outlines both the Skolem-Löwenheim Theorem and Skolem’s Paradox. This will provide background for Putnam’s argument against moderate realism, which is considered in the following section. The section after this speculates on the truth-value of mathematical claims and, in particular, whether they must be absolute. Finally, Zermelo’s refutation of Skolem’s Paradox is outlined since, although I do not agree with Zermelo’s position, it has some relevance to the philosophical implications of the paradox that I wish to emphasise.

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1. Benacerraf’s Dilemma Benacerraf, in his famous paper ‘Mathematical Truth,’ introduces one of the most important tensions in the philosophy of mathematics. He argues that any account of mathematical truth explains either the epistemology or the semantics of mathematics, each at the expense of the other, and that ‘we lack any account that satisfactorily brings the two together.’3 If a standard theory of truth and reference is adopted, this traditionally forces us to adopt a Platonic theory of mathematical entities. A standard semantics entails that mathematical terms have referents, and these are most often supposed to be Platonically abstract mathematical objects. At the same time, Benacerraf favours a causal theory of knowledge, and this seems to make mathematical knowledge impossible if mathematical entities are Platonic, and hence outside the causal realm. This dilemma is significant for philosophers of mathematics and can be resolved in either of two ways. The first is to stick with Platonism and give an account of mathematical intuition that explains the existence and legitimacy of mathematical knowledge. This was the route taken by Gödel4 and followed by Maddy,5 both of whom attempted to explain mathematical intuition in terms analogous to sense perception. The second is to start with a standard epistemology of mathematics and use it to determine the nature of mathematical entities. This is the tactic chosen by Putnam. It is also my preferred approach. However I favour an epistemically motivated version of mathematical structuralism, while Putnam prefers a nominalistic account based on the notion of proof. The theory I hold is a variety of what Putnam calls ‘moderate realism,’ since it ‘seeks to preserve the centrality of the classical notions of truth and reference without postulating non-natural mental powers.’6 The purpose of this paper is not to defend mathematical structuralism. Rather, it is to achieve the more modest aim of refuting Putnam’s claim that Skolem’s Paradox rules out any moderate realist approach and to evaluate his argument against moderate realism in light of Skolem’s Paradox. Putnam suggests that there exist three main positions on reference and truth: extreme Platonism, which is incompatible with causal epistemologies; verificationism, which replaces the classical notion of truth with an account given in terms of verification or proof; and moderate realism, which falls somewhere in between. It is this third position that attempts to preserve classical notions of truth and reference without

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positing unnatural mental powers, as the Platonist is forced to do. In order to evaluate Putnam’s refutation of this position I will first outline the Skolem-Löwenheim Theorem together with Skolem’s Paradox and then consider Putnam’s argument. 2. The Skolem-Löwenheim Theorem and Skolem’s Paradox The cluster of results generally referred to as the Skolem-Löwenheim Theorem suggests that there is a relativity with regard to set-theoretic results, since it follows from these theorems that if a theory expressible in first-order logic has an intended non-denumerable model, it also has a model with a denumerable domain (and vice versa). Skolem’s Paradox then arises since a sentence that says there exist non-denumerably many sets of natural numbers can be true, even though the domain of its interpretation contains only denumerably many sets of natural numbers. In other words, we can prove true a sentence that asserts the existence of uncountably many objects in a model that has only a countable domain. The reason this seemingly contradictory situation arises is that, even though the domain is countable, this cannot be seen from within the theory. In order to determine that a domain is countable, an enumeration function must be constructed. However this function is not constructible within the theory in question. Thus the intended nondenumerable model contains a true proposition, P, that states ‘there exist uncountably many objects,’ because in that model there is no enumerator function that can count the elements of the domain. In order for the domain of a model to be enumerable there must exist within the model a function that establishes a bijection between the elements of the domain and the natural numbers. Yet in the intended non-denumerable model there is no such function. An enumerator function for the domain of this model does exist but only in a different model. Thus from the point of view of another model, one which contains an enumerator function for the domain of the intended non-denumerable model, the domain is denumerable. This result suggests two very significant corollaries for set theory: first, that set-theoretical results are relative rather than absolute, and second, that no axiomatic system can fully capture our intuitive conception of set. One way to defuse the severity of these results is to reject the notion that a classical finitary language is adequate to provide a full axiomatization of set theory. This was the route taken by

Resolving the Skolem Paradox 67

Zermelo, who insisted on the infinitary nature of mathematics, in which case a finitary language will always be inadequate for capturing our intuitive understanding of set. (His objection is discussed in more detail in section 5.) Another potential way to evade the problem of the inability of axiomatic systems to capture our intuitive notion of set is to deny the existence of uncountable sets. Indeed, Jané7 claims that Skolem’s argument shows that there is no good evidence for the existence of uncountable sets. He notes that Cantor’s diagonal argument proves only that the set of all sets of integers if it exists is uncountable; it does not provide any evidence for the existence of this set. The set of all sets of integers arises from an intuitive handling of set-theoretical concepts, rather than a formal axiomatization of set theory. Due to the SkolemLöwenheim Theorem, the axioms will necessarily have a countable model. An uncountable model arises from an intuitive understanding of set rather than an axiomatization. The problem is now whether we accept or reject an intuitive (nonaxiomatic) conception of set. However this concern does not affect the point being made: that there is a gap between our intuitive understanding of set and any formal axiomatization. This is what Skolem revealed, and this point is not dependent on the existence of the set of all sets of integers. This set arises from our intuitive conception of set, and whether we accept or reject its existence, the fact remains that a formal axiomatization is unable to capture this intuitive understanding. 3. Putnam’s Argument against Moderate Realism In the first chapter of his Realism and Reason, Putnam argues that moderate realism – the view that tries to maintain classical notions of truth and reference without postulating unnatural mental powers – is the position most seriously threatened by Skolem’s Paradox. His claim is that Skolem’s Paradox forces a trade-off: we must either postulate an unexplained mysterious faculty of mathematical intuition (as Platonists do) or abandon classical truth theory. Putnam argues for the relativity of the truth-values of certain mathematical statements, claiming ‘Skolem’s argument ... casts doubt on the view that these statements have a truth value independent of the theory in which they are embedded.’8 This means that our intuitive notion of set cannot be captured by axiomatic set theory, and Putnam shows that since something must capture our intuitive notion of set (or else what is

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axiomatic set theory failing to capture?) we are led once more to Benacerraf’s Dilemma. The Skolem result can be used to argue for Platonism, since we must acquire our intuitive understanding of set somehow, and if it cannot be formalized by axiomatic set theory, then we must have some mysterious faculty by which we acquire this notion. As Putnam points out, this option will be rejected by ‘the naturalistically minded philosopher,’ who would never accept the notion that our mathematical knowledge originates from such an occult faculty. Putnam’s preferred solution to the problem is instead to reject classical causal theories of truth and reference in favour of verificationism. This view analyses truth and reference in terms of verification and proof rather than through truth-conditions or correspondence with reality. On this view, Putnam explains, our knowledge of the statement that a given set (such as the set of real numbers) is non-denumerable is no longer attributed to a mysterious faculty, but consists of our knowing what it is to prove that the set is non-denumerable. The relativity of set-theoretic results is no longer a problem under this analysis, because our understanding of notions like non-denumerability is based on ‘an evolving network of verification procedures.’9 The claim that the real numbers are non-denumerable is made true by understanding how to prove that it is true. Putnam prefers to analyse our understanding of language in terms of how we use it, so reference is linked to use rather than to the world. This approach does dispel the problem of Skolem’s Paradox; however it opens up the difficulty of how such a non-realist semantics can explain the objectivity of mathematical claims. Instead, I believe it is preferable to refute Putnam’s claim that Skolem’s Paradox is fatal for moderate realism, thus retaining a standard semantics for mathematical knowledge while avoiding the pitfalls of Platonism. Putnam’s claim rests on an argument for the relativity of ‘V = L,’ the claim that all sets are constructible.10 Gödel’s intuition, which Putnam notes is shared by many other set theorists, was that if set theory is consistent then V = L is false, even though V = L is consistent with set theory. Putnam then investigates the construction of a model for the entire language of science in which V = L is true. If Gödel’s intuition is correct, the model Putnam describes must not be the intended model, even though it satisfies all required theoretical and operational constraints. The only way that V = L could be false is if we added V z L to the axioms of ZF as a theoretical constraint. This means that the truthvalue of V = L depends on which theoretical constraints we adopt for

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the model in question. In other words, the truth-value of V = L will vary between different intended models. There is no objective way of deciding whether V = L is true or not. It depends only on whether we decide to adopt it as one of the theoretical constraints for a model. There are similar arguments for the relativity of both the axiom of choice (AC) and the continuum hypothesis (CH). We can find models for the entire language of science that satisfy AC or CH. Whether these are the intended models depends on whether the falsity of AC or CH is coded into ZF as a theoretical constraint. Putnam takes these arguments to show that (given our classical analysis of truth) realism must be false, since ‘the realist standpoint is that there is a fact of the matter ... as to whether V = L or not.’11 Putnam claims there is no fact of the matter as to the truth-values of statements such as V = L, AC and CH. His claim is ontological, a stronger claim than the epistemic one that we cannot know the truth-value of such statements. Putnam does not consider that the truth-value of these statements is merely unknowable to us; he claims that the statements have no truth-values. He does not consider another plausible possibility: that on a realist conception these statements (V = L, AC and CH) are neither absolutely true nor absolutely false, and that their truth-value depends in part on the theory in which they are embedded. They could be true in some theories, and false in others, without contradiction. A statement always exists within a theory, and every theory applies to a specific domain. This context determines the truth-value of the statement. My argument is that, contrary to Putnam’s anti-realist position, statements such as V = L, AC and CH do have truth-values. Their truth-value depends on their theoretical context and hence is relative. Although the truth-values may vary in different contexts, and for some contexts we may lack knowledge of the truth-values, this is quite distinct from the claim that such statements have no truth-values. 4. Truth-Value of Mathematical Claims It can be argued that we have access to mathematical reality in the same way that we have access to other aspects of the physical world. We observe basic mathematical structures, abstract from their physical instantiations, and using the concepts that this process provides, we derive further mathematical truths. Some of these turn out to be true of the physical world, some have value as instruments to gain greater

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insight into mathematical reality, and others are hypothetical or fictional truths that may or may not turn out to refer to the world. These latter claims are about mathematical structures that are not instantiated in the physical world (at least as far as we know), and because of our limited access to mathematical reality we sometimes do not know whether the assumptions we made in coming up with them correspond to mathematical reality or not. We can postulate various mathematical systems that all describe mathematical reality as far as we know, but in some cases it is difficult to tell which is the right one, even if they contradict each other. Thus we can come up with various mathematical models that are all candidates for being the privileged model, the one that reflects mathematical reality. This is how there can be mathematical statements that lack any absolute or fixed truth-value. Such a statement may have different truthvalues in different contexts, but its absolute truth-value is not known to us. We can postulate a model in which AC is true, and another in which AC is false, and if each of these could be the privileged model then AC has no absolute truth-value, only relative truth-values. AC is a claim about a structure that we have not found in our experience of the world, although it is possible that one day we may. Since there are models in which AC is true and models in which AC is false, none of which have been ruled out as candidates for being the model that accurately reflects mathematical reality, we cannot say whether in reality AC is true or false, or even if it has a truth-value. Various models that make differing claims about its truth-value can be useful for different purposes, so in some circumstances it may make sense to claim that AC is true or false in order to derive truths in a particular model. However we cannot claim that AC has a fixed truth-value across all models or know whether it has a fixed truth-value in reality. As long as there exist mathematical statements that are undecidable in this way, any formal system that we use to capture mathematical reality will fall short in some way. Gödel’s incompleteness results suggest that we will never be able to find the model that captures mathematical reality; however we can develop very rich models that are good candidates for being the privileged model and are still enormously useful. 5. Zermelo’s Refutation of Skolem’s Paradox Zermelo was one member of the mathematical community who believed he had a solution to Skolem’s Paradox. Van Dalen and Ebbing-

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haus12 provide an analysis of Zermelo’s refutation of Skolem’s Paradox, claiming that it fails to refute the paradox but that it provides a revealing insight into Zermelo’s epistemological convictions. Their discussion is useful in considering the implications of Skolem’s Paradox, since although I do not share Zermelo’s convictions, his position supports a wider claim I wish to make. In essence Zermelo’s refutation of the paradox centres on his infinitary convictions about the nature of mathematics. He concludes that Skolem’s Paradox is based on a ‘finitistic prejudice,’ namely the assumption that ‘every mathematically definable notion should be expressible by a finite combination of signs.’13 Van Dalen and Ebbinghaus reveal Zermelo’s views on the nature of mathematics, namely that it is infinitary in nature and can only be apprehended a priori, in a Platonic sense. He considered mathematics to be ‘the logic of the infinite’14 and thus believed that a first-order approach would fail to capture the richness of mathematics. For him, using finitary combinations of symbols is merely the way that our inadequate intelligence tries to approach what he considers to be true mathematics, which is ‘the conceptual and ideal relations between the elements of infinite varieties.’15 For Zermelo, Skolem’s Paradox reinforced his infinitary convictions about the true nature of mathematics. He believed that the paradox rests on the assumption that all of mathematics is expressible using a finite combination of symbols, the assumption that he refers to as the ‘finitistic prejudice.’ The fact that this assumption leads to a paradox confirms Zermelo’s infinitary convictions, since from a contradiction any absurdity can be derived. He considers that the contradiction apparent in Skolem’s Paradox confirms the erroneous nature of the finitistic prejudice and attempts to refute it by developing an infinitary logic. As van Dalen and Ebbinghaus explain, Zermelo’s failure to refute Skolem is due to his firm belief that true mathematics is infinitary in nature. This belief is based on intuition more than reason. Modern physics provides evidence to suggest that the physical world is not infinite and that the infinitesimal has no physical manifestation. It has been argued that we have evidence only for potential, rather than actual, infinity. Thus Zermelo’s refutation of Skolem, which rests on his infinitary conviction, may not be sound. 6. Philosophical Implications of Skolem’s Paradox Zermelo’s refutation of Skolem’s Paradox contrasts with Skolem’s own resolution of his paradox, yet both approaches can be taken to support

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the claim I wish to make. The paradox we face is that any theory can be represented in a denumerable model, so we can have a theory that contains the true proposition, P, that ‘there exist uncountably many objects’ even though this theory has a countable model. As mentioned, the reason for this is that the non-denumerable domain lacks an enumerator function, a function that maps the domain onto the natural numbers (effectively, a function which ‘counts’ the domain or establishes a bijection between the elements of the domain and the natural numbers). The way such a function can exist without refuting P is that the enumerating function need not be in the model in question. The model that contains P is an uncountable model in the sense that there is no function in the model that can count the elements in its own domain. However there is a way of enumerating its domain even though the function required to do so is not a part of the model. This is how P can be provable, even though there exists a different model containing the enumerator function, namely a countable model for the theory. This resolution to the paradox has some interesting philosophical implications, since it suggests both the impossibility of a genuinely uncountable theory as well as the relativity of set-theoretical results. We have seen that these results lead to the idea that we can postulate various mathematical systems that all describe mathematical reality as far as we know, but we have no way of knowing which is the right one, even if they contradict each other. This is how there can be undecidable mathematical statements, and any formal system that we use to capture mathematical reality will fall short in some way. On this view, our attempts at capturing mathematical reality are similar to our attempts at capturing any other aspect of the world via a formal system. An example of this situation is the failure of physicists (so far) to find a unified ‘Theory of Everything’ which encompasses both the large-scale phenomena explained by relativity theory and astronomy as well as the smaller-scale events described by quantum mechanics. These shortcomings parallel the relativity of set-theoretical results, as revealed by Skolem’s Paradox. Skolem’s Paradox tells us that set-theoretical results do not have an absolute truth-value; some will have different truth-values that vary relative to the interpretation under consideration. This is analogous to the lack of a unified ‘Theory of Everything’ in modern physics: Einstein’s laws of general relativity hold in most circumstances, but when the dimensions involved are extremely small these laws break down. In both cases the given formal system (general relativity or a specific axiomatization of set theory) is

Resolving the Skolem Paradox 73

unable to capture the phenomenon in question in its entirety (namely how matter behaves in the physical world or our intuitive understanding of set). Another example of the failure of formal systems to capture aspects of reality concerns our use of natural languages. It has been argued that formal languages are not powerful enough to capture our ordinary use of language. Haack,16 in a discussion of singular terms and the denotation of names, points out that it has been argued (for example by Schiller and Strawson) that there are subtleties in natural language that are beyond the scope of formal languages. She explains that often the pragmatic aspects of discourse simply cannot be captured by formal languages. It seems reasonable to expect that there would be a similar difficulty with mathematics and set theory, namely that there are some things that an axiomatic system cannot tell us about mathematical reality, and this is what Skolem’s paradox shows. The fact that a formalized set theory cannot capture our intuitive understanding of set is a perfect example of the inability of formal systems to capture all aspects of any given conceptual entity. Schiller and Strawson’s claim that formal methods are inadequate for capturing the subtleties of natural language parallels a similar claim, resulting from the incompleteness of arithmetic. If a formal system describing arithmetic falls short of capturing everything about arithmetical reality, then it follows immediately that it will not capture every true statement of mathematics. This is analogous to the failure of any one scientific theory to explain the world completely. Another manifestation of this principle is in first-order logic, which is undecidable. Reasoning is something that we do as a part of our interaction with the world, and first-order logic is an attempt to capture how we reason, at least in some circumstances. That this formal system is unable to classify every statement as either true or false (undecidability) is to be expected if we accept that formal systems will always fall short of capturing all aspects of the world completely. We develop systems of logic of increasing sophistication, power, and usefulness. However these still fall short of capturing human reasoning completely. In science, mathematics, logic, and linguistics we see this principle confirmed over and over again. Formal systems, despite their many advantages, are insufficient for capturing every aspect of natural phenomena. Putnam gives the following analysis of the philosophical problems arising from Skolem’s Paradox: we have explained our understanding

74 Lisa Lehrer Dive

of language in terms of how to use it, and then we have tried to find out what models ‘out there’ we can find for the language. He points out that something strange is happening here, because our understanding of the language is supposed to determine reference, and yet the language lacks interpretation. Perhaps this is the wrong way of approaching the matter. Rather than taking language as our starting point and analysing how we understand language, we need to consider what is ‘out there’ in the world as being at least as fundamental as our concepts. When we observe the world we need concepts, expressed in language, in order to make sense of what we observe, as well as logic in order to understand our observations. Just as we cannot have an isolated observation which is independent of concepts, it is impossible to understand a language meaningfully if it is considered in isolation from our experience of the world. Language is a tool for us to try to explain and pin down all the features of the world, and the fact that it comes up short and cannot completely capture various systems in the world is just how it is. Language is not as powerful as we might wish, but this is not such a serious problem. As well as his infinitary conception of mathematical reality, Zermelo holds a view that hints at this feature of formal systems. For Zermelo, mathematics is an inadequate way of trying to capture mathematical reality. Since the language of mathematics is finite and he believes that mathematical reality is actually infinitary, there are bound to be inadequacies. He was convinced that the nature of mathematics was infinitary and that our formalizations have not yet come that close to capturing the reality of mathematics, since we are limited by the symbols we use. His views have brought out the point that I wish to emphasize about formal systems: that they almost always fall short when attempting to capture some aspect of reality. He believed that the reason a formal axiomatization of mathematics fails to capture mathematical reality is that the axiomatization uses only a finite number of symbols and is therefore unable to reflect the infinitary nature of mathematical reality. Skolem’s paradox tells us that formal systems cannot capture mathematical reality, but for a different reason: that set-theoretical results are relative rather than absolute. Both these cases suggest that our attempts at explaining mathematical reality using formal systems have fallen short. However this does not imply that we have not come close. Using analogies with language, we can see that we can still get close to explaining mathematical reality, even if we cannot capture it in its entirety.

Resolving the Skolem Paradox 75

Language may be unable completely to capture things that we perceive about the world; however, this is not fatal for its usefulness. Reference is rarely if ever complete. I can successfully pick out people by using their names or by pointing to them or by describing them, for example, as ‘that blonde girl wearing a blue T-shirt.’ My reference can succeed without capturing everything about the person. I can use language successfully to refer to this person, but my reference will not tell me whether or not the person has a sister, and indeed my concept of that person might not include this information. The incompleteness of the reference does not prevent it from picking out the right person nor from making true statements about her. Similarly, in set theory our language is incomplete; it does not capture everything about the structures in question. However this need not be a fatal problem; we can still express many true statements. Skolem’s Paradox and Gödel’s incompleteness results do not require us to abandon classical notions of truth and reference. Although we cannot completely capture set theory within a formal system, we can still refer successfully to set-theoretical entities and say many true things about sets. The true claims that we can make are true in virtue of reflecting (true) facts about set-theoretical structures. This is why they can be considered objective truths. Once we start doing things like accepting AC as true, we become less certain of the set-theoretical results derived. Results proved using AC are contingent on its being true, and we can only be as certain of their truth as we are of AC. Our mathematical methods now become even more like those of empirical science, since we make assumptions (for example that AC is true) and continue working, deriving results that are contingent on the assumptions we make. The more evidence we have for AC, the more certain we are of results that follow from it. In addition, the more intuitive (or ‘correct-seeming’) results we derive using AC, the more convinced we are of its truth, even though we cannot be completely certain.17 It may actually be the case that AC has no objective truth-value, that there is nothing about the world that makes it true or false. The conclusion that the objective truth-values of some set-theoretical results are not known is not devastating to mathematics, since we can continue to do mathematics (albeit producing fallible theorems) without knowing whether certain results are true or false, or even whether they have no truth-value. This view fits with the incompleteness of reference and language, and this is simply a fact about our interaction

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with the world. Indeed it is a desirable outcome, since if a formal system is factually complete, it is untestable. If a formal theory is complete then it will be a maximally consistent set. In order to test a theory, we have to be able to add premisses to it and see whether the truths derived reflect reality. If the theory is already complete, then there are no true premisses left that we could add, so we would have no way of testing the theory. Taken with an inclination towards non-Platonic mathematical realism, the Skolem paradox reveals a phenomenon that is not restricted to set theory and mathematics – namely that formal systems fall short of capturing completely the phenomena they describe. The final point I wish to stress is that this is really an optimistic, rather than a destructive, finding. One reason is that it facilitates the integration of mathematical knowledge into a unified, naturalistic epistemology. Perhaps more important, my conclusion does not preclude formal systems – whether in logic, set theory, mathematics, or any other discipline – from being powerful tools of increasing sophistication. As we make progress in any of these disciplines, the formal systems come closer to being the privileged model, and they reveal new and interesting features of the system they describe. However, reconciling the Skolem result with moderate realism suggests that the privileged model will always be elusive. notes 1 Hilary Putnam, Philosophical Papers, vol. 3: Realism and Reason (Cambridge and New York: Cambridge University Press, 1983), 1–25. 2 Paul Benacerraf, ‘Mathematical Truth,’ Journal of Philosophy 70 (1973): 661– 79. 3 Ibid., 663. 4 Kurt Gödel, ‘What Is Cantor’s Continuum Problem?’ in Philosophy of Mathematics: Selected Readings, 2nd ed., Paul Benacerraf and Hilary Putnam, eds. (Cambridge: Cambridge University Press, 1983), 470–85. 5 Penelope Maddy, ‘Perception and Mathematical Intuition’ in The Philosophy of Mathematics, W.D. Hart, ed. (Oxford: Oxford University Press, 1996), 114–41. 6 Putnam, Realism and Reason, 1–2. 7 Ignacio Jané, ‘Reflections on Skolem’s Relativity of Set-Theoretical Concepts,’ Philosophia Mathematica 9 (2001): 129–53. 8 Putnam, Realism and Reason, 10.

Resolving the Skolem Paradox 77 9 10 11 12 13 14 15 16 17

Ibid., 22. L is the class of all constructible sets, and V is the universe of all sets. Putnam, Realism and Reason, 7. Dirk van Dalen and Heinz-Dieter Ebbinghaus, ‘Zermelo and the Skolem Paradox,’ Bulletin of Symbolic Logic 6 (2000): 145–61. Ibid., 145. Ibid., 150. Ibid., 153. Susan Haack, Philosophy of Logics (Cambridge: Cambridge University Press, 1978), 73. See A.D. Irvine, ‘Epistemic Logicism and Russell’s Regressive Method,’ Philosophical Studies 55 (1989): 303–27.

5 Are Platonism and Pragmatism Compatible? VICTOR RODYCH

In The Reach of Abduction, John Woods and Dov Gabbay note a parallel between Russell’s 1906–7 arguments for the ‘pragmatic’ selection of axioms and a similarly pragmatic criterion of axiom selection espoused by Gödel.1 In conversation, Woods has further suggested that Gödel’s Pragmatism may not be compatible with his Platonism. Specifically, Woods asks ‘whether Gödel can be any kind of Platonist if he holds any version of the view that there are bits of mathematics, logic, or set theory for whose truth there is not the slightest justification apart from the fact that they can be implicated in propositions whose truth is not in doubt.’2 In this paper I will try to answer the broader question, ‘Are Platonism and Pragmatism compatible?’ I will first argue that there is an essential tension between Platonism, an ontological theory, and Pragmatism, an epistemological or decision-theoretic theory. This tension raises the more specific question, ‘Can a Platonist be a realist about the physical world and a Pragmatist about mathematics?’ To answer this question, I will examine two variants of Pragmatic Platonism: Quine-Putnam Platonism and Russell-Gödel Platonism.3 Quine-Putnam Platonism, I will argue, rests upon a bad analogy between physical postulates, such as atom, electron, and quark, and mathematical ‘postulates,’ such as the set of real numbers. It also uses, I will suggest, a highly questionable criterion of ‘ontological commitment’ in conjunction with the famous, but dubious, ‘indispensability argument.’ Russell-Gödel Platonism, on the other hand, is pragmatic mainly (though not exclusively) about the selection of non-self-evident axioms. Gödel’s discussion of the probabilistic verification of new set-theoretic axioms is intended to make possible an endless rational (or justified) expansion of mathematics. Gödelian

Are Platonism and Pragmatism Compatible? 79

Pragmatic Platonism is not incoherent, I will argue, but mathematical intuition and Platonism are explanatorily vacuous. 1. Classical Platonism and Pragmatism Classical Platonism is the view that (a) there exists a realm of mathematical objects and/or facts, (b) mathematical propositions are true by virtue of corresponding to (or agreeing with) the objects and/or facts in this realm, and (c) mathematical objects and/or facts do not causally interact with physical objects or phenomena. The idea that Platonism and Pragmatism could be wed, or that they should be wed, will strike some at least as perplexing. Platonism is an ontological theory, whereas Pragmatism is either epistemological or decision-theoretic. If Pragmatism or pragmatic considerations are involved, we are usually concerned with decision making, from choices of action to theory or idea selection. If, for example, someone says s/he had to make a pragmatic decision, s/he means, roughly, that s/he decided primarily on the basis of practical considerations, as against, for instance, a strong desire or goal – s/he sacrificed one goal or desire for another, in the interest of maximizing utility. ‘I wanted to tell my boss off,’ Giselle says, ‘but I decided it wouldn’t be prudent, so I said that it was I who had accidentally deleted the files on his hard drive.’ If Pragmatism is essentially decision-theoretic in this way, how can what exists turn on human considerations of utility? Surely, human utility cannot be a criterion of existence. Whether it is useful to believe in electrons, or God, or real numbers, the utility of such a belief constitutes neither the existence of the believed entity nor a criterion or test for its existence. The chasm between existence and Pragmatism is perhaps most noticeable when we properly distinguish truth (or existence) from knowledge – ontology from epistemology. For example, one might be a Pragmatist about science – as were Poincaré and Duhem – and maintain that because we cannot know whether a physical theory is true, we must use pragmatic criteria to decide which theories to prefer. On this view, a scientific theory is true if and only if it agrees with reality; but because we can neither prove nor refute a theory, we must decide between theories on pragmatic grounds, such as predictive success and simplicity. Antithetical to this variant of realism-pragmatism stands the ‘pragmatic’ theory of truth, according to which a proposition (e.g., a theory) is true if and only if pragmatic considerations judge it to have the greatest utility. Realists with a robust sense of reality understand the former

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type of Pragmatism but find the latter’s conflation of ontology and epistemology repugnant. Clearly, they argue, though Hilbert may find it useful to believe ‘Sarah is a witch’ today, while Neurath finds it useful to disbelieve ‘Sarah is a witch’ today, the proposition and its negation cannot both be true, nor is relativistic truth (i.e., ‘true-for-Hilbert’ versus ‘true-for-Neurath’) compatible with realism. This raises the question, Can a Platonist be a realist about the physical world and a Pragmatist about mathematics? Perhaps s/he can, if her/his Pragmatism is a decision-theoretic foundation for an epistemology containing an ontology. That is, perhaps her/his point is that the best way to manage our phenomenal experiences is to construct an all-encompassing theory that advises us to accept or reject propositions, and endorse or reject ontologies, pragmatically, always observing the principle of ‘minimum mutilation.’ If so, Pragmatic Platonism is part of a decision-theoretic structure. 2. Quine-Putnam Platonism There are two main ways in which Platonism and Pragmatism have been married. The first way, which comes second chronologically, is the Quine-Putnam argument, which purports to show that if our best overall theory requires the postulation of certain entities, then we must accept the existence of those entities.4 Quine-Putnam Platonism has three components: (1) a criterion of (theoretical) ontological commitment, (2) the infamous indispensability thesis, and (3) the pragmatic or epistemological claim that we should posit entities to explain our phenomenal experiences. One might state the argument for Quine-Putnam Platonism, roughly, as follows. We believe in the existence of horses and houses, galaxies and electrons, and we have good reason for these beliefs. We have posited these various entities in order to explain our phenomenal experience. Our maxim is, essentially, pragmatic: choose the simplest ontology that explains experience. Our postulations, however, are governed by our theories about the universe, and our best theories collectively constitute our best theory of the universe. This theory is unavoidably mathematical – it employs the apparatuses of various mathematical theories. Each of these mathematical theories is a self-contained theory with its own truths and falsehoods, and each such theory asserts the existence of its own entities. Given that we must use these mathematical theories in our best theory of the universe, it follows that we must accept the existence of their entities in the very

Are Platonism and Pragmatism Compatible? 81

same way and to the very same extent that we accept the postulated entities of physics (e.g., electrons). Quine articulates his Pragmatic Platonism in his 1948 paper ‘On What There Is.’ Principle (1), Quine’s criterion of ontological commitment, is best stated as follows: ‘a theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true.’5 Thus, in Peano arithmetic (PA), we are ontologically committed to the set of natural numbers, since we must quantify over this set if axioms and theorems of PA are to be true. Quine’s principles (2) and (3) are also expressed in ‘On What There Is’: A platonistic ontology of this sort is, from the point of view of the strictly physicalistic conceptual scheme, as much a myth as that physicalistic conceptual scheme itself is for phenomenalism. This higher myth is a good and useful one, in turn, in so far as it simplifies our account of physics. Since mathematics is an integral part of this higher myth, the utility of this myth for science is evident enough. In speaking of it nevertheless as a myth, I echo the philosophy of mathematics to which I alluded earlier under the name of formalism. But an attitude of formalism may with equal justice be adopted toward the physical conceptual scheme, in turn, by the pure aesthete or phenomenalist.6

In 1971, Putnam stated the matter thus: [Q]uantification over mathematical entities is indispensable for science ... therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.7

It is worth noting that Gödel makes a similar claim in his ‘Russell’s Mathematical Logic’: [T]he assumption of [mathematical] objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions.8

Penelope Maddy’s misgivings aside,9 the central problem with Quine-Putnam Pragmatic Platonism is that the existence of mathemati-

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cal objects and/or facts is not indispensable to the successful employment of mathematics in physics. To see this, let us suppose that components of some mathematical theories must be employed in our current best theory of the universe. Let us assume further, following Quine, that these mathematical theories must quantify over the set of real numbers in order that their propositions be true (and false). What exactly does this latter supposition mean? It means that the language of real number theory asserts the existence of real numbers – just as, in PA, we can say ‘There exists a prime number between 11 and 15.’ Does it follow that the number 13 exists independently of the numeral ‘13,’ and the numerals ‘11’ and ‘15’? Quine says that we are committed to their existence by the truth and falsity of propositions of PA. But this presupposes that we need truth and falsity in PA, and in mathematics in general. But do we? If, with Frege (and Hilbert), we want our mathematical systems to be applicable to the physical universe, we will be wise to ensure that our mathematical systems include sequences of symbols and their syntactical negations, for this is a necessary condition of applicability. It is, however, hard to see why we need truth and falsity in mathematics, particularly if by ‘true’ and ‘false’ we mean anything like what we mean by these terms in physical discourse. This objection can be made rather succinctly as an objection to Putnam’s reasoning. When Putnam says that ‘quantification over mathematical entities is indispensable for science,’ he means that we say that real numbers exist, that we talk as if real numbers exist, and that we use the existential quantifier over the set of real numbers. We do these things when we do pure mathematics and when we apply pure mathematics to the real world. But do we need to say that real numbers exist – do we need to interpret the existential quantifier in a literal sense? Isn’t it rather the calculus of real numbers that it is crucial to its application in the sciences? That is, we need to employ real number theory in physics to make physics work well, but we do not need to say that real numbers exist. It is the formal system of real number arithmetic that is needed, not talk of the existence of real numbers. Put differently, no one disputes that using real number theory is useful, perhaps even indispensable, in physics, but our mathematical physics would work just as well if, in fact, real numbers did not exist. What this shows is that this manner of speaking, though useful and convenient, presupposes absolutely no ontology. We could, for example, teach mathematics to our children as entirely uninterpreted formal systems. It might be more difficult to do so – human beings might, for instance, learn arith-

Are Platonism and Pragmatism Compatible? 83

metic more easily when their teachers speak as if natural numbers really existed – but we could certainly do it. In this case, however, we would have generations of young physicists using these formal systems in their physics without ever speaking of the existence of real numbers. Mathematicians would still use the existential quantifier, but they would not use it to say that a real number with certain properties exists. Life would carry on much the way it does right now, and our physics would progress just as it does now, except that we would not talk about the existence of mathematical entities. The natural objection to this argument is that we can make the same point about so-called ‘physical postulates,’ including horses, people, stars, and electrons.10 We talk as if these entities existed in order to construct our best theory of phenomenal experience, but if such talk presupposes no ontology, then we are not committed to the existence of horses and stars and electrons. But surely we are committed to horses, stars, and electrons, so our talk must commit us to this ontology. The problem with this objection, however, is that it overlooks a crucial difference between the one kind of talk and the other: we don’t just talk as if horses and electrons existed, we posit their existence in order to causally explain our experiences. Much to the contrary, we do not posit the existence of real numbers, and we certainly do not posit the existence of real numbers in order to causally explain our phenomenal experiences. Yes, we do speak of 2 metres and of —2 metres, but such talk no more commits us to the existence of —2 than the contingent proposition ‘2 apples plus 2 apples yields 4 apples’ commits us to the existence of the number 2. The crucial point is that we commit to the existence of electrons when, and only when, they play a causal role in our causal theory. This is definitely not the case as regards real numbers: we do not presuppose or posit real numbers to causally explain phenomenal experiences or interactions between (perhaps posited) physical objects.11 3. Russellian Pragmatism Russell tells us that when he wrote The Principles of Mathematics, he ‘shared with Frege a belief in the Platonic reality of numbers.’12 Though Russell’s Logicism, unlike Frege’s, included Geometry, Russell agreed with Frege’s 1902 view that the axioms of arithmetic are ‘purely logical’ and ‘self-evident.’13 Russell’s discovery of the Russell Paradox altered his view of the nature of fundamental logical axioms, though Frege, apparently less of a pragmatist, still insisted in 1914 that ‘we

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cannot accept a thought as an axiom if we are in doubt about its truth[,] for [if] it is true but stands in need of proof [it] is not an axiom.’14 Perhaps for this reason, Frege, at his eleventh hour, turned to Geometry as a foundation for mathematics: mathematics would have the requisite self-evident and certain foundation, since the axioms of Euclidean geometry are self-evident, but Logicism is abandoned since Frege always maintained, with Kant, that Euclidean Geometry consists of synthetic a priori truths. In his response to the Russell Paradox, Russell instead abandoned the requirement that logical axioms be self-evident and unabashedly strove to provide ‘ordinary mathematics’ with an inductive foundation.15 In the 1910 Preface to Principia Russell and Whitehead unequivocally state ‘that the ideas and axioms with which we start are sufficient, not that they are necessary.’ ‘[T]he chief reason in favour of any theory on the principles of mathematics,’ they say, ‘must always be inductive, i.e. it must lie in the fact that the theory in question enables us to deduce ordinary mathematics’ (italics mine).16 In mathematics, the greatest degree of self-evidence is usually not to be found at the beginning, but at some later point; hence the early deductions, until they reach this point, give reasons rather for believing the premisses because true consequences follow from them, than for believing the consequences because they follow from the premisses.17

In the Principia, the less-than-self-evident Axioms were (especially) those of Infinity, Reducibility, and Choice (The Multiplicative Axiom). Russell and Whitehead did not claim that the Axiom of Reducibility was needed for a foundation for mathematics,18 but they did claim that some non-self-evident axioms would be needed. Thus, in justifying their use of the Axiom of Reducibility, Russell and Whitehead write: That the axiom of reducibility is self-evident is a proposition which can hardly be maintained. But ... self-evidence is never more than a part of the reason for accepting an axiom, and is never indispensable. The reason for accepting an axiom, as for accepting any other proposition, is always largely inductive, namely that many propositions which are nearly indubitable can be deduced from it, and that no equally plausible way is known by which these propositions could be true if the axioms were false, and nothing which is probably false can be deduced from it. If the axiom itself is apparently self-evident, that only means, practically, that it is nearly indubitable; for things have

Are Platonism and Pragmatism Compatible? 85 been thought to be self-evident and have yet turned out to be false. And if the axiom itself is nearly indubitable, that merely adds to the inductive evidence derived from the fact that its consequences are nearly indubitable: it does not provide new evidence of a radically different kind. Infallibility is never attainable, and therefore some element of doubt should always attach to every axiom and to all its consequences. In formal logic, the element of doubt is less than in most sciences, but it is not absent, as appears from the fact that the paradoxes followed from premisses which were not previously known to require limitations.19 (italics mine)

The earliest articulation of this pragmatic selection of logical axioms appears in Russell’s 1906 paper ‘On “Insolubilia” and Their Solution by Symbolic Logic’: The ‘primitive propositions,’ with which the deductions of logistic begin, should, if possible, be evident to intuition; but that is not indispensable, nor is it, in any case, the whole reason for their acceptance. This reason is inductive, namely that, [1] among their known consequences (including themselves), many appear to intuition to be true, [2] none appear to intuition to be false, and [3] those that appear to intuition to be true are not, so far as can be seen, deducible from any system of indemonstrable propositions inconsistent with the system in question.20

In his 1907 paper ‘The Regressive Method of Discovering the Premises of Mathematics,’ Russell similarly says ‘the method of investigating the principles of mathematics is really an inductive method, and is substantially the same as the method of discovering general laws in any other science.’21 4. Gödelian Platonism From 1944 until his death, Gödel argued for a hybrid of Platonism and Pragmatism. [E]ven disregarding the intrinsic necessity of some new axiom, and even in case it has no intrinsic necessity at all, a probable decision about its truth is possible also in another way, namely, inductively by studying its ‘success.’ Success here means fruitfulness in consequences, in particular in ‘verifiable’ consequences, i.e., consequences demonstrable without the new axiom, whose proofs with the help of the new axiom, however, are

86 Victor Rodych considerably simpler and easier to discover, and make it possible to contract into one proof many different proofs ... There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such powerful methods for solving problems (and even solving them constructively, as far as that is possible) that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory.22

It is no doubt true that Gödel agrees with Russell’s three criteria for axiom selection, quoted above. Unlike Russell, however, whose Pragmatism is driven by the desire to provide an acceptable foundation for known mathematics, Gödel’s Pragmatism stems primarily from the fact that putatively meaningful mathematical propositions are independent of the accepted axioms of set theory. Where Russell’s goal is to show that the ‘ideas and axioms with which [he] start[s] are sufficient’ for deducing ‘ordinary mathematics,’ Gödel’s principal philosophical goal is to provide an acceptable means for deciding putatively meaningful mathematical propositions, such as CH, which are independent of our axioms (e.g., ZFC). Where Russell takes accepted mathematics and attempts to give it an acceptable foundation, Gödel takes an acceptable but insufficient axiom set and attempts to show how we can rationally expand accepted mathematics. Like Russell, Gödel claims that ‘we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true.’23 This perception Gödel calls ‘mathematical intuition,’ which not only enables us to see the selfevidence of self-evident axioms, it ‘induces us to ... believe that a question that is not decidable now has a meaning and may be decided in the future.’24 In particular, given that ‘the set-theoretical concepts and theorems describe some well-determined reality, in which Cantor’s conjecture must be either true or false,’ ‘its undecidability from the axioms being assumed today can only mean that these axioms do not contain a complete description of that reality.’25 In light of Gödel’s First Incompleteness Theorem, the axioms of set theory will never be complete, but ‘the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation “set of.”’26 This extension will be achieved, according to Gödel, by means of a philosophical analysis and clarification of fundamental mathematical concepts (‘such as “set,”

Are Platonism and Pragmatism Compatible? 87

“one-to-one correspondence,” etc.’).27 In his 1961 publication, Gödel claims that the requisite philosophical method is Husserlian phenomenological analysis, which ‘should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us,’ and from which ‘new axioms ... again and again become evident.’28 We return now to our question, can Gödel maintain the pragmatic selection of new mathematical axioms and still maintain (a)–(c) – namely, that (a) there exists a realm of mathematical objects and/or facts, (b) mathematical propositions are true by virtue of corresponding to (or agreeing with) the objects and/or facts in this realm, and (c) mathematical objects and/or facts do not causally interact with physical objects or phenomena? Strictly speaking, the answer is ‘Yes.’ It is true, as Maddy says, that ‘Gödel’s Platonistic epistemology is two-tiered: the simpler concepts and axioms are justified by their intuitiveness; more theoretical hypotheses can be justified extrinsically, by their consequences.’29 It is also true that, in accepting ‘extrinsic’ justification for new mathematical axioms, Gödel explicitly grants that certainty is lost and he implicitly grants that such axioms are not a priori in character. What is crucial, however, is that neither of these two admissions is logically incompatible with (a)–(c). Nothing in (a)–(c) requires that all of our mathematical (or set theoretical) axioms be self-evident, or that all of the axioms in the smallest possible axiom set be self-evident. The fact that some of our (human) knowledge of mathematical axioms is probabilistic, and not a priori or certain, in no way clashes with the Platonistic claim that there exists a realm of purely mathematical facts. On Gödel’s view, we know that we are in touch with the mathematical realm because some of the true axioms ‘force themselves upon us as being true’ – because we directly intuit their truth. Moreover, we can be very confident that the new axioms we accept are also true of the mathematical realm because, even though a new axiom may have ‘no intrinsic necessity at all, a probable decision about its truth is possible ... inductively’ if it demonstrates its ‘success’ by means of its ‘fruitfulness in ... “verifiable” consequences.’ Gödel’s method is an extension of Russell’s, with mathematical intuition playing the crucial role in both. In both cases, mathematical intuition guides us in our selection of probable axioms – in both cases, the axioms selected may be false. The fact, however, that merely probable axioms may be false does not refute Gödelian Pragmatism, for as Rus-

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sell and Whitehead note, ‘[i]nfallibility is never attainable.’ This, after all, is the analogy Russell and Gödel make between the physical sciences and mathematics: just as we test scientific hypotheses and theories inductively, by their ‘successes,’ we similarly test mathematical axioms inductively. As Gödel says, in adding to Russell’s three criteria, we determine whether a new axiom simplifies proofs and contracts ‘many different proofs’ ‘into one,’ whether it sheds light on ‘a whole field,’ and whether it yields ‘powerful methods for solving problems.’ Evidence such as this constitutes inductive evidence that a new axiom is probably true. Gödel’s method, however, goes beyond Russell’s in another very important respect. In addition to helping us select non-self-evident axioms, Gödelian intuition (allegedly) enables us to determine whether some non-self-evident expressions (e.g., CH) are meaningful mathematical propositions (i.e., determinately true or false) and to ascertain the plausibility of such propositions by determining whether they or their negations are entailed by plausible propositions.30 Given that neither CH nor ™CH ‘appears to intuition to be false’ (Russell’s criterion #2), Gödel considers ‘strong “axioms of infinity”’ and notes that ‘it is very suspicious that, as against the numerous plausible propositions which imply the negation of the continuum hypothesis, not one plausible proposition is known which would imply the continuum hypothesis.’31 Thus, in 1947, Gödel’s own mathematical intuition estimated that ™CH is more plausible than CH. 5. Mathematical Disagreement and the Inadequacy of Mathematical Intuition Given that Gödel’s Pragmatism is logically compatible with his Platonism, we must ask instead whether the analogy that he and Russell draw between the physical sciences and mathematics breaks down. The answer, I believe, is affirmative, for unlike probabilistic scientific knowledge of theories, there is nothing approximating mathematical consensus about axioms because there is no mathematical counterpart to sensory perception. If the Gödelian method for adding new axioms is to work, with or without Husserlian phenomenology, it requires that mathematical intuition play the role that sensory perception plays in the physical sciences. The salient point here is that observation statements in physics can usually be intersubjectively tested by numerous people, and when they are so

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tested, observers typically agree. The perceptual experiences and reports of various similarly situated human observers tend to agree, and this, in large part, is what makes science work. The same simply cannot be said of mathematics and mathematical intuition. Though it may be true that mathematicians’ intuitions agree on what Maddy calls ‘the simpler axioms’ of mathematics, this certainly is not true of disputed axioms and of new axioms of set theory. More to the point, for as long as we have had axiomatic mathematical systems, mathematical intuition has failed to achieve consensus in important cases. For centuries mathematicians disputed the Parallel Postulate, some believing it to be self-evident, others thinking it true but in need of proof, and still others claiming, perhaps after the fact, that it is false. Just as this disagreement culminated in the construction of, and disagreements about, non-Euclidean geometries, the furor over the Axiom of Choice began. If anything, the protracted disputes over the Axiom of Choice reveal that contemporary mathematicians are far from intuitive and non-intuitive consensus about axioms and acceptable methods in mathematics (e.g., constructive methods and the Law of the Exclude Middle). Now, fifty-five years after Gödel’s 1947 ruminations about CH, and thirty-nine years after Cohen verified Gödel’s conjecture that CH is independent of ZFC, set theorists try out new axioms, some that entail CH and others that entail its negation. The newer Axiom of Determinacy contradicts Choice, and so one often reads of the ‘controversial’ Axiom of Determinacy – as if Choice weren’t controversial! What is perhaps most striking about all of this is that one often hears avowed Platonists saying that, really, CH is true in this system and false in that system! Such talk is striking precisely because it makes one think of Formalism, not Platonism. If, in fact, set theorists in 2002 are no closer to deciding CH now than in 1947,32 does this show that Gödel’s mix of Platonism and Pragmatism is untenable, that these intuitions are illusory or that they are not sufficiently strong in mathematicians to enable a consensus about CH and new set-theoretic axioms?33 Though it is impossible to predict how things will stand in one hundred or two hundred years, we can certainly say that, at present, mathematical intuitions do not enable consensus about CH and new settheoretic axioms. Mathematical intuitions are not a counterpart to sensory perception in the physical sciences. This does not show, as I have said, that Platonism and Pragmatism are incompatible, but it does strongly suggest that if one’s Pragmatism is like Gödel’s, one’s Platonism is explanatorily vacuous. If we are struck by human agreement

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about ‘the simpler [set-theoretic] axioms’ and human disagreement about new axioms and propositions such as CH, we explain nothing by invoking a realm of mathematical entities. In the face of these facts, Platonism has absolutely no explanatory value. We are far better off trying to explain agreement and disagreement about mathematical propositions in, say, terms of non-abstract meanings and human understanding, without any recourse to a mathematical realm of entities. Indeed, Gödel’s turn to Husserlian phenomenology to clarify ‘meaning’ (by ‘focusing more sharply on the concepts concerned by directing our attention ... onto our own acts in the use of these concepts’)34 seems an admission that a certain kind of introspective psychology offers more explanatory and mathematical promise than any postulation of non-physical, mathematical entities. Given, therefore, that Russell’s and Gödel’s pragmatic criteria ultimately turn on degrees of plausibility and intuitiveness, the postulation of mathematical entities is not only not needed, it in no way helps mathematics or our understanding of mathematics. notes 1 John Woods and Dov Gabbay, The Reach of Abduction: Insight and Trial, vol. 2: A Practical Logic of Cognitive Systems (intermediate draft, December 2001), 77. 2 Personal communication, 23 February 2001. 3 Or Russellian Pragmatism and Gödelian Platonism. See note 15, below. 4 It is important to note that the argument does not conclude, ‘then those entities exist.’ That is, the argument does not purport to show that mathematical entities exist, but only that we are committed to their existence, which may not even mean that we should believe in their existence. Quine offers us a criterion of ontological commitment, not a criterion of existence. 5 W.V.O. Quine, ‘On What There Is,’ Review of Metaphysics 2, no. 5 (1948): 21– 38; reprinted in From a Logical Point of View (Cambridge: Harvard University Press, 1953), 1–19; quotation from 13–14. See also 13: ‘we are convicted of ...’ 6 Ibid., 18. 7 Hilary Putnam, ‘Philosophy of Logic’ (1971) in Mathematics, Matter and Method: Philosophical Papers, vol. 1 (Cambridge: Cambridge University Press, 1979), 347. 8 Kurt Gödel, ‘Russell’s Mathematical Logic,’ in P.A. Schilpp, ed., The Philosophy of Bertrand Russell (Chicago and Evanston, IL: Northwestern Univer-

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9

10 11

12 13

14 15

16

17 18

sity Press, 1944), 125–53; reprinted in Philosophy of Mathematics, P. Benacerraf and H. Putnam, eds. (Englewood Cliffs, NJ: Prentice-Hall, 1964), 447–69 [2nd ed., 456]. Cf. Gödel, ‘Is Mathematics Syntax of Language?’ Version III [1953], in Kurt Gödel, Collected Works, vol. 3 (Oxford: Oxford University Press, 1995), 337. In her ‘The Roots of Contemporary Platonism,’ Journal of Symbolic Logic 54, no. 4 (1989): 1132, Penelope Maddy argues that Quine-Putnam Platonism disagrees ‘with the realities of mathematical practice,’ since ‘unapplied mathematics is completely without justification.’ Maddy’s objection seems to miss the fact that the issue is not that ‘unapplied mathematics is completely without justification,’ but rather that the ontologies of unapplied mathematical theories are left without justification. Note that Quine makes this very point in ‘On What There Is,’ 18, quoted above. It is arguable that even physical postulates carry no ontological commitments unless and until we obtain independent, empirical evidence for their causally efficacious existence. Bertrand Russell, The Principles of Mathematics, Introduction to Second Edition (London: G. Allen & Unwin, 1937), xiii. It is worth noting that in the Preface to volume 1 of the Grundgesetze [Michael Beaney, ed., The Frege Reader (Oxford: Blackwell Publishers, 1997), 195], Frege admitted that his Axiom (V) was not self-evident. Gottlob Frege, ‘Logic in Mathematics,’ 1914; the first nine pages reprinted in Beaney, ed., The Frege Reader, 311. In The Principles of Mathematics, 2nd ed., xiii, Russell says that the Platonism of the Principles ‘was a comforting faith, which [he] later abandoned with regret.’ I will not here argue that Russell maintains a variant of mathematical Platonism in Principia Mathematica, though I believe that he does. See Quine’s ‘Russell’s Ontological Development,’ Journal of Philosophy 63, no. 21 (1966): 661: ‘On later occasions Russell writes as if he thought that his 1908 theory, which reappeared in Principia Mathematica, disposed of classes in some more sweeping sense than reduction to attributes.’ Bertrand Russell and Alfred North Whitehead, Principia Mathematica, (Cambridge and New York: Cambridge University Press, 1973), 1: v (both quotations). Ibid. In fact, they claimed (59–60) that ‘although it seems very improbable that the axiom should turn out to be false, it is by no means improbable that it should be found to be deducible from some other more fundamental and more evident axiom’ (italics mine).

92 Victor Rodych 19 Russell and Whitehead, Principia Mathematica, 1: 59. 20 Bertrand Russell, ‘On ‘Insolubilia’ and Their Solution by Symbolic Logic,’ Rev. de Métaphysique et de Morale 14 (1906): 627; reprinted in D. Lackey, ed., Essays in Analysis, 1973, 194. 21 Bertrand Russell, ‘The Regressive Method of Discovering the Premises of Mathematics,’ in Essays in Analysis, D. Lackey, ed. (London: Allen & Unwin, 1907), 273–4. Noting that by ‘induction’ Russell means any ampliative inference, Woods and Gabbay, The Reach of Abduction, 2: 74, following the lead of Andrew Irvine and G.E. Wedeking, eds. Russell and Analytic Philosophy (Toronto: University of Toronto Press, 1993), have called this ‘regressive abduction,’ stressing that Russell is ‘able to plead the case for wholly non-intuitive axioms ... without having to concede that [their] truth is more than merely probable.’ See also Andrew Irvine, ‘Epistemic Logicism and Russell’s Regressive Method,’ Philosophical Studies 55 (1989): 303– 27. 22 Kurt Gödel, ‘What Is Cantor’s Continuum Problem?’ in Philosophy of Mathematics, 2nd ed., Paul Benacerraf and Hilary Putnam, eds. (Cambridge: Cambridge University Press, 1991), 477. 23 Ibid., 483–4. 24 Ibid., 484. 25 Ibid., 476. 26 Ibid. 27 Ibid., 473. 28 Kurt Gödel, ‘The Modern Development of the Foundations of Mathematics in the Light of Philosophy,’ in Collected Works, vol. 3 (Oxford: Oxford University Press, 1995 [1961]), 375, 377, 379, 381, 383, 385, and 387; quotations from 383 and 385, respectively. Gödel adds (385) that ‘this intuitive grasping of ever newer axioms ... agrees in principle with the Kantian conception of mathematics.’ 29 Penelope Maddy, ‘The Roots of Contemporary Platonism,’ 1134. 30 Kurt Gödel, ‘What Is Cantor’s Continuum Problem?’ 484–5. 31 Ibid., 480. 32 Which Gödel called ‘a question from the “multiplication table” of cardinal numbers.’ Ibid., 472. 33 Gödel’s inductivism is, in fact, much stronger than this. For a discussion of Gödel’s inductivistic claims, see my ‘Gödel’s “Disproof” of the Syntactical Viewpoint,’ Southern Journal of Philosophy 39, no. 4 (2001): 527–55. 34 Kurt Gödel, ‘The Modern Development of the Foundations of Mathematics,’ 383.

6 A Neo-Hintikkan Solution to Kripke’s Puzzle PETER ALWARD

In ‘A Puzzle about Belief,’1 Kripke argued that each member of the following pair of ascriptions to Pierre is true: ‘Pierre believes that London is pretty’ and ‘Pierre believes that London is not pretty.’ And he argued that the truth of these two ascriptions cannot be reconciled with the fact that Pierre – a leading logician – is rational. In this paper, I argue that the truth of the ascriptions in question can be reconciled with Pierre’s rationality. The apparent contradiction between these two theses stems from a pair of presuppositions regarding the analysis of belief sentences. Strictly speaking, a solution to Kripke’s puzzle can be found by the rejection of either of these presuppositions, but I argue that rejection of one of these theses yields a more fruitful account of belief sentences than does rejection of the other. And I sketch a semantic theory which resolves Kripke’s puzzle in this way. 1. Kripke’s Puzzle Kripke’s puzzle goes as follows. Pierre, a normal French speaker, on the basis of the good things he has heard about London, says ‘Londres est joli.’ Suppose we apply to Pierre’s utterance the French version of Kripke’s weak disquotational principle (if a normal English speaker, who is not reticent, sincerely and reflectively assents to ‘p,’ then he or she believes that p) and his principle of translation (if a sentence of one language is true in that language, then any translation of it into any other language also expresses a truth). This would imply the truth of ‘Pierre believes that London is pretty.’ Pierre later moves to an unattractive part of London and learns English not by translation but by the ‘direct method,’ and on the basis of what he sees, says ‘London is not

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pretty.’ From the weak disquotational principle we could, in these circumstances, infer the truth of ‘Pierre believes that London is not pretty.’ The trouble is that the truth of these two ascriptions seems to imply that Pierre has straightforwardly contradictory beliefs. But Pierre, a leading logician, would never have beliefs that were straightforwardly contradictory. To suggest as much would be to impugn his rationality. There is, of course, a sense in which Kripke’s puzzle is hardly puzzling at all. Pierre’s problem is that he conceives of a single city, London, in two different ways and fails to realize that these are two conceptions of a single city. He believes there are two distinct cities – which he calls ‘Londres’ and ‘London’ respectively – one of which is pretty, and the other of which is not so. Pierre is guilty of an error, but one that does not involve a failure of either logical acumen or rationality (and, as such, one for which he is not blameworthy). In order for his rationality to be at risk, he would have to believe that London is pretty and that London is not pretty while conceiving of London in the same way. That is, he would have to believe that there is a single object that both has and lacks a certain attribute (at the same time and in the same respect). But simply pointing this out does not solve the puzzle. As Kripke puts it, ‘[it] is no solution in itself to observe that some other terminology, which evades the question whether Pierre believes that London is pretty, may be sufficient to state all the relevant facts.’2 What is required by way of a solution is a theory of the semantics of belief ascriptions that can reconcile the truth of the two ascriptions in question with Pierre’s rationality. The issue is not so much the psychological question of what Pierre believes. The question is whether what Pierre believes makes the ascription ‘Pierre believes that London is pretty’ true (or, if you prefer, whether it makes it true that Pierre believes that London is pretty). 2. The Semantic Problem In order to adequately address the semantic problem, the assumptions which render the truth of ‘Pierre believes that London is pretty’ and ‘Pierre believes that London is not pretty’ incompatible with Pierre’s rationality need to be made explicit. And in order to do this, we need an ascriptive operationalization of irrationality. That is, we need to find an ascription the truth of which suffices for Pierre’s irrationality.

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And we need to show how the truth of this latter ascription might follow from the truth of the former two. What we want as an ascriptive operationalization of irrationality is an ascription whose truth guarantees that the believer believes of a single object that it both has and lacks a certain property while conceiving of it in the same way. I want to suggest that an ascription of the following form will suffice for these purposes: ‘T believes that a is F and not-F’ (or ‘T believes that a is and is not F’). So now the task is to show how one might argue from the truth of ‘Pierre believes that London is pretty’ and ‘Pierre believes that London is not pretty’ to the truth of ‘Pierre believes that London is pretty and not pretty.’ In my view, the most plausible version of this argument involves two separate entailments: (1) An entailment from ‘Pierre believes that London is pretty’ and ‘Pierre believes that London is not pretty’ to ‘Pierre believes that London is pretty and London is not pretty’; (2) An entailment from ‘Pierre believes that London is pretty and London is not pretty’ to ‘Pierre believes that London is pretty and not pretty.’ And these entailments are underwritten by two distinct assumptions: (a) The two-place predicative analysis (2-PPA): an ascription of the form ‘T believes that p’ is properly analysed as a two-place predicate – ‘Believes (T, that p)’ – where ‘T’ names a believer and ‘that p’ names a proposition. (b) The direct reference (DR) theory of proper names: the meaning of a proper name – that is, its truth-conditional contribution – in an ascription ‘that’-clause is its ordinary referent. It is worth noting that the proposition named by an ascription’s ‘that’clause – ‘that p’ – is the proposition expressed by the ascription’s complement sentence – ‘p.’ And, as a result, the DR assumption ensures that two ‘that’-clauses, differing only in that they contain distinct names of a single object, name the same proposition. The role these assumptions play in underwriting the entailments depends, in part, on what account of propositions one endorses. For expository purposes, I am just going to assume that propositions are sets of possible worlds. The 2-PPA assumption underwrites the first entailment because, if belief is just a relation between a believer and a

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proposition, belief is likely to be closed under conjunction (of, at least, simple propositions). That is, the following psychological principle is likely to have very wide application: if T believes that p and T believes that q, then T believes that p and q. And the DR assumption underwrites the second entailment for the following reason. If the truth-conditional contribution of a name in an ascription ‘that’-clause is just its actual referent, then the set of worlds named by the ‘that’-clause will consist of just those worlds in which the referent satisfies the predicative component of the complement sentence. As a result, a ‘that’-clause of the form ‘that a is F and b is G’ will name the same proposition as ‘that a [or b] is F and G’ whenever ‘a’ and ‘b’ co-refer. 3. Strategies As might be expected, corresponding to these assumptions are two distinct strategies for resolving Kripke’s puzzle. One strategy, associated most prominently with Nathan Salmon,3 involves rejecting the 2PPA assumption while retaining the DR assumption. On this view, the psychological belief relation is a three-place relation between believers, propositions, and ‘propositional guises.’ And the semantic relation – what is expressed by ascriptions – is, in effect, the existential generalization of the psychological relation. That is, an ascription of the form ‘T believes that p’ gets analysed as ‘(x)(Believes (T, that p, x)),’ where the existential quantifier ranges over propositional guises. An ascription is true, on this view, just in case there is some propositional guise or other under which the believer believes the proposition in question. And the reason the first entailment is blocked is because, on this view, belief is closed under conjunction only when the two propositions in question are believed under the same propositional guise. After all, it does not follow from that fact that one believes one set of worlds under some guise and another set of worlds under some guise that there is any guise under which one believes their intersection.4 While current fashion seems to be against me, I find the DR assumption untenable and so reject any solution to Kripke’s puzzle that retains it. My reason is that it renders ascriptions incapable of playing the role they do in explanatory inferences – inferences that rely essentially on how the believer in question conceives of the object of his/her belief. Typically, DR theorists attempt to resolve this difficulty by appealing to the distinction between information literally expressed by ascriptions and information pragmatically imparted by them.5 I have argued

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elsewhere, however, that this manoeuvre is ultimately unsatisfactory.6 As a result, my preferred strategy is to reject the DR assumption while retaining the two-place predicative analysis. On my view, ‘London’ differs in meaning – that is, truth-conditional contribution – as between the two ascriptions to Pierre. It is worth noting that this is a strategy Kripke explicitly rejects. And the reason he rejects it is because ‘the puzzle can still arise even if Pierre associates to ‘Londres’ and ‘London’ exactly the same uniquely identifying properties.’7 Now obviously Kripke is supposing that the alternative to the direct reference view is the sort of descriptivism that has fallen into disfavour since the publication of Naming and Necessity.8 But the rejection of the DR assumption does not force one to endorse descriptivism. In what follows, I will present a semantic theory – the neo-Hintikkan theory – which is neither a version of descriptivism nor a version of the direct reference view. Moreover, this is a view which advocates of the causal-historical theory of referring will find quite amenable. 4. The Neo-Hintikkan Theory: Preliminaries The guiding idea underlying the neo-Hintikkan theory of attitude ascriptions is quite straightforward: the reason co-referring names are not substitutable in ascription ‘that’-clauses stems from the fact that believers often put the (spatio-) temporal parts of the objects they encounter together in the wrong way. Consider the following pair of ascriptions: ‘The ancient astronomers believed that Hesperus is Hesperus’ and ‘The ancient astronomers believed that Hesperus is Phosphorus.’ The explanation of the falsity of the latter, despite the truth of the former, is that the ancient astronomers did not realize the temporal parts of the celestial body they encountered in a certain location in the evening sky were parts of the same enduring object as the temporal parts of the celestial body they encountered in a certain location in the morning sky. They put the temporal parts of Venus they encountered together in the wrong way. In my view, the best way to develop this idea is within the framework defended by Hintikka in ‘Semantics for Propositional Attitudes.’9 According to Hintikka, associated with a believer at a given time is a set of worlds – epistemic alternatives to the actual world for the believer in question – each of which is compatible with what the believer believes.10 And an ascription is true just in case the comple-

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ment sentence is true in each of the believer’s alternatives, or, as I’ve been formulating things here, the set of worlds which constitutes the proposition named by the ‘that’-clause includes all of the believer’s alternatives. So, for example, ‘Mary believes that Jane raises aardvarks’ is true, on Hintikka’s views, just in case ‘Jane raises aardvarks’ is true in all of Mary’s alternatives (or the proposition named by ‘that Jane raises aardvarks’ includes all of Mary’s alternatives). The meaning – or truth-conditional contribution – of a name in an ascription complement is, on this view, a function from worlds to individuals. And the meaning of an n-place predicate is a function from worlds to sets of ordered n-tuples. In light of this, an account of the truth-value of a complement sentence at an alternative can be given in terms of satisfaction in the usual way. Now where I differ from Hintikka is over his account of exactly which function serves as the meaning of a singular referring expression in an ascription complement on any given occasion of use. According to Hintikka, the meaning of a name in a belief context corresponds to one of the believer’s ‘methods of recognizing individuals.’11 There are, of course, problems concerning how a name in an ascription ‘that’-clause could come to have as its meaning one of a believer’s methods of recognition. But even if such difficulties could be resolved, the view remains uncomfortably similar to the sort of descriptivism rejected by Kripke as a solution to the puzzle. 5. The Neo-Hintikkan Theory: Details The fundamental difference between the neo-Hintikkan picture and Hintikka’s own view is that, according to the former, actual individuals can share (spatio-) temporal parts with the individual inhabitants of believers’ alternatives.12 More to the point, distinct temporal parts of a single actual individual can be shared by two distinct individuals in any given alternative. So, for example, the temporal part of Jane that exists on Tuesday might be shared by one individual in one of Mary’s alternatives, while the temporal part of Jane that exists on Wednesday might be shared by a distinct individual in that world. Exactly how one cashes out the ‘part-sharing’ relation will depend, of course, on one’s account of worlds. My preferred approach is to take believers’ alternatives to be sets of structured Russellian propositions and allow temporal parts of actual individuals to be constituents of such propositions. The details of this approach, however, are irrelevant for present purposes.13

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What’s important about the shared-parts assumption is that it facilitates a view according to which actual token uses of names can have referents in believers’ alternatives. As a result, the following theory becomes tenable: the meaning of a name in an ascription ‘that’-clause is the function whose value at a world is the referent of the name at that world. The first thing that needs to be done, however, is to explain how the reference of names in ordinary extensional contexts works. In my view, Kripke’s causal-historical picture is basically right-headed.14 According to this view, the referent of a name on an occasion of use is the object that stands at the beginning of a chain of appropriately causally linked events that culminated in the use of the name in question.15 My view differs from Kripke’s picture in two respects, however: (1) on separate occasions, a name can be used to refer a single object by means of distinct causal chains involving distinct initiating events; and (2) reference to a whole enduring object is secured by means of more directly picking out the temporal part of the object present during the chain-initiating event. The account of name reference I have in mind here is importantly similar to Nunberg’s theory of indexicality.16 According to Nunberg’s analysis, the extension of an indexical expression in a given context of utterance is determined by three distinct meaning components: a deictic component, a classificatory component, and a relational component. (Meaning in this context does not refer to a term’s truthconditional contribution, but to the rules determining its truth-conditional contribution modulo a given context of utterance.) The deictic meaning component is a function from contexts of utterance to an element (or elements) of the context, which Nunberg calls the ‘index.’ For example, the deictic component of both ‘I’ and ‘we’ would be the function whose value in a context of utterance is the speaker; and the deictic component of both ‘tomorrow’ and ‘yesteryear’ would be the function whose value is the time of speaking. The classificatory meaning component is a feature (or a set of features) that must be instantiated by the interpretation. The classificatory component of ‘I,’ in all (ordinary?) contexts, would be the property of being an individual person while that of ‘we’ is the property of being a group of people. Finally, the relational meaning component is a relation that has to hold between the index and the interpretation. For example, the relational component of ‘I,’ in all (ordinary?) contexts, would be the relation of being identical to while that of ‘we’ is the relation of being included in. This basic picture can be applied to the reference of proper names as

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follows. The deictic component of name meaning can be viewed as the function from the context of utterance to a temporal part of some enduring individual. In particular, the index will be a temporal part of an individual present during the event that initiated the chain of causally linked events that culminated in the use of the name in question. Suppose, for example, a use of ‘Mary’ is the culmination of a chain of events initiated by Fred’s perceptual interaction with Mary. In such circumstances, the index would be the temporal part of Mary that existed during this perceptual event. The classificatory component of name meaning would be the property of being an individual person, or a city, or what have you. Presumably this would depend on and vary with contextual presuppositions. And finally, the relational component of name meaning would be the relation of being a temporal part of. Given that Mary is the individual person of whom the part of Mary present during Fred’s perceptual event is a part, Mary is the referent of the use of ‘Mary’ in question. What is important to note is that, given this account of name reference, actual uses of names can have referents in believers’ alternatives as well as in the actual world. Suppose, for example, the temporal part of Mary present during Fred’s perceptual event is a part of an individual – call her ‘Terry’ – in one of Fred’s alternatives. While Mary would be the actual referent of ‘Mary,’ since Terry is the individual person of whom the part of Mary present during Fred’s perceptual event is a part in Fred’s alternative, Terry would be the referent in said world of the (actual) use of ‘Mary.’ The upshot of all this is that the way is now clear for the promised account of the truth-conditional contribution of names in ascription ‘that’-clauses: the meaning of a name is the function for worlds to individuals whose value at a world is the referent of the name at that world. Two potential problems that naturally come to mind are worth addressing at this point. First, two or more causal chains with distinct initiating events could have an equal claim to being the salient cause of a use of a name. And second, the causal chain that culminates in the use of a name could fail to bottom out in a chain-initiating event that involves a temporal part of an individual in the relevant way. Now the second problem is the problem of empty names, which provides difficulties for all theories of names. I do not, at present, have a general solution to this problem. What’s potentially troubling for the neo-Hintikkan theory about the first problem is that it suggests that a name in an ascription ‘that’-clause could be ambiguous even though there is no

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ambiguity as to its actual referent. In my view, however, names in the complements of opaque ascriptions often are ambiguous. The reason this is regularly overlooked is because people often focus on names, like ‘Superman,’ which are tied to particular guises of individuals. Such names, however, are the exception rather than the rule. 6. Kripke’s Puzzle Revisited What remains to be done is to show how the neo-Hintikkan theory reconciles the truth of our original pair of ascriptions with Pierre’s rationality. In each of Pierre’s alternatives, there are two cities. One, which he calls ‘Londres,’ is pretty, and the other, which he calls ‘London,’ is not pretty. Moreover, each of these cities is composed in part out of distinct (spatio-) temporal parts of the actual city London, in particular, those parts causally responsible for Pierre’s beliefs. Now suppose Mary is prompted by Pierre’s utterance of ‘Londres est joli’ to say ‘Pierre believes that London is pretty.’ According to the neo-Hintikkan theory, the truth-conditional contribution of ‘London’ in Mary’s ascription is the following function: the function from worlds to individuals whose value at a world is the city composed in part out of the temporal part of London involved in the initiating event which culminated in Mary’s use of ‘London.’ Given the causal connection between Mary’s and Pierre’s utterances, this temporal part is one of the parts of London shared by the city Pierre calls ‘Londres.’ Since the city of which this is a part is, in all of Pierre’s alternatives, pretty, Mary’s ascription to Pierre is true. And by parity of reasoning, if Mary were prompted by Pierre’s utterance of ‘London is not pretty’ to say ‘Pierre believes that London is not pretty,’ her ascription would be true, according to the neo-Hintikkan theory. But regardless of what might prompt Mary to say ‘Pierre believes that London is and is not pretty,’ her ascription would be false. After all, no temporal part of London is shared by a city in any of Pierre’s alternatives that is both pretty and not pretty. Pierre’s rationality is secure. notes 1 Saul A. Kripke, ‘A Puzzle about Belief,’ in Propositions and Attitudes, Nathan Salmon and Scott Soames, eds. (Oxford: Oxford University Press, 1988), 102–49. 2 Ibid., 123.

102 Peter Alward 3 Nathan Salmon, Frege’s Puzzle (Cambridge, MA: MIT Press, 1986). 4 If propositions are structured Russellian entities, then, arguably, one could always believe the conjunction of a pair of believed propositions under a hybrid guise constructed out of the guises under which the component propositions are believed. In this case, Kripke’s puzzle would have to be resolved by rejecting the second entailment. 5 See, e.g., Salmon, Frege’s Puzzle. 6 ‘Simple and Sophisticated “Naïve” Semantics,’ Dialogue 34 (2000): 101–22. 7 Kripke, ‘Puzzle,’ 125. The idea here is that even if we were to explicitly replace the names ‘London’ and ‘Londres’ with the identifying descriptions Pierre associates with those names, it could turn out that ‘Pierre believes that the F is pretty’ and ‘Pierre believes that the F is not pretty’ are both true. 8 S. Kripke, Naming and Necessity (Cambridge: Harvard University Press, 1972). 9 J. Hintikka, ‘Semantics for Propositional Attitudes’ in Reference and Modality, L. Linsky ed. (Oxford: Oxford University Press, 1971), chap 10. 10 One might object that in the case of people with inconsistent beliefs, all worlds are compatible with what the believer believes. In order to avoid such worries, one can retreat to talk of worlds that are complete by the believer’s lights, or something along those lines. 11 Linsky ed., Reference and Modality, 160. Strictly speaking, only ‘individuating functions’ – those functions which pick out the same individual in all of a believer’s alternatives – correspond to methods of recognizing individuals, according to Hintikka. Presumably, non-individuating functions correspond to those methods believers use to pick out individuals but which do not suffice for recognizing them. 12 Hintikka, more recently, has made claims similar to mine. See, e.g.,‘Towards a General Theory of Individuation and Identification,’ in The Logic of Epistemology and the Epistemology of Logic (Dordrecht: Kluwer, 1989). Closer examination reveals that the similarities between our views really are superficial. See my ‘Varieties of Believed-World Semantics: Hintikka, Stalnaker, and Me’ (unpublished manuscript) for more detail. 13 See my ‘A Neo-Hintikkan Theory of Attitude Ascriptions’ (unpublished manuscript) for more detail. 14 Kripke, Naming and Necessity. 15 I am equally happy to formulate things in terms of anaphoric chains as Robert Brandom does in Making It Explicit (Cambridge: Harvard University Press, 1994). 16 G. Nunberg, ‘Indexicality and Deixis,’ Linguistics and Philosophy 16 (1993): 1–43. Thanks to Anne Bezuidenhout for pointing this out to me.

Part One: Respondeo JOHN WOODS

It was unjustified optimism to propose over thirty years ago, in The Logic of Fiction,1 that a correct semantic theory of the fictional would prove to be neither complex nor hard to produce. How wrong can one be? One of the sharp virtues of Nicholas Griffin’s chapter is that it puts considerable pressure on initially attractive distinctions, creatively muddying the waters in the process. A case in point is the distinction between nonesuches, such as the present king of France, and non-entities (or non-existents), such as the solver of the case of the speckled band. Part of the appeal of the putative distinction is that we know a great deal about the latter, whereas, concerning the former, there is nothing to be known. Part of what appears to make this so is the indissoluble link between what is known about non-existent but fictional objects and what an author makes true in ways distinctive of an author’s semantic capacities. Even if we find these remarks clarifying, problems remain, as Griffin points out. One has to do with the strong ‘Meinongean’ bias built into the quantificational idioms of natural languages. In French we have, «Il y a des choses qui n’existent pas»; in German we have the same with ‘es gibt’ replacing «il y a». This makes it easy in such languages to attribute non-existence to identifiable objects. But try it in English and you get the appearance of anomaly that has completely fooled otherwise bright people: ‘There are things that don’t exist.’ The anomaly disappears once it is realized that, syntactic similarities apart, existential quantification in English does not impute existence, not even in the form ‘There exist things that don’t exist.’ This is a sentence that provides plenty of room for my distinction, or so I thought. It instantiates to the non-existence of Holmes but not of the present king of France. But why not? Griffin asks. Do we not

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have it in English that there are a lot of different nonesuches – the present king of France, the first woman pope, and so on? Do we not, therefore, have these as values of the variable of quantification in ‘There are lots of things x such that x is a nonesuch’? I think not. Saying that anything is a nonesuch is a façon de parler. ‘The present king of France is a nonesuch’ requires some paraphrasing (and semantic ascent): ‘There is nothing that “the present king of France” denotes.’ How, then, could it be, as Griffin and Hartley Slater aver (and Peter Alward and I, too, in a recent piece),2 that the present king of France – that very nonesuch – might appear in a work of fiction? How is this possible when nothing whatever is a nonesuch? The story would require us to believe that there are sentences true in fiction whose singular terms in subject position perform no semantic function whatever. Here there would be a kind of inconsistency (‘A person to whom no reference whatever is possible appeared one night at Bertrand Russell’s door, complaining of his utter irreferentiality’). But, complicated though the telling surely is, no theory of fictionality should be allowed to duck the challenge of true inconsistencies. So perhaps this one could be accommodated as well. Hartley Slater sees a way out from such entanglements by proposing an epsilon calculus as the natural logical home for a theory of fiction. A distinctive feature of such structures is that it allows sentences of the form ‘The present king of France Fs’ to be true even though nothing whatever is a king of France. There is much to admire in the epsilon calculi. They have instructive things to say about belief contexts, identity, individuation, and uniqueness. Since improper reference is saved by supplying an arbitrary referent, the underlying logic stays classical, and it is unnecessary to enter into the thickets of many-valued logic, free logic, or supervaluational semantics. But the epsilon calculus was invented by Hilbert to assist in explaining quantification over the transfinite. This should give us pause, I think. In Hilbert’s treatment there is no need of a distinction between nonesuches and non-existents. This being so, there is nothing to motivate the distinction between ‘the present king of France’ and ‘Sherlock Holmes.’ Indeed the epsilon calculi cannot preserve what (I say) is distinctive about nonesuch-terms, namely, that they are wholly and irredeemably irreferential. In Slater’s hands, ‘the present king of France’ is referential, and lots of things are true of its referent. And the main difference between it and the referent of ‘Sherlock Holmes’ is that the intuitive particularity of its referent can be secured by indexing the sentences of

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which it is true to some or other literary text. How, then, does it remain the case that ‘Sherlock Holmes’ has an arbitrary referent? There is nevertheless something strongly appealing about the indexing conventions, I have come to see, for all the reasons cited by Griffin. Another deserved victim of Griffin’s shrewd analysis is the distinction between history-constitutive sentences and fictionalizations, with which I sought to repair difficulties posed by the fictional to the logic of relations. I now concede that, even without some indexing to particular texts, this is a lost distinction, and, with it, it is far from problemfree. Griffin speculates that Routley was put onto the problem of the asymmetry of apparently symmetrical relations by me. I can confirm that this is so. It was sometime in 1971 that Routley spoke to me of my ‘Fictionality, and the Logic of Relations,’ which appeared in 1969.3 What he said was, ‘I think you have NO IDEA how serious a problem this is!’ Dale Jacquette is entirely correct in suggesting that The Logic of Fiction is a somewhat inadvertent provocation of his own contributions to modal actualism. Still, it is a modest etiology that pleases me. At the time that I was first working out what I wanted to say about fictional objects, I was also trying to get clear about what later I would call ‘semantic kinds.’ Right from the beginning, I thought it necessary to deviate from standard model theoretic structures; sets were too extensional for my purposes. Concurrently I was perplexed by the trouble created by the admixture of the identity sign and the alethic modals. Nearly everyone ‘blamed’ the modals. I thought it was the other way around; and this required that I give up on the standard model theoretic elucidation of contingency in identity contexts. The salient pieces are ‘Semantic Kinds’ and ‘Identity and Modality.’4 Neither of these pieces came close to articulating the kind of alternative to modal realism that Jacquette has worked out, and little of my heterodox inclination made its way explicitly into The Logic of Fiction. I remain tickled that it was, all the same, a provocation. Contrary to a modal realist approach to semantics, in which fiction is analysed via an antecedently developed modal logic, Jacquette proposes to reverse the arrangement. Since all possible worlds save the actual are fictions, it is appropriate to embed modal logic in an antecedently developed logic of fiction. I have recently considered a similar dependence-question. In Alward’s and my chapter in the Handbook of Philosophical Logic, we reflect on the relationship between a theory of (literary) fiction and fictionalism in the foundation of mathematics.

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Having claimed that, as usually supposed by philosophers of mathematics, the dependency runs from the former to the latter (the counterpart of Jacquette’s own position), we proposed that the dependency is best reversed (the counterpart of what Jacquette identifies as the standard position among modal realists). It is an interesting development, to say the least. It is possible that the differences between the philosophy of mathematics and the philosophy of possibility and necessity do justify these non-congruent dependencies; but it is not obvious to me that they do. Jacquette’s impressive defence of modal actualism is substantial occasion to pause and think again. We may agree that the standard approach to models also runs into difficulty in contexts other than the modal. The second of the Löwenheim-Skolem theorems is a case in point. How is it possible for firstorder theory with a non-denumerable ontology to have a countable model? Lisa Lehrer Dive proposes an attractive solution which exploits the fact that a theory with a non-denumerable ontology does not contain its own enumeration function. This leads her to suggest that what Skolem’s paradox tells us is that formal systems cannot capture mathematical reality because set-theoretic results are relative rather than absolute. It may be wondered whether this solution creates any less philosophical havoc than the paradox it was meant to subdue. Whether it does or does not – that is, whether it is an economical solution to the problem at hand – it is a suggestion that calls attention to a delicious methodological perplexity about the significance and limits of formal methods in philosophy. The problem is touched on briefly in my comments in Part Two. Suppose there are ideas – ideas such as perspective, continuity, probability, or set – in which we have a theoretical interest, yet for which conceptually satisfying articulations do not yet exist. There are philosophers and logicians (and economists!) galore for whom the appropriate procedure is to formalize the target notions. It is worth noting in passing that tendentiously minded formalizers are wont to represent their manipulations as ‘explications’ or ‘precisizations’ or ‘axiomatizations’ of their target concepts. Doubtless there are cases in which reflective and sober-minded people claim to find in such formalizations helpful elucidations. But these are precisely the cases in which there is a question about what to make of the residue, about those instantiations of the target concept that don’t find a safe harbour in the formal model. There are two main ways of thinking of this, with lots of variations in between. One is to reject the residues as conceptually incoherent or philosophically unusable. On this view, the formalization

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captures all that is clear and usable in the original idea. A second way of seeing the residue left by a formalization is as evidence that the formalization is defective or incomplete. So seen, the formalization fails to capture some philosophically tenable aspect of the original idea. Of course, a third, largely instrumentalist, response is also available, in which the theorist explains the formal model’s partial success as having captured all that is needed of the original idea for the theorist’s particular purposes. In its most basic form, the problem of a formalization’s residue is this: one party may see it as, in effect, condemned by the formalization that excludes it, whereas another party may see it as evidence of the formal model’s inadequacy. The problem is how to settle this dispute in a non-question-begging way. It is thus kin to a problem which, in Paradox and Paraconsistency,5 I called Philosophy’s Most Difficult. It is the problem of settling a dispute in which one party sees an argument as a reduction of its implying premiss-set and another party sees it as a sound demonstration of a surprising or highly counter-intuitive fact. Gödel, too, was drawn to the suggestion that no set of axioms had as yet (or, perhaps, would ever) capture the true nature of sets. He thus joins Dive in the creation of a residue problem. The residue problem is no less a problem for sets as for the truths of arithmetic, a fact of relevance to Gödel’s most celebrated result. It also leaves an arguable gap which Victor Rodych is bold enough to occupy. Even if the things the residue problem might get us to say about such entities as sets is that they are what they are independently of what our formal models say they are and irrespective of the extent to which the model is comprehensive in its coverage, that, says Rodych, is a form of Platonism which is not worth having. It is Platonism without explanatory force. Perhaps this is right; but if so, it makes it a good deal harder to adjudicate the significance of residues left by incomplete formalizations. In his chapter, Peter Alward offers a subtle solution of Kripke’s puzzle about belief. If I thought Kripke had presented us with a genuine problem, my relief in being able to take refuge in Alward’s solution would intensify accordingly. Kripke’s assumption that it cannot be rational to believe contradictions – even explicitly recognized contradictions – can surely be challenged by the likes of dialetheic logicians, who, like Graham Priest, are not noticeably irrational (except, in Priest’s case, in his affection for motorcycles). What makes Kripke’s puzzle even less puzzling is that it contains no presumption that the alleged contradiction is recognized as such by Pierre. Consider a case

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that resembles in the relevant respects the case that Kripke finds puzzling. Pierre has a flat in a street in East London. It is an awful place in an awful neighbourhood. True, it is possible that, whatever else Pierre chances to believe about his situation, Pierre believes that London is ugly. But it is hardly necessary that this is so. If Pierre were a bit more circumspect, his state of belief would be that this part of London is ugly. Suppose that Pierre has occasion to make regular visits to Mayfair. Mayfair is indeed lovely, and there is every reason to think that Pierre finds it so. Given what he believes, it follows that Pierre believes that this other part of London is lovely. Loose talk being what it is, Pierre might on his East End days say that London is ugly and say the opposite on Mayfair days. Each time he would be guilty of a kind of overstatement known as the fallacy of composition. notes 1 John Woods, The Logic of Fiction: A Philosophical Sounding of Deviant Logic (The Hague and Paris: Mouton, 1974). 2 John Woods and Peter Alward, ‘The Logic of Fiction,’ in Handbook of Philosophical Logic, 2nd rev. ed., Dov M. Gabbay and F. Guenthner, eds. (Dordrecht and Boston: Kluwer, 2004), 241–316. 3 Southern Journal of Philosophy 7 (1969): 51–63. 4 John Woods, ‘Semantic Kinds,’ Philosophia 3 (1973): 117–51; ‘Identity and Modality,’ Philosophia 5 (1975): 69–120. 5 John Woods, Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences (Cambridge: Cambridge University Press, 2003).

Part II Knowledge

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7 The Day of the Dolphins: Puzzling over Epistemic Partnership BAS C . VA N FR AA SSEN

It is a curious but profoundly important fact that general philosophical problems, no matter how traditional or venerable, lead us into sticky technical problems. In fact, the topic of this paper is a technical problem about subjective probability reasoning, but I got into it quite innocently by taking a position in philosophy of science. It was an unpopular position, so I had a lot to defend. Today – well, after many years of struggle to vanquish the foes, and much son et lumière on both sides, it is still unpopular and there are still more technical problems ... but hope springs eternal ... I will begin with a brief introduction to the general philosophical problems and then jump into subjective probability reasoning about what our future can be if we think we may have surprisingly alien new partners in the enterprise of knowledge. 1. Background: A Position in the Philosophy of Science The position I took is that full acceptance of science, even with no qualifications and no holds barred, need not involve belief that the sciences are true – that even such wholehearted acceptance requires no more belief than that they are empirically adequate. That means: what the sciences say about the observable parts of the world is true; the rest need not matter. I’m putting this very roughly, but it is enough to make you see the immediate challenge. Suppose that in accepting science I believe whatever it says about the observable. Doesn’t the line between the observable and the rest of nature shift continuously – with the invention of microscopes, spectroscopes, radio telescopes, and so on?1 What I mean by ‘observable’ here is just what is accessible to the

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unaided human senses. The word ‘observable’ is like ‘breakable’ and ‘portable.’ I would not call this building or a train locomotive breakable just because we now have instruments that can break them – nor call a battle tank portable because it can be carried using a Hercules transport plane. In the same way, the word ‘observable’ does not extend to what is purportedly detected by means of instruments. But this opens up the immediate second challenge: we humans change too, not just our technology.2 Evolution has not stopped. Who knows what we can yet grow into? A good point, and that is what I want to take up. 2. Observability Perspectival The first point I want to make is that ‘observable’ is redundantly equivalent to ‘observable by us’ – in that way too it is like ‘breakable’ and ‘portable.’ And yes, we can change. So thereby hangs a tale ... Observation and the Epistemic Community The ‘able’ in ‘observable’ does not look indexical. But it is; it refers to us – just as it does in ‘portable,’ ‘breakable,’ and ‘potable,’ though perhaps not in ‘computable,’ and certainly not in ‘trisyllable.’ Just now it does not seem to make too much difference whether we say ‘within our limitations’ or ‘within human limitations.’ But the range and nature of beings that we count as us is not fixed, either necessarily or even historically. The observable phenomena consist exactly of those things, events, and processes that are observable to us. We may very well be quite certain of who we are, and may have full beliefs about the characteristics that all and only we have in common. Suppose those common features are summed up in ‘human.’ Then we fully believe that the observable phenomena are exactly the humanly observable phenomena. But we realize that we are in evolution. We are also not so given to tribalism or species-chauvinism as to see those common features as essential. Even a modal realist could say ‘We are human, and humans are essentially X, so each of us is essentially X, but we may (could, might) in the future have beings among us who are not X.’ Epistemology and cognitive science part ways here. The cognitive scientist, in so far as s/he engages in empirical research and not merely in theorizing, studies human and animal information processing. In epistemology we must take the indexical seriously and reflect also on

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how we are to think of our own beliefs, opinions, and epistemic activity in general in the light of those contemplated futures where we and human do not coincide. The touchstone for the difference will clearly appear when we envision changes in the epistemic community, with consequent changes in the referent of ‘we,’ ‘us,’ and ‘our.’ This was noticed at once by various scientific realists who tried to exploit it in arguments against constructive empiricism. It will be instructive to diagnose the fallacies in those arguments for that will in fact lead us to genuine problems for antirealist epistemology. Smart Detectors and Bionic Persons Humans equipped with surgically implanted electronic devices, evolutionary stories of progeny who grow electron microscope eyes, or the eventual assimilation of dolphins or extraterrestrials into our community: these all are ways in which we, and our self-conception, could change. In such changes, what is observable by us also changes. Question: does that not change right now what we can give as reasonable and intelligible content to ‘observable’? Let us carefully consider the form of argument that challenges, by such illustrations, the observable/unobservable boundary on which constructive empiricism relies: (1) We could be or could become X. (2) If we were X then we could observe Y. (3) In fact, we are under certain realizable conditions, like X in all relevant respects. (4) What we could under realizable conditions observe is observable. Therefore: Y is observable. Certainly a valid argument. But what does it look like when instantiated to a particular content? Suppose we take as our example the alpha particles familiar from the earliest descriptions of cloud chambers. Let Y be alpha particles. Let X be an organism with special senses that operate like the cloud chamber and with a sensor that registers what corresponds to cloud chamber tracks. Then premiss (2) says, If we were (or were like) X, then we could observe alpha particles.

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What is the basis for this claim? It is clearly a theory which implies that those tracks in the cloud chamber are made by alpha particles. And they are indeed made by alpha particles if alpha particles are real and that theory is true. But our current acceptance of that theory – even if we in fact accept it – does not imply belief in that much. So the argument already assumes, for its polemical success, a different construal of acceptance of scientific theories, thus begging the question against constructive empiricism. Just to round out the picture: the same sort of question-begging presumption surrounds premiss (3), when given concrete content. We do not really need to appeal to AI, current electronics, or new physics, let alone molecular biology to provide the setting for the arguments. We only need the indubitable possibility of a smart detector of, for example, single electrons or single photons or single alpha particles. But that possibility is already easily within reach, if anyone cared to adapt the technology – always, however, on the assumption of the reality of those particles and truth of the relevant theory. For rather than waiting for the emergence of new kinds of smart detector, we can modify one we know already – me or you, for example. To do this we rig up a physical detector, coupled to an amplifier, which can register the impact or presence of a single electron. It emits the sound ‘Bingo’ whenever that happens. If we point an electron gun at it, designed via our current theory to emit one electron per minute, the apparatus emits the sound ‘Bingo’ at that rate too, and so forth.3 Now we detach the loudspeaker and link the output, perhaps a little less amplified, to an electrode in someone’s brain. The output change is reliably detected by him in the form of an indefinable feeling, perhaps only scarcely at the level of consciousness. Nevertheless this is sufficient for the next step – we condition him, by means of biofeedback techniques, to shout ‘Bingo’ when he gets that feeling. Now we have our smart detector of single electrons, and he was accepted as a member of us already. He can carry the apparatus strapped to his back; we can also bring about the existence of whole regiments of such smart detectors. And we can reliably evoke the shout ‘Bingo’ by coupling on an electron gun and pulling the trigger. So should we say right now that single electrons are observable? In fact, now that we have realized the crucial experiment so easily, it is not so convincing anymore. If we bracket the theory involved in such terms as ‘electron gun,’ we have simply a sequence of now (already) observable events with reliable predictions. And we realize

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actually that not only did we not need faith in AI, but we do not even need electrodes on the brain. For the relevant possibility was already there in Schrödinger’s famous remark that the emission of a single photon can sink a battle ship. All it needs is an amplifier whose output is coupled to an Exocet missile launcher. It would not take biofeedback techniques to train someone to shout ‘Bingo’ every time the apparatus sinks a battleship. And here again we have a reliable smart detector of single photon emissions in a specially arranged suitable context. (That is, the experimental arrangement requires a randomly operated off-on switch on the apparatus and suitably positioned battleship; he is trained to shout exactly if he sees a battleship sunk under those conditions. He will be very reliable even if there are a few other missiles flying around in the area.) But now of course the possible existence of potential believers who, according to our theory, are reliable single electron or photon detectors no longer looks like it could establish very much. The reason is that we have as usual produced a situation in which our predictions, even by us non-rigged-up people, are reliable and concern observable events in the present sense. That is, we can reliably predict that pulling the trigger on a macroscopic object we have classified as an electron gun in good working order will be shortly followed under the described circumstances by a shout of ‘Bingo’ from the first experimental subject. And we can reliably retrodict the position of the randomly operated switch, from the second subject’s ‘Bingo’ shouts. Changes in Our Epistemic Community The challenge in terms of humans who grow electron microscope eyes and the like takes also a second form, due to William Seager.4 This relates specifically to how we should think about our own epistemic future when we contemplate the widening of our epistemic community – the ‘us’ in ‘observable by us.’ In doing so we are contemplating the end of the equation of observable by us with humanly observable, at least in the sense in which ‘humanly’ refers to the sorts of animals that we early twenty-first-century humans are. This widening could bring into our community dolphins, extraterrestrials, or the children of Childhood’s End. Here is the challenge. Let us suppose that I now admit some positive probability for the admission – at some future date – of dolphins as persons, as bona fide members of our epistemic community. Suppose

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furthermore that I currently accept (but do not believe to be true, only believe to be empirically adequate) a science which entails that dolphins are reliable detectors of the presence of Ys. Here Ys are things that I currently classify as unobservable, since they are not detectable by us even if they are real. Add for good measure that at present we are ‘atheistic’ in this respect and believe that Ys are not real! Now it could be part of the supposition that dolphins themselves will claim evidence which refutes that present science. Let us not suppose that! Let us make it part of the story that after this widening of the community we shall still accept the theory. So at that point we will add, ‘Some of us observe Ys.’ We will add by implication that Ys are real. There is no great threat in the reflection that in the future we shall give up some beliefs we hold now and replace them by contrary beliefs. But this is a special case, and we can spell out the argument as follows: (1) The science I accept as empirically adequate entails that Ys exist. (2) The science I accept also entails that in some possible future our community will include members who observe Ys. Therefore: (3) I should now believe that Ys exist. Seager offers the following as an analogy to our worrying situation with respect to dolphins in the above case: (1*) We know that if we encountered rational creatures of type X who sincerely informed us that the earth would explode tomorrow, we would believe that the earth would explode tomorrow. (2*) We know that rational creatures of type X are possible. (3*) We know that if we were to encounter rational creatures of type X, they would in fact sincerely inform us that the earth will explode tomorrow. Therefore: We should now believe that the earth will explode tomorrow. The crucial supposition behind (1*) is that we accept a theory which entails that certain observable events (communications from the Xs) are reliable indicators of earth explosions (also observable events) to come. Thus we are appealing to the empirical adequacy of our background theory only.

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We have here in fact a good analogy for the dolphins, except for the current observability of the explosion: the Ys were not currently classified as observable. That introduces a disanalogy also for (2) and (2*): in the case of (2*), our acceptance of the background theory as empirically adequate leads us to a positive probability for the existence of reliable observers who can predict earth explosions. To display the equivocation in the original argument we need only emphasize the indexical character of ‘observable.’ In the envisaged scenario we see ourselves as truthfully saying at a certain later time, in a certain possible future: (A) Some of us are observing Ys; thus, Ys are observable. But it would be a mistake to infer anything like: (B) Ys will be observable at that time. For in (A) we have a sentence uttered truthfully at a later time, while (B) would be our statement now. The fallacy involved is the same as in When we are in Pisa next July we will truthfully say ‘The Leaning Tower is here.’ Therefore: The Leaning Tower will be here in July. As I pointed out to begin, ‘observable’ does not look indexical, but it is. That there are these hidden, subliminal indexical aspects to some of our discourse is precisely the lesson we learned from Putnam’s Paradox. However, Seager has in effect pointed us to a deeper problem that will provide us with a greater challenge. The New Riddle of Prevision The problem Seager posed is not so easily dismissed, for it carries a strong intuitive sense of puzzlement. What if we do experience such a change in what counts as us, as our epistemic community, and thereby see a profound sea-change in our relation to nature as a whole? Just how are we to conceive of ourselves as epistemic continuants, once we contemplate such radical changes in the range of epistemically accessible aspects of nature in our future?

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The question will become crucial at a particular point in the story: the point where we realize that the ceremony of admitting the dolphins to our community is about to be performed. Suppose it has not been performed yet, but we already know it will be soon – at that point we must surely change our mind about the Ys already. Let’s go back a bit further: the ceremony has not been decided upon, and admission to membership is still being debated but has become very likely. At that point should we not at least let go of our out-and-out agnosticism or ‘atheism’ and admit that it is very likely that our science is also right about some unobservable parts of nature? Now, going back still further to our present position, when all we have is the theoretical possibility, should we not think that such future likelihood, if it is to arrive, will have grown from a small likelihood in our minds already, or rather, a small likelihood that should already be there for us? In which case already now, before we have seriously encountered those dolphins or whatever as yet, we should not be completely agnostic about our science’s truth. I could continue to block this rhetoric by insisting that there is a gulf of principle between possibility and positive probability. But the gulf cannot be one uncrossable by a belief or rational positive probability that we will in fact admit such creatures. We need to look more deeply into general epistemology. How should we contemplate the possibilities of our own future opinions subject to such changes? As guiding analogy of a much more general sort, I want to ask about what I shall call epistemic marriage. Epistemic Marriage Consider the following conception of marriage. After the wedding, the two constitute a couple, and there is no longer personal but only communal opinion for them on all subjects to which both have equal access, including access through the partner’s reports of private experience (admitted on equal footing with memory of one’s own private experience). This is similar to the dolphins problem, except that the envisaged union is more intimate (and is clearly a matter of decision, not easily seen as forced by opinions about what things, people, etc. are in fact like). The question this raises is whether I can beforehand envisage such a change while maintaining that I shall always change my opinions and beliefs rationally. Perhaps the constraints of rationality on opinion do not affect much else. For example, they may give me no problem at all

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if I foresee perfect epistemic domination by myself of my partner – that she or he will submit himself or herself epistemically to my dominion. But then the question is whether that transition will count as rational opinion management for my partner. One doesn’t suppose that general epistemology is a respecter of persons. Consider then my present status before entering into such a union; suppose I consider it part of one of my possible futures. There are various possibilities for the type of person I shall marry, but some of them may have drastically different opinions from my own. Should I therefore expect that the couple of which I shall become part through fusion of this sort will have opinions drastically different from my own? Does that not mean that my present opinions must be accordingly ‘diluted’ – in the worst case – in which I foresee having to give up probability one or zero especially, it has to be given up right now? Example: I give probability zero to reincarnation. Do I now face the dilemma of either giving it a small positive probability or else rejecting as certainly false the supposition that I shall marry a believer in reincarnation? To use another example, suppose I am considering as spouse someone who claims to be psychic and I presumably do not believe in psychic powers. I don’t think I shall now say that I shall not marry someone whose beliefs disagree in some way with mine. I can also foresee that after the marriage, or at that time, or as part of the decision to marry, I shall acquire these contrary beliefs. But at the same time I need not hold that I endorse this as the way to acquire reliable opinions. In other words, I foresee a break of epistemic integrity. This is so even if (a) I am presently agnostic about psychic powers and (b) I believe that no phenomena which are observable to me will disprove the reality of psychic powers. This is a disturbing reflection. My epistemic integrity is compromised when I allow that I shall possibly go along with something like this. Once I realize that the married couple will legally inherit all my present contracts and obligations, it appears that I am ready to incur a certain loss for the me-couple. It is a bit like what Americans call marriage tax, only worse.5 Clearly we have reached a fundamental difficulty in our epistemology, and we need to go back to fundamentals. We have to examine the terms of our discussion – opinion, belief, and the constraints of rationality on how we manage them – on a more fundamental level.

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3. Self-Prevision: Its Logical Laws, Its Subjective Sources So the topic we have to investigate is what might be called our reflective opinion: this includes both how we view our current opinions and how we envisage what they may be in the future. Hintikka’s Problem As a start I want to remind you of a mistake made in those heady days when modal logic seemed to provide a royal road to philosophical enlightenment. There was a logic of everything, so of course there was a logic of belief. The seminal text was Jaakko Hintikka’s Knowledge and Belief.6 What is the logic of belief? That means: what inferences about belief are valid? Consider X thinks that A; therefore X thinks that B. This relationship of entailment can presumably be captured in a logical system, the logic of belief. Unfortunately that entailment relation was trivialized by what we might call ‘the problem of the moron.’ Whatever sentences A and B are, no matter how closely logically related, there was a conceivable person of sufficiently low logical acumen who wouldn’t get it. So we need a new approach to the logic of belief and opinion.7 Logic in the First Person On purely logical grounds we can see that if someone is of the opinion that A, that may bring with it a commitment to or responsibility for B, on pain of incoherence. The paradigm example here is Moore’s Paradox. If I were to say It is snowing and I do not think that it is snowing then I would display an incoherence in my state of opinion. I cannot say this, but not because it could not be true – I cannot say it on pain of incoherence. We can describe this fact about my opinion in terms of an inferential relationship: It is snowing >> I think that it is snowing I think that it is snowing >> It is snowing

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But both in its meaning and in the logical laws obeyed, this is quite different from the standard logic, for we certainly would not infer from the above that IF I think that it is snowing THEN it is snowing is a valid sentence.8 The first-person character of these sentences is of course crucial to this relationship. There is nothing wrong with ‘It is snowing and Paul does not think that it is snowing.’ Indeed, there is nothing wrong with ‘It WAS snowing and I DID not think that it WAS snowing’ or ‘There will be times when it WILL BE snowing and I SHALL NOT think that it is snowing.’ The reason would appear to be that ‘think’ in the firstperson present tense has the linguistic function of expressing my opinion. The difference between stating what our emotions, values, and intentions are on the one hand and expressing them on the other is of course familiar. That contrast is crucial also for opinion. Also, we can express our opinion only in indexical, self-attributing fashion. Opinion is perspectival. There is a possible ambiguity here. In a therapy session a person may come to the realization of a surprising autobiographical fact: he discovers what his opinion really is. In that context, even ‘I think’ may play the other, simple fact-stating role. So I propose that for our present inquiry it is best to make a syntactic distinction that we do not see in English, and I will italicize the words when they play the expressive role and use boldface for ‘think’ in the fact-stating role: It is snowing >> I think that it is snowing I think that it is snowing >> It is snowing I think that it is snowing >> I think that I think it is snowing are all valid. But I think that it is snowing >> I think that I think it is snowing makes no sense; the expressive use does not iterate (the fact-stating use can). However we should note also that the ‘I think’ accompanies every thought, as Kant said; and for this very reason we can usually let it go without saying. So in fact in ordinary discourse it is often left out.

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Subjective Probability We need now to complicate the picture just a bit by allowing for more nuances of opinion. Mostly I don’t just believe that this or that will happen – it only seems more or less likely to me. In our examples, we already allowed for this. Now let’s make it official by replacing that ubiquitous I think with It seems ... likely to me that, with the blank filled in with some degree. Do not think numbers right away: our opinion is typically too vague for that, though we have natural ways of being more precise. Compare: It seems likely to me that it will snow. It seems very [extremely] likely to me that it will snow. It seems twice as likely to me that it will snow than that it will rain. It seems twice as likely to me as not that it will snow. The last one is numerically precise; it translates directly into My subjective probability that it will snow is 2/3. Symbolically: P(it will snow) = 2/3 To accommodate the vaguer examples, we must allow for intervals like [0.1, 0.4] to replace the number. Before going on I want to give us an intuitive grasp on how we do actually reason in this format. My full beliefs together give me one picture of the world, not a very complete one obviously – but that is where I say ‘That is what things are like!’ About all the alternatives left open by these full beliefs, though, I am not so definite: that is where I say those sorts of qualified things illustrated above. Now, one way to keep this scheme before our eyes is by means of what I call the Muddy Venn Diagram. Just as in elementary logic class, we depict the space of all possibilities by means of a Venn Diagram: B A

C

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But then we smear and heap mud on it, to indicate proportionally how much credence we give to these various possibilities: B

A x

C

Suppose that the total is 1 kg of mud; then if A has 1/3 kg on it, that indicates that my probability for A being the case is 1/3. This is not just a mnemonic: using this we can immediately see what the logical rules for consistency must be. For example, just thinking about the amount of mud on the various parts, we can see that: P(A and B) + P(A or B) = P(A) + P(B) which is the Axiom of Additivity for probabilities. Notice also that to teach ourselves to reason with vague probabilities we should just learn how to do it with precise ones. When children learn how to deal with such judgments as John is about 5’9” and Julia about 6’2” they do not need to study a special calculus of approximate numbers – their school arithmetic is all they need to understand that Julia is taller than John. Similarly with our vague degrees of belief. Opinions can be stated as well as expressed, of course. We can also make state attributions saying that so and so has some such epistemic attitude, to describe his or her opinion (in part). To revamp some of our previous examples: It seems likely to me that it seems unlikely to Jeremy that it will snow. We must allow for both precise and vague probability here. Here is an example with several of the above features: P(pJeremy(it will snow) = [.5, .75]) = .8 The initial ‘P’ needs no subscript for it expresses always the speaker’s current opinion. The boldface is again to be used for biographical and autobiographical statements of fact.

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To continue our discussion of reflective opinion, we should now ask about expressions of the form P(pME(it will snow) = ...) = —. Until I have spent some time with a therapist I may not be too sure of what I think, so this makes sense. But in view of the sorts of logical relations we saw above, what are the constraints of coherence for some thing like that? Let’s ask concretely to what extent I could coherently express some lack of confidence in my own opinion. What of the possibility that some proposition A that seems unlikely to me is in fact true? How likely does that seem to me? You understand that we are in Moore Paradox land here. Coherence requires precisely that if we say something of this form P(It will snow and pME(it will snow) = x) = y then the number y must be no greater than x. So here too we have a significant logical point although solely about what expressions of opinion will and will not display an incoherence in that opinion. Rational Change in View I’ll stick with the fiction of sharp, numerical probabilities for now, and leave the more realistic (hence more complicated) presentation for another time. The simplest case of a change in opinion is the one which some newly acquired bit of belief triggers modus ponens. For example, I come into the kitchen and I see small black droppings and note bite marks on the cheese. If I immediately conclude that we have a mouse, some people think I have made an inference to the best explanation. But I had no need of any such move or maneuver: I already thought all along that if there are such changes in the kitchen scene then it was visited by a mouse. That is just modus ponens, you’ll note. If I am not quite as dogmatic in my beliefs, then this evidence will not take me that far, but I will just go to a pretty high probability that there are mice. We can again make this visually intuitive with the Muddy Venn Diagram. The space of possibilities before I came down the stairs was divided into two parts: the part that has my kitchen with small black droppings and bite

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marks on the cheese and the part that does not. When I see the evidence I simply wipe off all mud from the ruled-out part: B

A x

C

This is called conditionalization, the probability analogue of modus ponens. But even this simple logical updating is already a bit more complicated with degrees of belief. First of all, I may only wish to raise my degree of belief that there was a mouse, not raise it to certainty. Second, if I raise that, I cannot leave all the rest alone, for that will affect my views on how our house relates to the local fauna in general. Enter here the probability calculus: it is the logic that spells out what coherence requires on my opinion in general. It even gives us at least the resources for describing what purely logical updating in response to new evidence can be like. Enter here also a major epistemological rift. The old-fashioned idea that we must proportion our belief directly to the evidence – as propounded by Locke and oft repeated since – has as its descendant orthodox Bayesian epistemology. This position implies: Purely logical updating in response to new evidence – i.e., conditionalization – is the sole rational form of changes of opinion. You can see at once that this position will rule out epistemic marriage in all but the trivial cases. There are more liberal alternatives. The Bayesian will say that if you are willing to depart from pure liberal updating, then anything goes. That is not so; we can again insist on coherence constraints, at least in our present opinion about the future opinions we may come to. Specifically, I consider the following to be mandatory for present coherence:9 General Reflection Principle: My current opinion about event E

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must lie in the range spanned by the possible opinions I may come to have about E at later time t, as far as my present opinion is concerned.10 As an example, think of what would violate this principle. You are going to buy a lottery ticket, and I ask you ‘if the number ends in a 0, will you think that you are likely to win something?’ and you say NO. The same for my questions about 9, 8, ..., 1. After all that you still say ‘But I am feeling lucky! I will buy the ticket because I think I am likely to win this time!’ Well, that violates the principle. One important consequence of this principle occurs immediately if we apply it to numerically precise subjective probability. That is the ‘ordinary’ or ‘simple’ Reflection Principle: P(It will snow, given that pME[t](it will snow) = x) = x. For example: On the supposition that an hour from now it will actually seem K times as likely to me as not that it will snow, it does seem K times as likely as not to me that it will snow later today. where the autobiographical attribution now concerns my opinion at some later time t, and t here is a relative time (such as ‘tomorrow’ or ‘later today’ or ‘one hour from now’). How can this be deduced? We have to read the General Principle as applying not only to probability but to expectation in general. By this I mean that opinion has, in this context, the following as its most general form: My expectation of my salary increase is 4 per cent, because it is equally likely that I will be evaluated as deserving a bonus (in which case the increase will be 6 per cent) or as deserving no bonus (in which case it is 2 per cent) My salary increase is a quantity which can take various possible values, and my expectation of it is my subjective average for the possible scenarios I can envisage. Then the trick is to treat one’s own future opinion as such a quantity. Applying the General Reflection Principle to that quantity will then yield the ‘simple’ Reflection Principle. But this ‘simple’ Principle has looked very suspicious to many people (even though the orthodox Bayesian clearly satisfies it and has not for that reason looked suspicious to anyone!) so we should make sure it does not say too much.

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This principle does not make it impossible to express either confidence or lack of confidence in my future opinions, but not in one direction or another. I may expect that my future probability for something will be out by some factor, either too low or too high, but not be sure that it will be too low, nor be sure that it will be too high. The principle does entail a more general form also of the nuanced Moore Paradox point. I can certainly say:11 P(It will snow and pME[t](it will snow) = x) = y For example: It seems N times as likely as not to me that the following are both true: It will snow later today and it actually seems K times as likely to me as not that it will snow. But y must be no greater than x! If y is greater than x (or N than K) then the reflection principle is violated. One uncompromising limit, however: if I am now sure that I will have a certain opinion in the future, then I must have it now – on pain of present incoherence. Wesley Salmon mentioned someone’s comment on the principles of Scientology: ‘I am an empirical scientist so I won’t say they are false before the evidence is in. But when it is I will!’ What does this anecdote illustrate? If this scientist’s opinion is coherent, then of course he has signalled that he already disbelieves those principles. And he does so because he already fully believes that the evidence will go one way and not another. I realize that this principle makes no sense if we simply see it as concerning factual prediction. There may come to be serious physiological and psychological deficiencies in my future. But if I express an opinion that violates the General Reflection Principle then I display a deficiency either in my current opinion or else in the way I shall go about managing my opinion in the future. As analogy, imagine me saying: ‘There will be arithmetical mistakes in my budgeting for next year.’ The problem is not that this sentence cannot be true – it can – but that I am expressing something that violates norms I should be expected to uphold. We want to reply ‘So do something about it!’ – and that is just what the General Reflection Principle signifies. Finally, how much weaker is this principle than the orthodox Bayesians’ insistence that the right rule is always to conditionalize? The two coincide precisely when the person is sure that s/he can canvass all the possible outcomes and say what his/her posterior probability would be in each of those cases.12 So the Reflection Principle has all the bite there is to be had for exactly those people who cannot foresee how

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they will make up their minds under all possible circumstances. Not exactly an implausibly conjured class! 4. Epistemic Marriage Revisited13 Recall our concept of epistemic marriage, which could include assimilation of dolphins or extraterrestrials into our epistemic community as full and equal members. In marriage one hopes for a certain degree of symmetry and equality as well as harmony. This is also what various studies have looked for in the pooling of opinions and preferences. We can begin modestly by suggesting that any views already held in common should be preserved in forming the views of the unit. Such conditions are called Pareto conditions and can take various forms. Let the partners be X and Y, forming the unit U: [P1] If A and B seem equally likely to both X and Y, then A and B are to seem equally likely to U. We arrive at conditions [P2] and [P3] by replacing ‘A and B seem equally likely’ by ‘A seems at least as likely as B’ and ‘A seems more likely than B’ respectively, adjusting mutatis mutandis. A stronger condition is this: [P4] If A seems at least as likely as B to either X or Y, and seems more likely to the other, then A is to seem more likely than B to U. These conditions can all be satisfied, provided X and Y have coherent states of opinion to begin. In fact it is quite easy to see how: they simply settle for a degree of belief somewhere between the initial two. To do this systematically so that the result will be coherent, they choose a linear combination: If they have sharp probability functions p and p9 then U can be given any mixture of these; that is, any combination q = ap + (1 – a) p9 . If, in addition, as in any good marriage, we require symmetry, the proper combination would be half and half: (½)p + (½)p9 . So it can be done.14 But will it count as rational? We have here precisely one of those episodes in which we feel ourselves called to account for the change in view.

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Imagine that I marry someone who believes in reincarnation, and we settle on the common opinion that reincarnation is precisely as likely to be real as not real. If she was initially [almost] certain about it, and I [almost] certain of the opposite, this would be halfway between. But did I conditionalize on new evidence? Something new has happened – the marriage. But did I have the prior opinion, Reincarnation is as likely as not to be real, on the supposition that I marry someone who believes in it, which would be required for conditionalizing to lead me to the new opinion? Most likely not.15 So after all this we seem to have only succeeded in making our problem clear. While there seems to be a way to form the epistemic union, there seems to be no way to rationalize the consequent individual change of opinion along traditional lines. But it is not irrational to be so struck by the appearance of rival opinions to one’s own. Indeed it seems rational to accept the appearance of such a rival as requiring an attempt to stand back somewhat from one’s own point of view. The proper response would seem to be something like this: stand back, bracket the differences between the two, and then let the resulting ‘neutral’ opinion evolve to a less neutral one in response to the evidence. Mixing may be an attempt to do something of that sort, but it miscarries for it actually results in a sharply discontinuous change of opinion – prejudging what the outcome should be. Nevertheless, there must logically be many different ways to do this. Are there any constraints on this? The Reflection Principle Applied Even just given the General Reflection Principle, mixing is also not an acceptable prospect. For think of any situation once I have decided on this marriage and the ceremony is about to begin. At this point I am [almost] certain that reincarnation is not real, yet certain that very soon now I shall instead judge it to be as likely as not. That violates reflection. But we can imagine slightly different situations in which reflection is not violated. Imagine that there are two partners I may marry: one strongly believes in reincarnation and the other strongly disbelieves. I foresee that if I marry the one, my degree of belief in reincarnation will go down, and if I marry the other it will go up. Since I do not privilege

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one direction over another I may satisfy reflection. This works only as long as I remain suspended. There is also another way in which I may satisfy it even if I can marry only one of these two. Suppose the epistemic unit is formed the moment it is decided upon – and suppose the decision will be unforeseen and unsettled until the very last moment. Then again there may be no incoherence. This solution we could call that of ‘epistemic elopement.’ Reflection may be unviolated in the case of sudden, unpredictable epistemic elopement. To Envaguen However, the fact remains that foreseeing that one’s opinion will change to a mixture of one’s current opinion with that of another is a clear violation of reflection. Prospects for epistemic marriage seem dim. But in fact there is a solution. It won’t help those who insist on conditionalization but will satisfy the Pareto conditions and reflection. The solution for the partners is not to settle on a specific spot in between, but to envaguen (to make their opinion vaguer).16 This is easiest to illustrate if they begin with sharp subjective probabilities. Suppose that I and my partner have probabilities 0.01 and 0.99 for reincarnation. Rather than settle on 0.5, we agree that The probability of reincarnation is no less than 0.01 and no more than 0.99 shall encapsulate our entire assessment of how likely reincarnation is. (If X and Y have probability functions p and p9 then U will have the function P: P(A) = [p(A), p9 (A)].) It can, and should, be part of their commitment that they will have a common epistemic policy as well. Both can then hope that as they follow that policy to manage, amend, and update their common opinion, it will converge on the prior opinion s/he brought to the marriage. How will this affect the dolphins problem? Before union we do not think that dolphins observe Ys when we are still agnostic about whether Ys are real at all. After the union, our common opinion will be at least as vague on the matter as either of us was. Together we will go over the evidence, once we are truly both contained in the ‘us’ of ‘observable to us.’ What will happen? We can’t say without violating reflection. 5. Conclusion Let me quickly recapitulate and draw a moral. Real anti-realism must be a position that can only be expressed in the first person (preferably

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the first-person plural). But that will be no more than an empty sound if we don’t then also exploit that to change our understanding of traditional philosophical problems. It would be a great boon for epistemology if it got itself definitively out of the ‘X knows that p’ mess as well as the skepticism syndrome and all other such sado-masochistic dead-horse entanglements. But there are two ways out, one illusory and the other fruitful. The illusory one is what Quine called naturalized epistemology. Certainly, philosophers should study scientific models of information processing, especially in physics and computer science. But these models represent only the physical correlate of the epistemic process. If we simply transpose them to the human case we are forgetting that the basic philosophical questions apply to the sciences as well. To be a good way out of the past the way out needs to do justice to the past and to recapture what was valuable in it even while rejecting it. So, in going back to the topic of observability I wanted not only to solve an outstanding puzzle but also to illustrate the good way out. That way is to ask what a philosophical problem looks like once we really put ourselves back into the picture. You may not immediately have appreciated this when I brought in subjective probability. I won’t blame you if you thought ‘Been there, seen that, had enough of it!’ For this subject was another one treated with great technical virtuosity together with such a lack of critical concern with traditional philosophical issues that I cannot blame you. There has been a sort of subjective probability slum in philosophy, and its inhabitants, me included, have not convinced many other philosophers that what happens there is anything more than technical selfindulgence. But I think this will change if subjective probability is put in the first person and its problems recast at a fundamental philosophical level. For then it will become clear that we have there, however imperfectly still, a way of representing opinion that shows up the naiveté and oversimplification inherent in much of traditional epistemology. I’ve meant to comment on the day of the dolphins as only one example of how a philosophical question may be transformed when we switch in descriptive epistemology from the simple trichotomy of belief/disbelief/neutrality to subjective probability as our framework. I submit that there will be a similar transformation of other philosophical questions if approached in this way, creating in each case a new array of problems and puzzles to be addressed, solved, dissolved, or shown up as further illusions of reason.

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notes I happily dedicate this essay to my friend John Woods, who has, ever since our Toronto days together, inspired me with his sustained inquiry into the mysteries of reason, both formal and informal. 1 See Ian Hacking, ‘Do We See through a Microscope?’ in Images of Science: Essays on Realism and Empiricism, with a Reply by Bas C. van Fraassen, Paul M. Churchland and Clifford A. Hooker, eds. (Chicago: University of Chicago Press, 1985), 132–52, and my reply in the same volume. 2 See especially Paul Churchland, ‘The Ontological Status of Observables: In Praise of the Superempirical Virtues,’ in Images of Science, Churchland and Hooker, eds., 35–47. 3 A TV or computer monitor basically consists of an electron gun at one end and a sensitive screen on the other. The computer can read the screen, respond if a particular spot lights up, and emit a sound. Not exactly advanced technology today, except for the degree of sensitivity we are imagining here. 4 William Seager, ‘Scientific Anti-Realism and the Epistemic Community,’ in PSA 1988, vol. 1: Proceedings of the 1988 Meeting of the Philosophy of Science Association, A. Fine and J. Leplin, eds. (Philosophy of Science Association, 1988), 181–7. This was part of a symposium on Realism at the Philosophy of Science Association Biannual Conference of 1988, in which I acted as commentator. 5 When two people with incomes marry, their joint income can go into a higher bracket, with a higher taxation rate. 6 Jaakko Hintikka, Knowledge and Belief: An Introduction to the Logic of the Two Notions (Ithaca, NY: Cornell University Press, 1962). 7 The approach outlined here is that of chap. 7 of my Laws and Symmetry (Oxford: Oxford University Press, 1989). 8 It is to be remarked that the same applies to It is true that if the language has truth-value. 9 See my ‘Belief and the Problem of Ulysses and the Sirens,’ Philosophical Studies 77 (1995): 7–37. The less general Reflection Principle, also noted below, I introduced in ‘Belief and the Will,’ Journal of Philosophy 81 (1984): 235–56. 10 ‘Opinion’ here covers both probability and expectation. Semantic and settheoretic paradoxes threaten if such a principle is left with the range of applicability unrestricted. 11 Note that unless x = 1, I cannot conditionalize on the statement [It will

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12 13

14

15

16

snow and p(it will snow) = x], with p indexed to myself now; for if I gave it probability 1 then I would be in violation of the Reflection Principle. But the content of that statement, equally expressed by some eternal sentence, of course, is a proposition which I could give probability 1 at some other time. See my ‘Conditionalization, a New Argument For,’ Topoi 18 (1999): 93–6. In this section I implicitly refer to two papers: Teddy Seidenfeld and Joseph Kadane, ‘On the Shared Preferences of Two Bayesian Decision Makers,’ Journal of Philosophy 86 (1989): 225–44, and Philippe Mongin, ‘Consistent Bayesian Aggregation,’ Journal of Economic Theory 66 (1995): 313–51. More stringent Pareto conditions are not as easy to satisfy; and if preferences are to be balanced as well as probabilities, we run into unsolvable problems. See the papers cited in the preceding note. Moreover, even if we leave values and preferences out of account, there is a problem about preserving agreed-on correlations, due to Simpson's paradox; see e.g., my Laws and Symmetry, 204–5. If we made such prior opinions a requirement for rational epistemic union, the dolphin problem would not be problematic either. For then we would not accept them unless already beforehand we had concluded that whatever they called observable was in fact observable. That would mean: ‘What they say is observable to them is observable to us’ pronounced at the earlier time when ‘us’ still excludes them. Vague probability is itself a topic with much technical literature and remaining problems. See, for example Richard Jeffrey, ‘Bayesianism with a Human Face,’ in Testing Scientific Theories, J. Earman, ed. (Minneapolis: University of Minnesota Press, 1984), 133–56; my ‘Figures in a Probability Landscape,’ in Truth or Consequences, M. Dunn and A. Gupta eds. (Dordrecht: Kluwer, 1990), 345–56; and Joseph Y. Halpern and Riccardo Pucella, ‘A Logic for Reasoning about Upper Probabilities,’ Proceedings of the Seventeenth Conference on Uncertainty in AI (forthcoming). Available online at: http://www.cs.cornell.edu/home/halpern/papers/up.pdf.

8 Cognitive Yearning and Fugitive Truth JOHN WOODS

Who is this that darkeneth counsel with knowledge? ... Where wast thou when I laid the foundation of the earth? Book of Job

We simply lack any organ for knowledge, for ‘truth’ ... Nietzsche, The Gay Science

No lesson seems to be so deeply inculcated by the experience of life as that you never should trust experts. If you believe the doctors, nothing is wholesome; if you believe the theologians, nothing is innocent; if you believe the soldiers, nothing is safe. They all require to have their strong wine diluted by a very large admixture of insipid common sense. Lord Salisbury

1. Introductory Remark The proximate cause of this essay is an account of plausibility to be found in various writings of Nicholas Rescher.1 Rescher’s plausibility logic possesses considerable interest in its own right, but its more immediate appeal for me derives from work that Dov Gabbay and I have been doing on the logic of abduction.2 Of course, there are lots of historically important abductions in which hypotheses are introduced on grounds other than their plausibility, indeed despite the total lack of it. But there are also cases galore in which the plausibility of a hypothesis plays a central role in an abducer’s reasoning. So I think that we might say a logic of abduction should seek to subsume a logic of plausibility.

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One of the more interesting features of Rescher’s plausibility logic is a certain view of knowledge that is presupposed by it. On this view, knowledge arises from the transmission of information from an authoritative source. Quite apart from the logics of plausibility and abduction, this is an interesting approach to epistemology, and I want to tarry with it a while here as a kind of stalking horse for an issue that will occupy me rather more centrally, concerning which a brief word now. I want to begin with a claim of no originality, a claim that is widely taken as obvious. It is the claim that on any account in which truth is a condition of knowledge, the distinction between reasonably believing A to be true and A’s being true, and the distinction between thinking that one knows that A and knowing that A, are distinctions that are phenomenologically inapparent to the would-be knower on the ground in the here-and-now. I shall call this the fugitivity thesis with regard to truth and knowledge. All this is old hat, of course; we have known it long since. So here, too, past is prologue. For what I want to do in due course is to ask and answer the following question: ‘Why hasn’t our widespread and confident acceptance of what I am calling the fugitivity thesis led us to modify our standard epistemic practices? Why, in plain English, do we persist in making knowledge claims and truth claims?’ 2. Epistemology There is a strong tradition in the theory of knowledge according to which a small part of what we know, we know by direct apprehension of our interior states. This is sometimes called infallible knowledge. The objects of this knowledge are said to be mental entities such as sensedata and basic or primitive concepts, or the propositional contents of certain kinds of complexes of primitive concepts. The rest of what we know are the fallible products of our own efforts, which can best be seen as inferences from these interior states (again, sense-data, concepts, etc.). Later on I shall propose a certain thesis about knowledge. For now it suffices to say that, with some misgivings, I do not intend it to apply to infallible knowledge, that is, to unmediated knowledge, such as may be, of our inner states. This is the contrast class for the kind of knowledge to which I intend my thesis to apply. I will say in passing, and without further ado, that I regard this contrast class as small (if not null) and philosophically vexed. But this is not my issue here. The vaunted distinction between empiricism and rationalism cuts across this traditional characterization in a particular way. Empiricists

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restrict the inferences that result in knowledge to those from sensedata and, in most variations, certain of the propositional contents of certain complexes of primitive concepts. For rationalists, however, all knowledge requires the presence of an irreducible non-sensory component. A further feature of this traditional approach is that except for the interior states from which knowledge arises, knowledge is, at least in principle, a solo affair. What this means is that at a minimum human knowledge attains its highest grade to the extent that it is achieved unaidedly. Opposed to the solo conception of knowledge is what we might call the oracular conception of knowledge; in doing so, I appropriate the attractive figure of the oracle from Jaakko Hintikka’s writings on interrogative logic.3 On this view, some of what we know, we know not by inference from sense-data and not by inference from extrasensory concepts, but rather on the basis of the say-so of others. When conditions exist that dignify as knowledge what is accepted on the say-so of others, we will say that the source of this knowledge is an oracle. Thus OK: S knows that A when S believes that A on the basis of information that has been transmitted to S by an oracle. Oracles are epistemically authoritative sources of information. When information is transmitted by an oracle to another party, the authoritativeness of its source endows its receipt with epistemic significance. The second party now knows what the source has disclosed to him. Since knowledge arises out of an interaction with an oracle, it has a dialogical rather than a solo nature. A principal difference between them is that solo knowledge is taken to be knowledge achievable in principle solely on the basis of one’s own cognitive devices and such stimulations as the world chances to produce, without the involvement of any other cognitive agent or information-producing device (such as a wristwatch or a thermometer). Dialogical knowledge is the reverse of this. It is knowledge that could not, even in principle, be achieved solo. For this to be an interesting contrast, it is necessary to grant some latitude to the idea of what is possible ‘in principle.’ But we shouldn’t want to give it a reach that would make it the case that what for a knower is a solo possibility in principle is a possibility that holds for beings that bear no plausible approximation to how beings like us actually are. In particular, we don’t want to conflate what might be the

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case in principle with what is logically possible. This is important; it means that for beings like us, some of what we know is irreducibly non-solo. So part of what is interesting about oracular epistemology is that it imbibes this idea that some of what we know is irreducibly nonsolo. In its purest form, an oracle is another being, whether God, the advice-giver at Delphi, a human being, or artifact, whose circumstances qualify Him, him, or it as an authoritative source. Testimony presented by experts in a criminal or civil trial is one good and commonplace example of this authoritativeness, and it captures nicely part of what Aristotle meant by endoxa, which are beliefs held by all, or by the many, or by the wise. As I read him, Rescher’s plausibility logic is developed in the context of what we could call an oracular epistemology or, for short, an Omodel of knowledge. The O-model presents itself in various forms, two of which could fairly be called ‘extreme.’ One might hold that all knowledge whatever is subject to the condition that, in some rather literal way, it arises from an authoritative source or oracle. This would mean, among other things, that perceptual knowledge should be thought of as knowledge vouched for by the testimony of our senses, that non-empirical knowledge should be understood as that which is sanctioned by the deliverances of reason, and that nature herself is her own oracle, the book of nature, and so on. A second way in which the Omodel could exhibit a kind of extremity is by giving up on truth as a condition of knowledge. This is the form in which (although it is not Rescher’s way) I shall here examine the question of oracular epistemology. I want, however, to make it as clear as I can at the outset that, although I shall be investigating certain features of this knowledgewithout-truth aspect of oracular epistemology, it is not a feature of what might be called mainstream oracularism. So I should not want to leave the impression that oracular epistemology as such is my central interest. My project rather is the fugitivity thesis, which is occasion to investigate effects of the presence and absence of truth in various accounts of knowledge. Oracles need not be individuals; they can also be collective or institutional agents. Here are two examples, each corresponding to the two other things that Aristotle means by endoxa. Corresponding to Aristotle’s category of what is believed by all is the common knowledge possessed by the whole population (e.g., that water is wet). Corresponding to what is believed by the many is popular knowledge in a community

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or population (e.g., that water is H2O). The authoritative source in the first instance is the whole population and, in the second, is some large part of it. A further example of an oracle is someone (or some entity) that is in a special position to know. So, you know that you have a headache because you are positioned to know it just by having it. Or, to vary the example, you know tomorrow’s weather forecast because you have read this evening’s paper and I haven’t. Common and popular knowledge can be defined as follows: CK: Proposition A is common knowledge in a population P if and only if for every S and S9 in P, 1. S believes that A; 2. S believes that S9 believes that A; 3. S believes that S9 believes that S believes that S9 believes that A.4 PK: Proposition A is popular knowledge in a population P if and only if A is a belief widely held in that population; hence is common knowledge in a subpopulation of it. Now why should we regard these sources of belief as oracular, that is, as authoritative? One not unattractive answer is this inference to the best explanation: IBE: The best explanation of the commonness of common knowledge and of the popularity of popular belief is that at the time of our subscription to them they are, defeasibly, beliefs that it is reasonable for us to hold. So, under those conditions, they are reasonable for us to hold. 3. Truth As we have formulated them here, neither OK, CK, nor PK assigns a role to truth. This is not to say that truth has no role; OK, CK, and PK might be understood as having left its function tacit. Whether they do or not, it certainly cannot be said that truth lacks all presence here. Here is why. There is no S and A satisfying these conditions for which it is possible for S to utter with pragmatic consistency that he believes that A yet that A is not true. Should the truth of A chance not to be a condition on knowing that A, this is nevertheless not something that

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anyone claiming to know that A can admit to. Doing so is pragmatically inconsistent. Pragmatic inconsistency creates what Sorenson5 calls a blindspot. If S says to a second party S9, ‘A but A is not true,’ then, in the absence of further information, S has made it impossible for S9 to determine S’s epistemic (or doxastic) position with regard to A. Clearly, it is something of a pickle for someone not to be able to tell the truth of the situation he’s actually in without falling into pragmatic inconsistency. Perhaps this is a good point at which to suggest that the idea that truth not count as a condition on knowledge is not an idea demanding harsh and outright dismissal. It has been proposed in various forms. In one of them, the truth-condition is replaced by a plausibility condition. In another, the truth-condition is collapsed into the reasonable believability condition. In yet another version, the truth-condition is merely cancelled. In what follows, I shall be concerned with the first and third variations, not the second. 4. An Objection and a Reply I want now to pause to consider an objection to dispensing with truth as a condition on knowledge. It is a rather obvious complaint from the perspective of traditional epistemology. The TE Objection: In addition to the concept of endoxon, we owe to the ancient Greeks the crucial distinction between appearance and reality. Of course, this is an important distinction for metaphysics, but it also has an expressly epistemological instantiation. This is the contrast between belief and knowledge or, in some variations, between true belief and knowledge. It is easy to see that in what the oracular epistemologist calls common knowledge, popular knowledge, and, more generally, knowledge from say-so, we have a clear betrayal of this distinction. For the best that we can say of these things, epistemologically speaking, is that they are beliefs, not knowledge, perhaps even in some cases true beliefs. How is the oracular epistemologist to answer this complaint? Consider this as a possible rejoinder: 1. The traditionalist is minded to discourage the claim that oracular

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knowledge is actually knowledge. His discouragement arises from what he takes knowledge to be. There are several inequivalent analyses of knowledge espoused by traditionalists. But all converge on the point at hand; all provide that oracular knowledge isn’t knowledge. 2. Suppose the traditionalists are right. Then most of what beings like us think we know we don’t. Most of the decisions we take in life are taken ignorantly, that is, in the absence of knowledge. This is a hugely counter-intuitive consequence of traditionalism. In general, when, in the absence of empirical checkpoints, a theory stands in deep contradiction to what is widely believed, it has a correspondingly weightier onus of proof. But no known traditional theory of knowledge has met that onus. 3. Even so, the traditionalists might still be right. If they are, then knowledge has scant value. Every sector of the cognitive economy is shot through with oracularity, whether particle physics, or that which guides you reliably to Central Station or to your large success at the top of your profession. Beings like us are thoroughgoing ignoramuses if the traditionalist is right. It doesn’t matter. We survive, we prosper, we do particle physics, we construct magnificent civilizations. Why, then, would we put such a premium on knowledge? The rejoinder claims that the traditionalist’s view implies that the oracularist has stuck himself with the position that what he takes for knowledge is (largely) non-existent and (virtually) useless, or, for short, with the NU hypothesis. And the ocularist’s response is, in plain words, ‘So what?’ 4. A failure to satisfy the traditionalist’s conditions on knowledge is plainly no impediment to reasonable belief and well-considered action. Our massive ignorance is compatible with our substantial rationality. Not only is knowledge not much of a practical good, it is not much of an epistemic good either. So we must not be discouraged by the disapproval of traditional epistemology. Interesting as the oracularist’s rejoinder certainly is, it is not without its difficulties. They are in fact vitiating difficulties. Neither the traditionalist nor the extreme oracularist is committed to holding that most of what we think we know we don’t. The ocularist need not claim that, if knowledge were indeed held to the traditionalist’s conditions, most of what we think we know (and do indeed know) we wouldn’t. The oracularist, in turn, has nothing against knowing propositions that chance

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to be true. Neither is the traditionalist committed to holding that, by the oracularist’s conditions, most of what we think we know (and do indeed know) we don’t. For this to be so, it would have to be the case that most of what is attested to by reliable or authoritative sources is false. These considerations are enough to disarm the rejoinder, but, as it turns out, the rejoinder contains the kernel of an important idea, to which I shall return in due course. For now let me say that this kernel of an idea is suggested by the following: Epistemic Anomaly: In order to survive, prosper etc., it is necessary to quest for knowledge, not to attain it. Before leaving this point, let us note that there has arisen a large constituency in the pragmatic tradition ensuing from Peirce, for whom the sting of this epistemic anomaly might well be drawn. What would draw it and cause the inflammation it produces to die down is the assumption that the gap between quest and attainment is at all times both small and diminishing, and that our questing efforts are crowned, if not with knowledge, then with progressive verisimilitude.6 It would be interesting to assay the rich suggestion of progressive verisimilitude in all the detail that it so clearly requires. This is something that pagecount constraints rule out here. Instead I shall now turn our attention to the dominant model of traditional epistemology, the JTB – or Justified True Belief – model. 5. The JTB Model The most ancient of the approaches to the philosophy of knowledge is one arising expressly in the Theaetetus. It sees my knowing that A as my having the justified true belief that A, where justification is that which brings to heel the distinction between the appearance of knowledge (true belief) and its reality. Here is a view that has held a central place in virtually all of western philosophy and has survived various rivalries and attacks, even, dare I say, the Gettier Problem. The JTB approach has been fruitful in another way. It throws up, in its invocation of justification, truth, and belief, three irreducibly important notions whose need to clarify has furnished whole legions of philosophers gainful employment, century in and century out. This is noble and wholly necessary work, needless to say; but it is not my work. I want to approach the JTB model in a different way. I want to raise the

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question of the transparency of knowledge, but not before making an important clarification. It is well to note that the JTB and O models of knowledge are not natural contraries of one another. The JTB model actually is a capacious schema for a large number of different and incompatible theories of knowledge. Each of its three conditions admits of construals that make it possible for most of the going epistemologies to be JTB theories. Thus the JTB model cuts across the grain of most of the standard rivalries without the intrinsic necessity to favour one side at the expense of the other. This is true of the realist–idealist divide, the externalist– internalist distinction, the reliabilist–anti-reliabilist controversy, the causal–anti-causal rivalry, and the oracular-antioracular distinction. One does not get a non-JTB theory merely by interpreting any or all of its three defining conditions in ways that differ, even radically, from other interpretations. Any interpretation that leaves these conditions standing is one that sees knowledge as justified true belief. This does not stop being so when justification is understood to include or even to be exhausted by authoritative say-so. A genuine rival of the JTB model must deny one of the three conditions. We said earlier that absence of truth presents the epistemic venturer with a pickle. Perhaps there is occasion in this to wonder on the extreme oracularist’s behalf whether it is wholly advisable for him to persist with his knowledge-sans-truth approach to epistemology. Here is how I propose to proceed in the next few sections. I want to try to determine whether there are costs intrinsic to the JTB model which, on a fair assay, can be judged too costly for an epistemologist to bear. In particular, I want to determine whether the JTB position is comparatively problem-free in precisely the respect in which the extreme oracular approach has a blindspot problem. But first there is a certain distraction that it will be necessary to get rid of. This is the business of the section to follow. 6. Opacity and Fugitivity It is not uncommon for a claim that A is the case to be met with the challenge of an interlocutor, ‘But do you know for a fact that A?’ Often, if not invariably, this is a perfectly natural occasion to want to know whether we know. The provocation need not be interpersonal; if someone is epistemically fastidious, he might all on his own want to know whether what he thinks he knows he does know. Philosophers familiar

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with the modal tradition in epistemic logic might be drawn to a certain view of this matter. It is the wrong view, however. Consider sentences of the type KK: Is it sometimes the case or possibly the case that for agents S sentences of the form
KsKsA are true? It is easy to see that on any variant of the JTB-model, the leftmost operator is eliminable without loss under replacement by the propositional contents of the three JTB-conditions on knowledge. So, if the answer to our question
Could it be that KsKsA? is Yes, we have it that JTBK: It is possible that S has the justified true belief that he knows that A. JTBK also embeds an occurrence of the knowledge operator. It is the same occurrence in the same place in the original context
KsKsA. In each context the
Ks in question stands in direct apposition to A, hence is within the scope of a modal operator –
JTBs in the sentence JTBK and
Ks in the sentence KK. Each of these supplies an opaque context for
KsA. In those contexts, the rightmost occurrence of
Ks is not open to the substitution of equivalents, assuming, of course, that knowledge is indeed equivalent to justified true belief. So even if
JTBsKsA is true,
JTB JTB A need not also be true, and is not indeed true in lots of pars s ticular cases. So
Ks... is not a transparent context for
KsA. Our interest in wanting to know whether what we think we know we do know is, as we said, an interest in determining whether knowledge is transparent. The mistake that I am trying to diagnose is the mistake present in the following inference: TransTRANS: Since
Ks... is not a transparent context for
KsA, knowledge is not universally TRANSPARENT. What we have here is an equivocation on transparency. In one sense, transparency is a property of a context within which the substitutivity of equivalents is secure. In the other sense, TRANSPARENCY is the property of not knowing something without knowing that you do. The two properties are independent of one another. For let it be the case that I believe with justification that A and that A is true. On the JTB model this is necessary and sufficient for my knowing that A. Consider

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now that I believe with justification that I know that A. Then I know that I know that A, since we already have it that it is true that I know that A. The lack of transparency in the second sense does not follow from the lack of transparency in the first sense. What this shows is that in order to know that I know that A it is not at all necessary that I know the propositional contents of the conditions that make it the case that I know that A. The terms ‘transparent’ and ‘opaque’ have long since been fixed by modal logicians and similarly minded philosophers of language as contexts in which respectively substitution is truth-preserving and is not. I propose not to disturb that settled usage. Accordingly, when we think that A is true without our knowing whether it is, I will say that truth is a fugitive property, and when we think that we know that A without knowing whether we do, I will say that knowledge is likewise fugitive. And if, for example, when we think we believe that A we do indeed believe that A, then I will say that belief is a manifest state. Thus my epistemic fugitivity displaces the old modal term ‘opacity’ and my epistemic (or doxastic) manifestness displaces the old modal term ‘transparency.’ I have said that it is a mistake of reason to confuse – as we may now say – manifestness with transparency and fugitivity with opacity. There is a related confusion that we should also try to avoid. It is the confusion of the thesis of the fugitivity of truth (and knowledge) with skepticism about truth (and about knowledge). So we must say a word or two about skepticism. 7. Skepticism Consider the open sentence Skept: It is compatible with everything that S knows that it is not the case that A. Of course, some (indeed most) instantiations of Skept give wholly unsurprising results. Skepticism is precisely what one would expect to be the case with regard to all sorts of propositions – that Caesar ate a Pompeian fig at exactly 3:47 a.m. exactly 812 weeks before his death, to say nothing of the set of all falsehoods (waiving diagonalization problems). Skepticism becomes more interesting when instantiations of ‘A’ produce surprising or even vastly counter-intuitive results, results that call into question the externality of the world, or the existence of other

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minds, and so on. It will suit our purpose here to consider the result of replacing S in Skept with ‘I,’ and A with
A9 is true. This gives us Skept*: It is compatible with everything that I know that it is not true that A9 is true. Now either
A9 is true is part of what I know or it is not part of what I know. If it is, then it is not compatible with all that I know that it is not the case that A9 is true. So Skept* is false for such cases. Similarly, if we put it that A = I know that A9, if part of what I know is that I know whether A9, then it is not compatible with all that I know that I don’t know whether A9. Skept also fails here. So there are lots of cases in which the fugitivity thesis is true and Skept is false. Hence the fugitivity thesis stands or falls independently of skepticism with regard to
A9 is true and
S knows whether A9 . Even so, there is a connection with fugitivity. The fugitivity thesis asserts that there is a problem with knowing whether A9 is true, and a problem with knowing whether I know that A9. There is a problem therefore, in knowing whether conditions that constitute a counter-example to Skept actually hold. But since problems do not preclude the realization, skepticism and fugitivity are still inequivalent theses. 8. The Irrelevance of Pragmatic Inconsistency If Harry asserts that A or asserts that he knows that A, it is pragmatically inconsistent of him also to assert that A is not true. It would also appear that, if Harry has claimed to know that A, then he cannot assert on pain of pragmatic inconsistency even anything like, ‘The evidence for A’s truth may be insufficient’ or ‘But A might be false’ or ‘But I might be wrong.’ It hardly matters whether everyone would agree that these are indeed blindspot-creating utterances. What I wish to concentrate on is a potentially attractive inference that should not be drawn from the assumption of pragmatic inconsistency. Doing so would be a mistake of reason. 1. Let C be the class of conditions in virtue of which, in a given situation in which Harry finds himself, clauses (2) and (3) of the JTB model are satisfied.

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2. Then, on pain of pragmatic inconsistency, C constitutes a commitment of Harry to the proposition expressed by claims (1). 3. Commitment is satisfaction-preserving. 4. Therefore, the conditions C that satisfy (2) and (3) likewise satisfy (1). The problem, of course, is with (3). If I utter ‘I am your legal guardian,’ I am committed to make true the proposition that I pay for any special fees legally imposed upon users of the school system (as would be the case, for example, if my ward were sent to a school outside his own school district). But some legal guardians fabricate a legal address in a neighbouring school district precisely in order to leave this commitment unhonoured. Yet they do not on account cease being the legal guardians of their wards. 9. Brittleness and Elasticity If the fugitivity thesis is correct, then we might well expect that, viewed from the inside, although would-be knowers aim to get at the truth of things, and although they might be aware that their reasons for thinking that they know that A are reasons, certainly, for thinking that it is reasonable of them to believe that A, they can never have better reasons, then and there, for thinking that A is true; and yet in lots of cases A might not be true in fact. Here is why. Most of what we say we know we also regard as fallible. (The contrast class, such as it is, is not my concern here, as I have said.) This is an interesting fact both epistemically and dialectically. Its epistemic importance is that it implies the truth of ‘But perhaps A is false.’ Its dialectical importance lies in the further fact that there are interpretations under which this truth cannot be uttered without pragmatic inconsistency. Call these falsifying considerations F. Granted that F would falsify A, hence would counter-satisfy JTB’s condition (1), would it falsify anything else? Would it counter-satisfy condition (3)? The answer is No; and the reason is that (1) deploys a brittle property and (3) deploys an elastic property. Think of the analogous case of a generalization when construed as a universally quantified conditional proposition (e.g., ‘All grizzlies are four-legged’) and a generalization when construed as a generic statement (e.g., ‘Grizzlies are four-legged’). Construed the first way, the generalization is falsified by a single true negative instance. Construed the second way, the generalization’s truth can tolerate some number of true negative instances of certain

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kinds. The first generalization is brittle; the second is elastic. It is the same way respectively with the properties of truth and reasonability of belief. A’s falsehood makes it the case that A is not true, and hence that condition (1) is countersatisfied, whereas the conditions F that cancel A’s truth do not, just so, cancel A’s reasonable believability. There was a time when it was reasonable to believe that Euclid’s axioms had universal empirical applicability, never mind that, by traditional epistemological lights, it was false all along that they did. We have it, then, that I might think that I know that A, and be reasonable in thinking that I know that A, when A is actually false. It is also a fact to which, under certain interpretations, I cannot give pragmatically consistent expression in any context in which I claim knowledge of A. But the two facts are wholly compatible with one another. The brittleness of truth and knowledge has a direct bearing on their fugitivity. Truth and knowledge are fugitive because we can think that something is true without its being true and can take something for knowledge without its being knowledge. Thinking and being having different falsification conditions. Of course they do. Taking for true, taking as known, having good reason to take for true or to take as known are elastic; truth and knowledge are brittle. This ends the first, rather commonplace, part of my task. I have reminded the reader that there is a fugitivity problem for truth and knowledge. I now approach the second part of the project. Let me do so simply. Why, I ask, does our knowledge of the fugitivity of truth and knowledge not induce compensating adjustments to our standard epistemic practice? 10. Doxastic Irresistibility Truth is both a fugitive and a brittle property. It can be absent even when we have every reason to think not. Some people think (although it is disputed) that truth is disquotational. But no one doubts that belief is quotational. In believing A, I believe that A is true. In believing that A is true, I, like all people who aren’t philosophers of a certain stripe, also believe that A reflects how the world is independently of what I chance to think about it. In so believing, I, like everyone else, am taking the realist stance. The realist stance is not something I elected to take; it is rather something that I am built for. It is a dangerous world out there, and we must take care and pay attention. Realism is a particularly efficient way of paying attention. It arranges things so that when I have an on-rushing grizzly experience, I take a grizzly to be rushing on. This is

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a natural prompter of evasive action, and a good thing too. The alternatives are discouraging. If, upon having an on-rushing grizzly experience, I yielded to the philosophers’ temptation to reflect on whether my experience might reasonably be externalized, or if I elected to wait to see whether I would go on to have a grizzly-tearing-my-throat-outexperience, this would be unfortunate. It might even be fatal. These are two importantly linked traits. Belief is quotational and truth is taken realistically. To this a third factor must be added. By and large, beings like us cannot stand agnosticism. I can’t survive, never mind prosper or do particle physics, if I can’t manage to achieve a state of belief with regard to numberless important things: for example, whether I have feet, whether fire burns, whether you have a mind, whether the floor will support your weight, whether you have weight, whether the Nikkei has gone up or down, and on and on. Beings like us are greedy devourers of belief. We are awash in belief and are constantly in process of revising it. Beliefs are indispensable in orienting us to the world and serving as goads to action. We are wholly subservient to this doxastic centredness; and being so leads to an interesting arrangement of facts: Fact 1: Beings like us are doxastic push-overs. Fact 2: Belief is quotational. Fact 3: Beings like us take the realist stance.7 Conclusion: We are persistent and voracious doxastic realists. This is not to say that our thralldom to belief is either promiscuous or indeterminate. In seeking to be in states of belief, we have little interest in being in any old state of belief. We are drawn to beliefs that satisfy us. (This is the principal difference between your beliefs and my beliefs.) Beliefs that satisfy us are beliefs that quell quite particular yearnings. Having the requisite beliefs is as natural and as unavoidable as breathing. Believing something is believing something to be true. Believing something to be true is believing that we have got at how things actually are, independently of our believing it to be so. This sets up a serious-seeming problem. How in the world is my extreme ocular epistemologist going to peddle his line to beings like us? 11. Champagne and Sekt Beings like us have a huge stake in and large appetite for getting at the truth of things, of coming to see things as they really are. Taking, as we

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do, the realist stance, the natural way of expressing these states and dispositions is as an appetite for knowledge. Consider an analogy. Harry has a compulsive fondness for champagne. But, unbeknownst to Harry, everything to date that has satisfied this desire is in fact a lowgrade German sekt, which Harry takes for champagne. Harry will naturally think that his appetite has been satisfied by what it is an appetite for. But he will have been wrong. Harry was seduced by an erroneous inference, which in schematic form is 1. S desires that x. 2. Something, y, satisfies S’s desire for x. 3. So y = x. This is one of the ways in which Harry and the rest of us are similar. Harry thinks that what satisfies his desire for champagne is champagne. And we think that what satisfies our thirst for knowledge is knowledge. So far, Harry has been mistaken in every case in which his desire for champagne has been satisfied; and so far as we can ever determine, the same might be true of the satisfyings of our desire to know. But there is also an important difference between Harry’s libational case and our epistemic case. Once Harry comes to realize that sekt isn’t champagne, he has the means of determining that what now satisfies his desire for champagne is champagne, not sekt. Epistemically it is different. Even though we come to realize that what satisfies our thirst for knowledge is belief, which often enough is not knowledge, knowing this does not enable us to determine that what now satisfies my thirst for knowledge is knowledge, not belief. Knowledge is fugitive; knowledge is brittle. People aim at the truth of things, but in fact the process ends with belief, because belief is a condition of alethic satisfaction. People strive to know what is what, but the striving ends with belief, because belief is a condition of epistemic satisfaction. Because belief satisfies our compulsion to know, and since knowing things includes knowing what to do, it is essential that we have beliefs in order that we do anything at all deliberately. This is not to say that action is possible only on the basis of being in states of utter confidence that satisfy our compulsion to know. There are cases in which I am wholly tentative about what to do and substantially in the dark about what is up. But there are large subsets of these very cases in which being in those states satisfies my desire to know whether I know what to do and whether I know what’s up. The belief that I don’t know what

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to do, and the belief that I don’t quite know what’s up, may entirely satisfy my desire to know these things. So I do not want to suggest for a minute that our large proclivity for being in states of belief brings with it a condition of epistemic serenity. It is clear that if beings like us fell into a systemic agnosticism, then, in the absence of some mighty rejigging of the causal order, we would surely perish. We can say the same thing anthropically. Under even slight systemic changes to conditions under which belief is caused to occur, there is a good chance that epistemology would not and could not exist. We see in these considerations an interesting asymmetry in our cognitive lives. In our compulsion to know, two aspects are discernible – the quest and the attainment. Concerning what we quest for, we are epistemic maximizers. We aim for what is true, and we will not be satisfied by anything less, unless it is the second-order truth that what we originally quested for can’t be had. Attainment is another matter. Although it is not what we quest for, mere belief constitutes satisfaction of the compulsion to know, and hence is the appearance of the attainment of what was quested for. At the level of attainment, we are epistemic satisficers. And we are so without being aware of it (except after the fact, in moments of reflection on the fugitivity thesis). This gives us two gaps to take note of. One is the gap between thinking that we know and knowing (or thinking that something is true and its being true), that is, the gap that reflects the fugitivity of truth. The other is the gap between questing-maximization and attainment-satisficement. These gaps are not unconnected to one another, needless to say. Perhaps their most important point of similarity is that neither gap is phenomenologically apparent to the would-be knower in the here and now. These, in any event, are gaps that call to mind Hume’s celebrated remark that reason is and ought to be slave of the passions. 12. Fallacies Let S be a would-be knower and DCS be any causal source of beliefs of S or, as we might say, a doxastic causal source for S. The principal value of a DCS for S is the creation of beliefs without which S simply cannot manage. A second benefit that, in the absence of countervailing considerations, S’s compulsion to know, even filtered through the objectivitypresumptions embedded in the realistic stance, is satisfied, even if S does not in fact know these things. Satisfaction is the key factor. For

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any proposition B there is the propositional content B*, of some or other compulsion to know, such that believing B suffices for the satisfaction of the compulsion to know whether B*. In some cases, in fact rather frequently, B and B* are identically the same proposition. But let me again say there are also cases in which they are distinct from one another. If B is the proposition ‘I believe but am not sure that I know that Z,’ then believing B satisfies my compulsion to know, for example, whether believing something is knowing it. It also bears repeating here that the realist stance is not optional. We might become entirely convinced that the gap between reasonable believability and a truth, and the gap between belief and knowledge, are gaps that constitute boot-strapping problems for us. But even thinking this is thinking it to be true, and, the realist stance being what it is, believing this is believing that it is true of how things really are. Believing this also satisfies our compulsion to know whether there is a belief-knowledge boot-strapping problem for epistemology or, more generally, what it takes not to be in the epistemic dark. This is Hume’s point about his own skepticism. We can think we know that induction is indefensible or that causal necessity leaves no empirical trace without there being the slightest chance of terminating our own inductive practices or abandoning our habit of causal judgement. Persisting with the disposition to take what we think we know as a marker for what we do know is a mistake of a kind that resembles the traditional approach to the concept of fallacy.8 On this view, a fallacy is a mistake that is attractive to make (since untutoredly it appears not to be a mistake), is a universal mistake (since everyone is disposed to make it, and most do make it), and is an incorrigible mistake (since even the acknowledgement of it as a mistake is slight discouragement of recidivism). I have recently come to a different view of fallaciousness for reasons that are sketched in Gabbay and Woods.9 Here is an even briefer sketch. Cognitive agents come in various types or grades, depending on their command of the requisite cognitive resources – resources such as information, time, and computational capacity. Beings (or devices) at the low end of the scale operate with a comparative scarcity of these assets. Agents at the higher end of the scale command a comparative plenitude of them. Lower-end agents are individuals like us. Higherend agents are institutional agents, such as NASA. What counts as competent cognitive performance will vary with the available resources, hence with agency-type. Since individual agents operate under condi-

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tions of scarcity, it is reasonable to postulate for such agents scarceresource compensation strategies. If the empirical record is anything to go on, these strategies include a number of what traditional theorists would call fallacies. Chief among them perhaps is the fallacy of hasty generalization. For beings like us, leaping to conclusions is as natural as scratching an itch. Apart from the frequency of the practice, it is notable that it seems not to play us false, by and large. For we survive and we prosper. What this suggests is that in our cognitive efforts we and NASA have different standards to hit, standards whose difference reflects the difference in our command of the requisite resources. So it would seem that hasty generalization is, as such, a fallacy for NASA but not for us. An attractive byproduct of this view is that it affords us a very natural way in which to elucidate the ancient idea that a fallacy has the reality but not the appearance of a mistake. If hasty generalization is a mistake for NASA, then it is a mistake. If it is not a mistake for us, it is not a mistake. It doesn’t appear to be a mistake because it isn’t a mistake. So it is a mistake (for NASA) that doesn’t appear to be a mistake – and isn’t – for us. If our analogy were still to hold, even under this non-traditional approach to fallacies, we could admit to the shared features of attractiveness, universality, and incorrigibility and jointly dismiss the blanket presumption of error. For one thing, this would give us some understanding of why the habit of taking what we think we know for knowledge is incorrigible (it is uncorrectable because, among other things, it is not an error). The reasons-causes controversy also enters this picture. It does so in an interesting way. Suppose someone were able to believe sincerely, strongly, and systematically that his beliefs lacked for good reasons, that they were just a certain sort of mental state induced by various factors, none of which is probative. Then, if belief held its trait of quotationality and if the person in question persisted in the realist stance, he would be in a constant state of cognitive dissonance. His every belief that B would commit him to and involve him in sincere espousal of its real truth; yet the attendant sincere self-assurance that there are no reasons for such beliefs would put him in the position of holding for each B, ‘B is true, but I have no reason for thinking so.’ In fact, however, since there is no reason to think that actual reasoners are in anything like this state of blindspotted supersaturation, there is every reason to think that for any B, just being in the state of mind in which B is believed is also being in a state of mind in which it is believed that

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there are reasons to believe B, never mind whether the agent in question has the slightest idea of what, in a given case, those reasons might be. Some philosophers are of the view that the cause of a belief cannot be a reason for it. I demur from this; but even if I am mistaken, there remains an important connection between them. What causes a belief to occur in beings like us causes it to be believed that there are reasons that support it. But here too there is a boot-strapping problem. Believing that there are reasons to believe that B is one thing; there being reasons to believe that B is another. Any evidence in support of the former may, as far as we know, support no more than the latter. I shall come back to this point. 13. Dispensing with Truth Let us pause to collect our bearings. I have been talking now for several pages about theories of knowledge in which truth is a necessary condition of it, and I have been wanting to point out some of the peculiarities of such approaches that arise for us would-be knowers in the here-andnow. But I began this chapter with a brief attempt to motivate a discussion of an oracular approach to epistemology. I also said earlier on that I would not be much involved with oracular epistemology as such, but rather would be attending to that extreme version of it in which truth is dispensed with as a condition on knowledge. I want now to redeem this pledge, and to do so in such a way that I can, after all, show something quite general about oracular epistemology, that is, about the mainline variant of it in which for some classes of true propositions it fully satisfies the third condition of the JTB model of knowledge that those propositions be the propositional contents of information transmitted by an authoritative source or oracle. Suppose, then, as with one of the extreme forms of oracularism, that truth is not a condition on knowledge. This triggers straightaway a serious problem. Either the property of being an oracle is a manifest property or it is not. Suppose that it is. Then there is the problem posed by the plain fact that in cases galore oracles contradict themselves concurrently. On the other hand, if oracularity is not a manifest property, the would-be knower in the here-and-now is beset with the very problem that, on the JTB model, besets the would-be knower in regard to truth. In each case, the knower is faced with the problem of fugitivity: the fugitivity of oracularity in the one case and fugitivity of truth in the other. This carries two important consequences. One is that even if one

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is an extreme oracularist, giving up on truth does not solve the fugitivity problem for the would-be knower. The other is that the fugitivity problem is one that would afflict all variations of the O-model. But what if (as, I believe, contrary to fact) oracularity is a manifest property? Consider two oracles O1 and O2, each concurrently advising with regard to some proposition B. O1’s assurance is that B is true. O2’s makes the opposite call. Since truth is not, on the present assumption, a condition on knowledge, it cannot be inferred from the falsehood of either B or
™B that one at least of these knowledge claims is false. For some S and S9 we could have it that S now knows that B and S9 now knows that
™B. Or consider a single would-be knower S*. O1 might reliably inform S* that B and O2 might reliably inform S* that
™B. This sets up S* for knowing that both B and
¬B. This consequence would be averted if S* chanced not to believe one of his conflicting oracles. But, under present assumptions, this could never happen if oracles could concurrently disagree and are judged to be oracles by S* (remember: our present assumption is that oracularity is a manifest property). Of course, the greater likelihood is that actual reasoners will not agree that bona fide authorities who concurrently contradict themselves lay an adequate claim on buying all that they disclose. In that case the actual reasoner is committed to the consistency of the following set of claims: 1. 2. 3. 4. 5.

O1 is a bona fide authority. O2 is a bona fide authority. O1 informs me that B O2 informs me that ™B I know at most one of the pair {B,
™B}.

This being so, even though any oracle’s say-so is sufficient for the knowledge that B, this is not something that the present reasoner concedes. He concedes that O1 and O2 are bona fide authorities, yet he insists that the disclosure of at least one of them fails to produce knowledge. If the oracularist is nevertheless correct, this will be a case in which, for either B or
™B, the knower thinks that he does not know it. So the fugitivity problem now besets knowledge even if the oracular approach to knowledge is correct. I conclude, therefore, that if nonfugitivity were ever to be considered a condition on knowledge, there would be nothing to choose between any variant of the JTB model and even my extreme version of the O model.

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14. Concluding Remarks I have taken it as given that most philosophers would agree that a fugitivity problem exists for truth and knowledge. What is surprising is not that the fugitivity thesis is true, or widely accepted as true. What is surprising is that recognition of its truth seems to have no effect on standard epistemic practice. Our standard epistemic practice is simply replete with points of utter alethic satisfaction, never mind that – infallible knowledge aside – every one of those points is caught in the embrace of the fugitivity thesis. You would think that we would have learned to be more circumspect and more modest. Why haven’t we? Why have we not adjusted our behaviour to the facts of fugitivity? Why do we not honour these facts more attentively? Hume thought that it is a matter of how we have been habituated. Kant thought that it had to do with the metaphysical structure of empirical knowledge. Of the two, I think that Kant had the more nearly correct kind of answer. For the answer does appear to turn on how we are constituted. The answer, then, is that the reason we don’t reflect these fugitivity facts in our epistemic practice is that we cannot, that we are not built for it. We are fated to take the realist stance, just as belief is fated to be quotational. In our philosophical moments we see that our doxastic satisfactions are vexed in precisely those ways we are unable to reflect in our epistemic practice. This does not demonstrate the ancient canard about the utter illusoriness of the human condition. But it does establish about half of it. We should tell the existentialists. Given the ways in which we experience the world, we could be said to be Can’t-Help-It-Realists. Can’t-Help-It-Realism is an essential part of the problem of our epistemic indifference to the fugitivity of truth and knowledge. Given the ways in which we experience the world, we are also Can’t-Help-It-indeterminists. Even those among us who have been persuaded that our participation in the causal order precludes our freedom, none have yet to figure out how to experience themselves in the world as in a condition of across-the-board causal bondage. It has been suggested that Can’t-Help-It-Realism and Can’t-Help-ItIndeterminism can be subdued by the right training. The ancient Pyrrhonists claimed some success in experiencing the world as illusory. To do so, they said, requires lengthy and highly disciplined training. What would this be like? It would be a generalization of a common enough kind of occurrence, the way in which we experience an optical illusion as illusory. Perhaps with the right training we could be got to

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experience the illusion of putting our clothes on in this same way. Perhaps more advanced students could adapt these techniques to their experience of their next-door neighbours. And so on. Let us suppose, then, that after years of arduous training doughty Pyrrhonists were able to experience the world as false. Then most of what you and I assent to they would dissent from (in pecore, perhaps). Every time you and I were disposed to utter ‘Here’s Harry coming,’ the Pyrrhonists would be prone to say ‘Here is faux-Harry faux-coming.’ We can all agree that this would be a rather impressive achievement on the Pyrrhonists’ part. Certainly it would reflect a considerable relaxation of the grip of what I have been calling the realist stance. But it would not and could not have subdued it entirely. As long as the Pyrrhonists are giving sincere voice to what they are experiencing, then when they report that faux-Harry is faux-coming they are telling us what they believe. But belief is quotational. The Pyrrhonists are telling us what they take to be true. We might well imagine that these things that they are telling us (actually faux-us) are true are things that they actually know to be true. But truth and knowledge are fugitive properties, as we would well expect a Pyrrhonist to know. Should not he, of all people, reform his epistemic practices? He cannot. That is, he cannot so long as he believes what his experience tells him. For it tells him that the world in reality is such that here is faux-Harry faux-now coming along. notes I am grateful for helpful comments from Mark Migotti, Dale Jacquette, Bas van Fraassen, Dov Gabbay, David Hitchcock, Kent Peacock, Miriam Thalos, and David deVidi. Research for this chapter was supported by the Social Sciences and Humanities Research Council of Canada and the Engineering and Physical Sciences Research Council of the United Kingdom. My thanks to both. 1 Especially his Hypothetical Reasoning (Amsterdam: North-Holland, 1964) and Plausible Reasoning: An Introduction to the Theory and Practice of Plausible Inference (Assen and Amsterdam: Van Gorcum, 1976). 2 John Woods, ‘The Problem of Abduction,’ Tijdschrift voor Wijsbegeerte 93 (2001): 265–73; Dov M. Gabbay and John Woods, The Reach of Abduction (Amsterdam: North-Holland, 2005), to appear in the series The Practical Logic of Cognitive Systems; see also Dov M. Gabbay and John Woods, ‘Filtration Structures and the Cut Down Problem for Abduction’ in this collection.

Cognitive Yearning and Fugitive Truth 157 3 Jaakko Hintikka and James Bachman. What if ...? Toward Excellence in Reasoning (Mountain View, CA: Mayfield, 1991); Jaakko Hintikka, Ilpo Halonen, and Arto Mutanen, ‘Interrogative Logic as a General Theory of Reasoning,’ in Handbook of the Logic of Argument and Inference: The Turn Toward the Practical, vol. 1 of Studies in Logic and Practical Reasoning Dov M. Gabbay, Ralph H. Johnson, Hans Jürgen Ohlbach, and John Woods, eds. (Amsterdam: NorthHolland, 2002). 4 Ruth M. Kempson, Presupposition and the Delimitation of Semantics (Cambridge: Cambridge University Press, 1975), 167. 5 Roy A. Sorenson, Blindspots (Oxford: Clarendon Press, 1988), 37. 6 For a recent development, see Theo Kuipers, From Instrumentalism to Constructive Realism: On Some Relations between Confirmation, Empirical Progress, and Truth Approximation (Dordrecht: Kluwer, 2000). 7 A similar notion can be found in Kent Peacock, ‘Quantum Holism and the Incompleteness of Knowledge’ (in progress), which attempts to locate a notion of realism compatible with quantum theoretic discouragements. 8 John Woods, ‘Who Cares about the Fallacies?’ in Argumentation Illuminated, F.H. van Eemeren, Rob Grootendorst, J. Anthony Blair, and Charles A. Willard, eds. (Amsterdam: SicSat Press, 1992), 22–48. Reprinted in John Woods, The Death of Argument: Fallacies in Agent-Based Reasoning (Dordrecht and Boston: Kluwer, 2004). 9 Dov M. Gabbay and John Woods, ‘The New Logic,’ Logic Journal of the IGPL 9 (2001): 157–90. John Woods, Ralph H. Johnson, Dov M. Gabbay, and Hans Jürgen Ohlbach, ‘Logic and the Practical Turn,’ in Handbook of the Logic of Argument and Inference, Gabbay et al., eds., 295–337.

9 The de Finetti Lottery and Equiprobability PAUL BARTHA

1. Introduction The axiom of countable additivity (CA) plays an essential role in modern probability theory. The axiom states: (CA) If we have a countable infinity of outcomes H1, H2, ... that are mutually exclusive, then P(H1 › H2 › ...) = P(H1) + P(H2) + ... Kevin Kelly,1 following de Finetti,2 questions whether CA is an indispensable constraint on subjective interpretations of probability. In such interpretations, particularly as applied to the justification of scientific hypotheses, CA assumes great epistemological significance because of its role in deriving the convergence theorems.3 In essence, CA forces us to adopt the biased view that if there is ever going to be a counterexample to a universal hypothesis, we are far more likely to find it in some finite segment of the future than in the entire remainder of history. For this reason, Kelly believes that the principle ‘should be subject to the highest degree of philosophical scrutiny’ (323), rather than being adopted purely for its mathematical merits. The opposing point of view, that CA is no more problematic for the subjective interpretation than any other axiom of the probability calculus, is represented by people such as Howson and Urbach4 and Williamson.5 Their main argument is that a Dutch Book justification can be given for CA, just as for any of the other standard axioms. Dutch Book Arguments can be criticized,6 but I do not wish to call them into question in this paper.

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A crucial and much-discussed test case for these opposing positions is an example due to de Finetti, which (in slightly altered form) I refer to as the de Finetti lottery. The example requires that we have a uniform probability distribution over the positive integers – something that seems to be mathematically impossible. Section 2 describes the de Finetti lottery. Section 3 argues that CA is indeed inapplicable and that we can define a binary relation of equiprobability, which does the work of a uniform probability distribution over the positive integers. I defend this position against three objections and draw some lessons for our intuitions about probability. 2. The de Finetti Lottery De Finetti claimed that we should be able to make sense of a uniform probability distribution over a countably infinite set, such as the natural numbers. To make this vivid, let us imagine a lottery in which the number of tickets issued is countably infinite, one for each positive integer, and each ticket has an equal (subjective) probability of winning. Such a lottery is conceivable. The assumption that each ticket is equally good seems reasonable, or at least not a priori false. As de Finetti pointed out, however, there is no way to assign an equal probability to each ticket’s winning if we accept countable additivity. If we let pn be the probability assigned to ticket n, then the pn’s have to satisfy two conditions: (Equiprobability) (Countable additivity)

pn = pm for all n, m p1 + p2 + p3 + ... = 1

Equiprobability is just the desired assignment of a uniform probability distribution over all tickets. Countable additivity tells us that the probability that some ticket wins (which is 1) is the infinite sum of the probabilities that each individual ticket wins. The two conditions cannot, however, both be satisfied. If each pn = 0, the infinite sum will be 0, but if each pn is the same positive number, then the series diverges. Countable additivity compels us a priori, as de Finetti says, ‘to assign practically the entire probability to some finite set of events, perhaps chosen arbitrarily.’7 This is deeply puzzling: What is strange is simply that a formal axiom, instead of being neutral with respect to evaluations ... and only imposing formal conditions of coherence, on the contrary, imposes constraints of the above kind without

160 Paul Bartha even bothering about examining the possibility of there being a case against doing so.8

De Finetti abandons countable additivity and retains equiprobability by letting each pn = 0. His argument rests on intuitions about symmetry. Any two tickets are interchangeable; any two ticket-holders are in an epistemically indistinguishable position. Similar intuitions support a uniform probability assignment in two analogous situations: a finite lottery and a lottery over the real numbers in the interval [0, 1]. In these analogous lotteries, equiprobability is perfectly reasonable and unproblematic (for there is no conflict with countable additivity). Why can’t we retain it in the case of a countably infinite set? 3. Equiprobability and Countable Additivity I think that de Finetti is right not to give up on equiprobability, but wrong to let each pn = 0. Sections 3.1 and 3.2 present two arguments for keeping countable additivity and dropping equiprobability. Section 3.3 shows how we can retain equiprobability without directly rejecting countable additivity – a mysterious-sounding claim, but the mystery will be resolved shortly. Section 3.4 develops and responds to a final objection. 3.1 The ‘No Random Mechanism’ Argument The most popular direct objection to a uniform probability assignment in the de Finetti lottery, originally formulated by Spielman,9 is that any mechanism for choosing a positive integer (i.e., a particular ticket number) will inevitably yield a biased distribution. As Howson and Urbach write: ‘it is not at all clear what selecting an integer at random could possibly amount to: any actual process would inevitably be biased toward the “front end” of the sequence of positive integers’ (81). Even if we concede this point,10 the natural response, as Williamson points out, is that it applies only to physical chance, not to subjective probability. There is no need to exhibit a physical mechanism that could select each integer with equal probability. Since we are dealing with subjective probability, we can abstract away from the mechanism used to select the winning ticket. Indeed, the point de Finetti

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wants to make is that, lacking any knowledge about the mechanism, the only reasonable thing to do is to regard all of them as equally likely to win. If we are not persuaded by this response, it might be because we believe that subjective probabilities should reflect our knowledge of physical chances. One familiar attempt to represent this connection is Lewis’ Principal Principle:11 (PP) Prob(A / P(A) = r) = r. Your subjective probability for A, given that the chance of A is r and no other inadmissible information (e.g., that A is or is not true), is r. We might expect, similarly, (PP9 ) Prob(A / P(A) z r) z r, or, with still greater generality, what we might call the Unprincipled Principle: (UP) Given that the actual physical chance distribution cannot have certain features, our subjective probability distribution ought not to have those features. If we know that there is no physical mechanism that gives each integer an equal chance to be selected, then our subjective probabilities should not be uniform. Although there might be some way to salvage this argument, it won’t be via UP. Even the special case PP9 is clearly wrong. Choose randomly between two indistinguishable coins, one with a bias P(heads) = 0.9 and the other with a bias P(tails) = 0.9. Our subjective probability for heads should be 0.5, despite our knowing that the chosen coin is not fair. We obtain this subjective probability as a weighted average of the two probability distributions we could have if we knew which coin we had chosen. In the case of the lottery, a similar sort of averaging supports equiprobability, although it is not quite so easy to formulate. For every mechanism that favours integer m over integer n, there is another just the same except that the values pm and pn are exchanged. So our subjective probability assignments should ensure that pm = pn for any m and n. Of course, there is a history of paradoxes that attend such symmetry-

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based arguments. We know that symmetry arguments can sometimes yield conflicting probability values for the same outcome. But sometimes they work. Harmless applications of uniformity require considerable analysis and defence, but they exist. Section 3.3 provides a justification for the appeal to symmetry in the de Finetti example; section 3.4 responds to one final objection. 3.2 The Dutch Book Argument There is a straightforward Dutch Book Argument for countable additivity as a constraint on subjective probability that, if successful, appears to rule out a uniform probability assignment in the de Finetti lottery.12 The argument is a simple generalization of the usual Dutch Book Argument for finite additivity. Suppose that pi is our fair betting quotient for the proposition that ticket i wins, and 1 is our fair betting quotient for the proposition that some ticket wins. Suppose that countable additivity is violated, so that p1 + p2 + ... < 1.13 Each of the following bets is fair: bet against ticket i with a stake of $1 and betting quotient pi. This bet pays pi dollars if ticket i loses (we win our bet), and (pi – 1) dollars if ticket i wins (we lose our bet). The system consisting of all these bets taken simultaneously is fair. Suppose now that ticket N wins, as must happen for some N. We win pi for all tickets other than N, and pN – 1 for ticket N. Our net gain is therefore (p1 + p2 + ...) – 1, which, by assumption, is negative. So no matter what happens, we lose money. This constitutes a Dutch Book. This argument shows that if we assign a standard real-valued betting quotient to the proposition that ticket i wins, for each i, then these betting quotients must sum to 1 on pain of vulnerability to a Dutch Book. In particular, de Finetti’s own solution, which sets pi = 0 for each i, is unacceptable within a Dutch Book framework. There are, however, at least two ways to avoid the conclusion that countable additivity is forced upon us. One is to use non-standard probabilities. Assign to each proposition that ticket i wins an equal but

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infinitesimal degree of belief. Bartha and Hitchcock14 show how this may be accomplished in such a way that the hyper-finite sum of these probabilities is 1 even though any finite sum is infinitesimal.15 Arguably, the total net gain should also be computed using hyper-finite summation rather than standard countable summation. But I will not dwell on this point because I want to focus on an alternative approach. The second way to avoid the conclusion that countable additivity is rationally required is just to deny that we have a real-valued degree of belief, or fair betting quotient, for the proposition Ai that ticket i wins. Any positive number is too large (given our desire to assign the same betting quotient to each ticket number), while 0 is too small (the bet would cost nothing and might pay off). The Dutch Book argument never gets off the ground. We might have a betting quotient of 0.5 for ‘an even-numbered ticket wins.’ We might also have, as explained in the next section, a relative betting quotient of 1 for the pair of propositions Ai (ticket i wins) and Aj (ticket j wins). The crucial point, though, is that we lack betting quotients for these propositions taken in isolation. If we have no betting quotients for these propositions, and hence no subjective probabilities, then countable additivity is inapplicable rather than violated. 3.3 Equiprobability and Relative Betting Quotients This section shows that a relationship of equiprobability between two outcomes16 can be defined independently of the existence of any probability function. We define the relationship in terms of relative betting quotients and then apply it to the de Finetti lottery.17 A relative betting quotient for a pair of propositions tells us, roughly speaking, how to trade off a bet for one and a bet against the other. The (fair) betting quotient for A is a real number p between 0 and 1 such that a bet on A that costs pS and pays S if A is true and nothing if A is false is subjectively fair, for any stake S. To define relative betting quotients, first consider a special case. If two outcomes A and B have well-defined betting quotients p and q respectively, and p z 0, then the relative betting quotient of B to A, written RBQ(B; A), is just q/p. Suppose that this ratio is k, so that (informally) we consider outcome B to be k times as likely as outcome A. Table 9.1 represents a bet for A with stake k and a simultaneous bet against B with stake 1.

164 Paul Bartha TABLE 9.1 Betting quotients A

B

For A (stake k)

Against B (stake 1)

Net gain

F T F T

F F T T

–pk (1 – p)k –pk (1 – p)k

q q –(1 – q) –(1 – q)

0 k –1 k–1

TABLE 9.2 Relative betting quotients A

B

Payoff to the agent

F T F T

F F T T

0 kS –S (k – 1)S

This system of bets is subjectively fair, and the betting quotients p and q disappear in the final column. Only the ratio, k, matters for the net payoff. No money changes hands if neither A nor B is true. We now generalize this idea to encompass cases where A and B lack betting quotients. An agent’s relative betting quotient for B relative to A, written RBQ(B; A), is a non-negative real number k such that the bet described by table 9.2 is subjectively fair for any stake S. If there is no such unique k, then RBQ(B; A) is undefined. The idea is simple. If neither A nor B is true, no money changes hands. If A is true, the bookie pays out kS. If B is true, the agent pays the bookie S. Note that the stake S may be negative, in which case the direction of gains and losses is reversed. The agent regards a bet on A with payoff kS to be of equal value to a bet on B with payoff S. If A and B have well-defined betting quotients p and q with p non-zero, k is just the ratio q/p. There are two special cases. If A is a contradiction but B is not, then no value of k makes the bet fair, so that RBQ(B; A) is undefined. If both A and B are contradictions, then any value of k makes the bet fair (since no money will ever change hands in any case), so once again we leave RBQ(B; A) undefined.

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We can define a Dutch Book for a system of relative betting quotients in exactly the same way as for ordinary betting quotients. The following results may be derived for any system of relative betting quotients that is not vulnerable to a Dutch Book in a manner parallel to the usual Dutch Book arguments:18 (R1) (Positiveness) RBQ(B; A) t 0 whenever defined. (R2) (Reflexivity) RBQ(A; A) = 1 for any A that is not a contradiction. (R3) (Tautologies) If T is a tautology, then RBQ(A; T) d 1 whenever defined. (R4) (Contradictions) If A is a contradiction and A is not, then RBQ(A; A) = 0. (R5) (Finite additivity) If B and C are mutually exclusive and RBQ(B; A) and RBQ(C; A) are defined, then RBQ(B › C; A) is defined and RBQ(B › C; A) = RBQ(B; A) + RBQ(C; A). (R6) (Generalized conditionalization) If RBQ(B; A) and RBQ(C; A) are defined and RBQ(B; A) z 0, then RBQ(C; B) is defined and RBQ(C; B) = RBQ(C; A) / RBQ(B; A). These results are analogues of familiar properties of probability. We do not, however, get countable additivity of relative betting quotients – at least not in general. Unlike simple betting quotients, relative betting quotients have no upper bound. There is no reason why an infinite sum of relative betting quotients should converge at all. But consider the special case where RBQ(Bn; A) is defined for all n, the Bn’s are exclusive, and RBQ(B; A) is defined where B is the infinite disjunction of the Bn’s. In this special case, the Dutch Book Argument of section 3.2 can be adapted to show that f

(CA*)

å RBQ(B ; A) = RBQ(B; A). n

n=1

If RBQ(B; A) is defined and the assignment of relative betting quotients is coherent, we call this value a relative probability, which we write as R(B, A). We are most interested in the case where R(B, A) = 1, in which case we say that A and B are equiprobable. If we exempt contradictions, then the relationship of equiprobability is reflexive (by (R2)), symmetric (by (R2) and (R6)), and transitive (again by (R6) and (R2)); hence, it is an equivalence relation.

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If R(B, X) is defined, where X represents the entire outcome space, then write PrR(B) for this value. It may turn out that PrR is a (countably additive) probability function. In this case, we say that PrR is the monadic probability function that can be obtained from the relative probability function. In general, PrR(E) will not be defined for every E representing a set of outcomes. Return to the de Finetti lottery. We are finally in a position to assert that any two propositions of the form ‘ticket i wins’ and ‘ticket j wins’ are equiprobable, because their relative betting quotient (and hence their relative probability) is 1 even though they have no well-defined betting quotients (and hence no real subjective probability value). With this analysis in hand, we see that our original question about the necessity of countable additivity should be answered negatively in one sense and affirmatively in another. If the issue is whether all subjective probabilistic reasoning must always be constrained by countable additivity, then the answer is negative. On this point, which was their main concern, de Finetti and Kelly are correct. If the issue is whether monadic subjective probabilities are subject to countable additivity whenever they can be defined, then (contrary to de Finetti and Kelly) the answer could still be affirmative. The de Finetti lottery does not provide a counter-example. To appreciate this point, consider equation (CA*) above, letting Bn stand for ‘ticket n wins’ and A stand for ‘some ticket wins,’ that is, the entire outcome space. If each relative betting quotient RBQ(Bn; A) were 0, we would have our original puzzle all over again. Each Bn would then have monadic probability 0, and we would have a violation of countable additivity for the monadic probability function. The correct view is that none of these relative betting quotients is defined! Putting it in more colourful language, the events ‘ticket n wins’ and ‘some ticket wins’ are incommensurable. 3.4 The Relabelling Paradox Our solution in 3.3 is threatened by the following example, which purports to show that positing a relationship of equiprobability between individuals in a countably infinite population leads to paradox.19 Example 1: 1. Let A be a countably infinite population. Label its members a1, a2, a3, ... One individual an is to be selected. Suppose, for reductio, that any two individuals are equally likely to be selected.

The de Finetti Lottery and Equiprobability 167

2. We should then have PrR(EVEN) = PrR(ODD) = ½, where EVEN { selected an has an even label; ODD { selected an has an odd label. We should also have PrR(ONE) = PrR(TWO) = PrR(THREE) = PrR(FOUR) = ¼, where ONE { selected an has a label n = 4k+1, etc. 3. The original labelling should not matter. Let us re-label as follows: • b4n = a2n for each n. • b2n+1 = a4n+1 for each n. • b4n+2 = a4n+3 for each n. We can think of the relabelling as rearranging or shuffling the order imposed in the original list. Here is a picture of what happens for the first few individuals. The list a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 ... becomes rearranged as a1 a3 a5 a2 a9 a7 a13 a4 a17 a11 ... The set EVEN is compressed, while the set ODD expands to fill the newly created vacuum. Let ODD-NEW { selected individual’s new label bn is odd EVEN-NEW { selected individual’s new label bn is even 4. Since the new labelling is just as good as the old one, the reasoning in step 2 shows that PrR(ODD-NEW) = PrR(EVEN-NEW) = ½. But ODD-NEW is equivalent to ONE, so that we also have PrR(ODD-NEW) = PrR(ONE) = ¼, a contradiction.

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The key assumptions in deriving the contradiction are equiprobability and Label Independence: relabelling the members of a countably infinite population should make no difference to probability claims. In my view, Label Independence is the culprit. Relabelling does make a difference. To resolve the paradox, we need to look at the grounds for assigning fair betting quotients, or equiprobability in particular. The basic idea20 is that two outcomes are equiprobable if they are epistemically symmetric. We define a relation of equiprobability directly from a given set of acceptable symmetry transformations. Symmetries are construed as bijections (1-to-1 and onto mappings) on the outcome space X with certain properties that reflect epistemic constraints. A set S of symmetries is regular if it has the following two properties: (G) S is a group under function composition; and (M) For no non-empty subset C of X and positive integers m > n are there symmetries T1, ..., Tm and $3 > $2 > $1. Then for any population and any set of payoffs, the disposition with which it would be most rational to invade that population is whichever of the following is highest: UC = $3(u – 1) + $3c + $1r + $1s CC = $3u + $3(c – 1) + $3r + $2s RC = $4u + $3c + $3(r – 1) + $2s UD = $4u + $2c + $2r + $2(s – 1). And, as Danielson rightly points out, on this score the more predatory RC comes out as categorically equal or superior to any of its three competitors, including its more recognizably moral cousin CC. So the gap between rationality and morality that Gauthier took himself to have closed – Morals by Agreement (MBA) showed that CC is categorically equal or superior to UD (a.k.a. Hobbes’ Foole) – is revealed as remaining, alas, wide open!

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Clearly – or at least hopefully – something’s gone amiss. But what? What’s gone wrong, I submit, is that the naiveté of MBA’s dispositional dualism is replicated in AM’s ‘round monism.’ That is, MBA, AM, and APr are of a mind that morality cannot be modelled as strategic responses to iterated Prisoners’ Dilemmas.9 But MBA and AM confine themselves to one-round, non-iterated tournaments. What, I wondered, would Danielson’s apparatus reveal were it extended to multi-round, non-iterated encounters? That is, with what disposition would it be most rational to invade a given population were one concerned with the impact of the disposition with which he invades on subsequent dislodgments and invasions, and the effect of all of this – given his epoch of interest, and the epochs of interest of these anticipated subsequent invaders – on his cumulative take? And APr’s answer is that: though caution coupled with predation (RC) continues to fare as well as or better than its competitors in any given round, under some by no means uncommon circumstances, caution simpliciter (CC) can and does emerge as the superior strategy. And how does APr show this? It does so by asking us to consider a population consisting of 3 UCs, 1 CC, and 1 UD, an epoch of interest of 2, an immigration quota of one per round, a dislodgment threshold at anything less than a cumulative score of 7, and a scoring schema of 4/ 3/2/1. In such a situation an RC invader accumulates 17, whereas a CC scores 20.10 Quod erat demonstrandum. APr offers us better advice than AM. And since AM is just APr in the ‘special case’ of our having only one day to live, so to speak, AM can now be replaced with APr and replaced without fear of remainder! Of course all the above inputs – the population, epoch of interest, immigration condition, dislodgement threshold, and payoff structure – were cooked to generate this result. That is, the case was concocted only to show that there are values for these variables under which RC can be bested. What remains – and this is what APr is all about – is to tease out the mathematical regularities underpinning more general recommendations as to which disposition to adopt under what conditions, or – what, on the supposition that human beings are by and large rational, amounts to the same thing – the regularities that account for the dispo-

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sitions that have been adopted under the circumstances in which we must have found ourselves. 3. Complicating the Model What remains as well, however, is to show that anything which complicates real-world decision making can likewise be captured by the model. Here I offer only a representative sampling of how APr can capture such complications: 3.1 Translucency, Scrutiny, Autophany, and Dissimulation Costs Both MBA and AM presuppose full dispositional transparency.11 In the real world, however, we wear our dispositions on our sleeves, that is, phenotypically/behaviourally, rather than our foreheads, that is, genotypically/algorithmically. Virtual ethicists are concerned, therefore, to model translucency. That is, virtual players strategize the dispositions with which to invade a population with attention to probabilities of accurate readability. But since translucencies vary, those confidence levels must be adjustable. As, in the real world, indeed they are.12 Likewise, then, must they be adjustable in APr. Furthermore, neither MBA nor AM adjusts for scrutiny costs, nor for what we might call ‘autophany’ costs. To explain: Neither UC nor UD scrutinizes. CC and RC do. And if their scrutiny costs are high enough, they might be better advised to simplify to UC or UD. Scrutiny costs are most often borne by the scrutinizing player alone. But often enough too they’re shared. For example, if by the time you’ve checked me out, so to speak, the flag has fallen on our interaction, we both lose. But all four dispositions need to ‘autophanize’ – that is, reveal their decision-procedures to each other – since failure to do so will force these others to treat them as – and so they might just as well be – UD. But, of course, this autophanizing introduces the possibility of dissimulation. Dissimulation incurs costs of its own, sometimes costs that are prohibitive.13 So APr needs to, and can, adjust for differential scrutiny, autophany, and dissimulation costs. 3.2 Toggle-Ability and Plasticity It will likewise have been noticed that the model so far presupposes that moral dispositions are hard-wired, not only in the sense that we

496 Paul Viminitz

pre-commit to them, but also that they remain in force categorically. For without this categoricalness I couldn’t provide you with sufficient assurance that ‘If you scratch my back I’ll scratch yours!’ But in the real world – and we know this from attending to the debate between consequentialists and deontologists – categorical imperatives run the dual risks of courting disaster and precluding windfall. So what we’ve been calling a disposition is really just a sub-algorithm, one that takes conditional payoffs as toggle-points or transducers. A categorical disposition – such as those developed by MBA and AM – is therefore nothing more than a ‘special case’ of this. And the like can be said, then, about dispositional plasticity.14 And just as we’re plastic with respect to populations, our dispositions can be – and in large measure are – indexed to domains of interactivity. Virtual ethicists model toggle-ability thresholds and plasticity triggers with very little difficulty. One way is to simply front-end our virtual players’ moral algorithms with a situational transducer, in much the way, for example, many cognitive scientists think the human brain first disambiguates speech from mere sound and then modularizes each for processing. This, of course, increases our co-players’ reading burden. And if that burden becomes too heavy we may have to lighten it for them. And so, once again, just how plastic we are is likewise just a solution to an equilibrium problem. 3.3 Dislodgment and Invasion And with like alacrity can we model dislodgment thresholds. An academic department might have leave from its dean to invite one invader per retiree. Not so a company undergoing downsizing. But what’s especially instructive are the effects of differential dislodgment thresholds. Tenured professors are undislodgeable. Untenured require, say, four publications per review period. In the Calorie Game the average man needs to average 1800 calories a day, the average woman only 1200. It should not surprise us, therefore, that women tend to be – because they can afford to be – more broadly compliant.15 3.4 Coalition and Positioning Games Coalition strategies are especially important in the study of war, as are positioning strategies in marketing. But one needn’t leave the suburbs for the battlefield or marketplace for phenomena with a like logic. By agreeing to share their proceeds, an RC can free ride on his UC part-

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ner’s lower scrutiny costs, while she, in turn, takes a share of his windfalls against other UCs. Think of the good cop/bad cop routine, or dispositional complementarity in marriage. 3.5 King-Makers/Breakers and Satisficing Coalitions have to do with sharing payoffs. But another coalition-esque strategy is for a disposition to transduce first for, say, race. That is, a racist doesn’t share her payoffs with members of her race, but she does interact differentially with them. So, for example, Malcolm Murray has shown that, provided her race-recognition costs are low enough, a racist UC will score better than a UC simpliciter, unless, that is, there are enough anti-racists in the population.16 This ‘unless’ raises another important difference between AM and APr. Holly Smith has demanded that even kamikaze dispositions, like Shaft-RC and anti-racist-UC, be accommodated by the model. But Danielson rejects dispositions other than the canonical four because such king-makers/breakers are not themselves maximizing strategies, and because accommodating non-maximizing strategies would so complicate the model that we could learn virtually nothing from modelling them.17 But, one might counter, aren’t racism and anti-racism among the phenomena we want our model to help us understand? Accordingly, APr opts for a compromise between Smith’s insistence on ‘parametric robustness’ and Danielson’s worry about obfuscation. That compromise is to accommodate any disposition that can satisfice.18 That is, pace Smith, APr disallows invasion from mutations that are themselves unviable. But pace Danielson it allows invasion from possible king-makers and/or -breakers, like Shaft-RC and anti-racistUC, provided they won’t be dislodged within their epoch of interest. Or, to put the matter more instructively, APr, unlike AM, is concerned less with rational superiority than rational adequacy. And it is this contentedness with adequacy which, I submit, makes APr the more suitable tool for, among other things, modelling evolutionary ethics.19 4. In-Principle Limitations 4.1 Infinite Input Problems, e.g., Infinitely Future Generations Differential scrutiny, autophany, and dissimulation costs, toggle-ability, plasticity, and dislodgment thresholds, invasion conditions and coalition strategies, racism and anti-racism, are, as I say, just a repre-

498 Paul Viminitz

sentative sampling of the kinds of complications APr can incorporate with relative alacrity. But, I’m loath to confess, there are features of the real world that APr can’t capture. These fall into three categories: features involving infinite inputs, features involving looping, and features involving infinite regresses. Let’s look at each in turn. Representative of limitations arising from infinite inputs is the problem of infinitely future generations. As already noted, one surprising bonus of APr is that it solves the erstwhile intractable problem of grounding our concern for future generations. That is, since future generations can neither return an injury nor repay a kindness, game-theoretic reductionists (a.k.a. contractarians) have been at a loss to explain why we should care about them at all. But since, according to virtual ethics, moral considerability is reducible to dispositional considerability, and since APr demonstrates that the ‘X’s disposition is considerable to Y’ relation is transitive, the problem of future generations is now solved. But assuming, as at least some people attest, concern for infinitely future generations is a real-world concern, APr can’t solve it because any algorithm faced with an unpatterned infinite input will simply run forever and hence never reach resolution. Thus the contrast between the considerability of a) finitely future generations and b) infinitely future ones is akin to that between c) the perfect chess-playing machine and d) the perfect chess-playing machine were the fifty-move rule removed. The former is doable; the latter is not.20 Is the intractability of the considerability of infinitely future generations, and similar limitations, just a purely theoretical embarrassment for Apr, with about as much practical import for virtual ethics as has Gödel’s incompleteness proof for quotidian logic? Would that I could say yes. But more about this momentarily. 4.2 Looping Problems A second source of intractability with very clear real-world implications, however, is looping.21 That is, real-world agents – for example, a couple trying to decide which movie to go to – do entertain co-referring preferences. But these can only be modelled by co-referring algorithms; and yet, as is well known, co-referring algorithms loop. In a recent conference dedicated to nothing but tuism, the emergent consensus was that this is not a problem for contractarians, since interactive dilemmas involving co-referring preferences are, if games at all, games of pure coordination. They’re not mixed-motive games and therefore can’t be the grist of morality.

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4.3 Infinite Regress Problems What is a problem for programs as ambitious as APr, however, are real-world cases of infinite regresses. Representative of these is the problem of simultaneous attackers. Simultaneous attackers can only be accommodated by the model on pain of reintroducing the very strategic dilemmas the model is designed to render parametric.22 That is, if APr did not prohibit simultaneous attackers, at some point, if need be by fiat, it would be faced with an infinite regress of Entry Prisoners’ Dilemmas (EPDs) and so could never run to completion. How much damage this does to the power of the model remains to be seen. 5. Limitations in Fact In-principle limitations are not so much embarrassments as restrictions to any model. But the real constraints on the model – the ones about which I think we should be particularly exercised – arise not out of computational impossibility but out of mere computational exhaustion. That exhaustion arises not from any of the myriad sources of complexity that worried Danielson. All of these complexities are such that whether we rule them out by fiat or accommodate them will depend on just how fine-grained we need the model’s advice to be. For the purposes of refuting Hobbes’ Foole, MBA is just exactly enough. For the purposes of showing that MBA’s advice can’t be categorical, AM does quite nicely. And for the purposes of showing that neither can AM’s advice, Apr’s concocted case would seem to do the trick. For the purposes of advising real-world agents on what dispositions to adopt for resolving real social problems, none of the abovenoted complications can be ruled out by fiat. That might present us with a data entry problem. Data entry is expensive. But the affordability of an expense, any expense, is a function of just how improved our lives could be with, and/or just how dire the consequences of doing without. A week of deliberations over whether to pay or renege on a ten-dollar loan is stupid. As is a year dithering over whether to exact revenge for a minor social slight. But even a thousand data entry clerks working 24/7 to keep our nuclear deterrence strategy in constant update is probably a bargain. So when skeptics claim – as they’re wont to do – that our models can never be fine-grained enough to make good on their promise of prediction and control, what they’re really claiming – indeed all they could be claiming – is that they see no

500 Paul Viminitz

promise of these models ever becoming cost-effective. With respect to, say, courtship, they’re probably right. That is, as to whether I should pay for dinner on a first date or insist on going dutch, I think I’d just as soon rely on my folk-psychological intuitions rather than postpone the date until I can reserve the university’s mainframe for a couple of weeks. But tax policy is surely another matter! 6. Limitations-in-Fact Turned Regress-Problematic But what I want to look at now is a particular, and peculiar, limitationin-fact which produces a curious – and, as it turns out, highly embarrassing – limitation-in-principle. To wit: It’s a well-rehearsed phenomenon that one, if not the, important distinction between idealized models and real human agents is that algorithms for the former can be exhaustive whereas those of the latter must almost invariably be merely heuristic. Heuristic algorithms are saddled with glitches, glitches that it’s meta-rational, albeit not firstorder rational, to court. Insofar as it’s almost invariably rational in mixed-motive games to attend to the algorithms of one’s co-players, it follows that it’s likewise almost invariably rational to attend to these glitches. But it’s almost invariably the case that the algorithms for accommodating these glitches in one’s co-player’s decision-heuristics will themselves be merely heuristic and therefore will themselves court glitches, which can themselves be exploited, albeit only by a glitch-ridden heuristic, and so on, almost, but not quite, ad infinitum. I say almost ad infinitum because finitude is built into the metaphysics of the real world. It follows that, since how deep one goes determines to whose advantage the information thus gleaned accrues, it’s the interests of one’s client which determine how deep one goes. So, it would seem, one can’t lay claim to political neutrality in the depth of one’s analysis. Consider, then, two organizations pitted against each other – the Association of Casino Owners and the Association of Gamblers – each of which has hired a game theoretician to identify and exploit the other’s glitches. The winner, it would seem, is the theoretician who goes one level deeper than his opponent. More to the point, however, the code of ethics for most professions would prohibit the same theoretician working for both sides. Why? Because he’d be in a position to money-pump. But whereas this money-pumping would be for him a windfall, for us it’s a deep and worrisome embarrassment.

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7. A Whiff of Conscience? To come clean, I don’t know how to ameliorate this embarrassment. Erstwhile we’d have wanted to claim that the game-theoretic reduction of ethics and politics is itself ethically and politically neutral. But what emerges now is that this can’t be said of any particular reduction. For what’s just been observed about glitch-embedding can likewise be said about just how many of the factors (cited in section 3) we might want to allow to complicate the model, and at just what level of finegrainedness. But if game-theoretic reduction is itself a weapon in social conflict, then likewise can it be a weapon in class conflict. The theoretician might try to assuage his conscience by reminding himself that conceptual innovations and their attendant apparatuses can be no more monopolized than any other technological innovations and their attendant apparatuses. But this is naive. Not only can casinos outspend gamblers, some ‘players’ have less conceptual plasticity than others. That is, the extended metaphors of war and bargaining that make up game theory are more ‘user friendly’ to people whose lives have been spent in the marketplace – men generally and particularly businessmen – than to people who haven’t – women generally and particularly mothers. So it should not surprise us that, as I noted in section 1, ‘much of the resistence to our reduction’ – from feminists and others concerned with asymmetries of power – ‘is driven not by skepticism but ... fear!’ As I say, I don’t know quite what to do with this worry, except to say that it is one. notes 1 Though the game-theoretic reductionist program got off the ground with J. von Neumann and O. Morgenstern’s Theory of Games and Economic Behavior (Princeton, NJ: Princeton University Press, 1944) and R.D. Luce and H. Raiffa’s Games and Decisions (New York: Wiley, 1957), what fuelled and promoted it since was in largest measure the nuclear arms race. 2 David Gauthier, The Logic of Leviathan: The Moral and Political Theory of Thomas Hobbes (Oxford: Clarendon Press, 1969). 3 David Gauthier, Morals by Agreement (Oxford: Clarendon Press, 1986). 4 David Gauthier, ‘Deterrence, Maximization, and Rationality,’ Ethics 94 (1984): 474–95. 5 As my friend and erstwhile colleague Karen Wendling once remarked.

502 Paul Viminitz 6 See ‘Artificial Prudence and Future Generations,’ manuscript. 7 Peter Danielson, Artificial Morality (London: Routledge, 1992). 8 That is, any interactive dilemma in which the payoff for unilateral defection is higher than for mutual cooperation, mutual cooperation higher than for mutual defection, and mutual defection higher than for unilateral cooperation. 9 By an iterated game is meant one in which players are free to alter their dispositions between rounds. By a non-iterated one is meant one in which they’re not. As Danielson, Artificial Morality, 45, rightly observes, ‘iterated games are not morally significant problems because they can be solved by straightforwardly rational agents.’ That is, if your cooperation in round n of a multiple-round Prisoners’ Dilemma is conditional upon, say, my having cooperated in all previous encounters, then provided I can anticipate a greater cumulative take by cooperating in rounds 1 through n – 1 than I can by defecting in one or more of those rounds, it’s straightforwardly rational for me to cooperate in those rounds. And if it’s straightforwardly rational for me to cooperate in those rounds, I’ll do so without the aid of morality. If, on the other hand, in order to elicit your cooperation in round n I must hard-wire myself to cooperate in that round – if, that is, my maximization must be truly constrained – then whether I cooperated or defected in previous rounds will be entirely irrelevant. 10 The reason, of course, is that the adoption of RC drives the UCs immediately below their dislodgement threshold, whereas the adoption of CC keeps them around to cooperate with in the second round. 11 Though for expositional simplification Gauthier opts for transparency over translucency, on page 174 of Morals by Agreement, he points out, quite rightly, that his counsel doesn’t depend on this simplification. 12 Consider, for example, a Swede with his fellow Swedes, versus one visiting Zaire. 13 That is, often enough the effort expended to appear other than one is outstrips the benefits of this dissimulation. 14 By toggle-ability I mean the propensity to revert to the default condition of UD and by plasticity the propensity to switch from one disposition to another. Feminists (and others) object that this privileging of UD is question-begging. I suspect they’re right. But one can duck this charge by making toggle-ability just a ‘special case’ of plasticity. 15 In a Bargaining Dilemma an agent is said to be broadly compliant if she’ll settle for even a marginal improvement in her take over what she’d get in the absence of mutual cooperation. An agent is said to be narrowly compliant if he –note the gender – will settle only for all but this marginal improve-

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16 17

18

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ment in the take of his co-bargainer. Since it’s Pareto-optimal to exhaust the bargaining space, it shouldn’t surprise us that pig-headedness expands to fill the room available to it in any partnership, especially marriage. In an as-yet-unpublished manuscript. Danielson, in Artificial Morality, declines to ‘follow ... Holy Smith’s demand [for] what we might call parametric robustness, an ability to do well against [a] wider variety of strategies, rational or not – like her contrived kingbreaker, which, by refusing to cooperate with CC, does worse than UC’ because doing so ‘would introduce an unlimited variety of players ... and quickly overwhelm [our] ability to manage complexity and advance our understanding of the issues’ (95–6). A strategy is said to satisfice just in case it keeps the agent above its dislodgement threshold. I’ve argued elsewhere – see my manuscript, ‘The Uses and Abuses of Maximizing/Satisficing Debate’, – that satisficing can itself be a meta-moral disposition. A further complication is what to do with dispositions that are viable only by the leave of those who are viable in their own right. Down’s syndrome might be a case in point. APr could stretch to accommodate such dispositions but, like most classical political philosophies – e.g., Hobbes, Locke, and Kant – it treats them instead as non-agents. Unsurprisingly feminists (and others) find this treatment offensive. Our defence, of course, is that non-agency doesn’t entail non-patiency. But this is likely to be small consolation to those with recalcitrant linguistic-turned-moral intuitions. A game of chess is declared drawn if neither a piece has been taken nor a pawn promoted for fifty moves. So the set of all possible chess games is finite. And since it’s finite, a sufficiently powerful computer can – and some day will – be built. But if the fifty-move rule is removed, the set of all possible chess games is infinite. For a detailed discussion of looping problems in virtual ethics, and solutions to them, see my ‘No Place to Hide – Campbell’s and Danielson’s Solutions to Gauthier’s Coherence Problem,’ Dialogue 35, no. 2 (1996): 235–40. For a full discussion of this problem see my ‘Simultaneous Attackers in Artificial Prudence,’ in New Studies in Exact Philosophy: Logic, Mathematics and Science, vol. 2, B. Brown and J. Woods, eds. (Oxford: Hermes Science Publishing, 2001), 273–8.

Part Five: Respondeo JOHN WOODS

What got me thinking about engineered death was, especially in the case of abortion, the massiveness and sheer speed of the collapse of received opinions, a paradigm shift in moral thinking about death. In Engineered Death,1 I set myself two questions about abortion. I was interested in reflecting on the conditions that gave rise to this paradigm shift. I also wanted to determine whether the dialectical wherewithal existed with which to defend the old way of thinking. In the first instance, I conjectured that the displacement of the theological conception of death by what I called the secular conception threw up some tricky metaphysical issues which, among other things, made it surprisingly difficult to give sense to human killing as an intrinsic wrong. As I now perceive, this conceptual change was part of a larger transformation in which a divine command morality was replaced by a secular motley of would-be contenders. This gives rise to interesting possibilities. One is that in the transition from theologically backed ethics to secular ethics, the nature of the warrant of ethical principles changed, but the content of them did not, or anyhow not much. I pressed my second question about the defensibility of the wrongfulness of abortion in the context of an affirmative presumption about the preservation of moral content. I made this assumption because, in the 1970s, there was ample evidence that on many ‘basic’ questions, moral content had not changed. This was indicated, in particular, in the widespread persistence of the view that murder is an awful and intrinsic moral crime. Under this assumption, human life does indeed have a trumping value, albeit defeasibly. The dialectical task of the pro-abortionist would then take one of two courses. It could be argued that the moral presumption in favour of human life doesn’t apply to fetuses –

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that fetuses don’t have human life, or aren’t persons, or whatever else. It could also be argued that there are defeasing conditions on that same presumption that we have mistakenly rejected or overlooked in the past; thus a new relevance was thought to be discernible in such factors as the sheer inconvenience of unwanted pregnancies or the demands of sexual equality. My view was, and is, that this second line of argument was bound to fail, as long as the principle that human life has trumping value is retained. This has led Michael Stingl in his closely examined chapter to criticize some of my arguments. He seems to express real surprise that I had simply not been able to see that human life is not a trump. Well, of course, perhaps it isn’t. Perhaps I should have seen that, in the abandonment of theological ethics, moral content – even about very basic things – was bound to change, and that those who lament the intrinsic awfulness of killing are somehow ludicrously passé. Still, I did make that assumption and did base my arguments against the defeasing force of inconvenience and sexual equality on it. Perhaps it was a silly assumption, although even today I tend to think not. Somewhat different things need saying about my RR-argument. My RR (Resolution Rule) resembles a minimax strategy. Stingl thinks, with Rawls, that minimaxing is a good move only when momentous matters are at issue, such as the social and political constitution of a country. Rawls’ saying so doesn’t make it so, of course, although its employment in Pascal’s Wager surely conforms to Rawls’ criterion. In my use of it, there was a further pair of presuppositions at work. It was assumed, first, that if the pro-abortion position chanced to be incorrect – a mistake of reason, if you like – then the wrong that attends indiscriminate feticide would be momentous. The second assumption was that there is a subclass of disputants about abortion for whom their opponents’ positions could be described as ‘real possibilities’ for them. Pascal’s Wager is a case in point. It is directed to lapsed Catholics in relation to the truths of Christian doctrine. If the real possibility condition is met, then even the Catholic atheist may well opt for the prudence of trying to reacquire Christian belief. But it won’t work for this same target audience with regard to, say, the teachings of Druidism. In my deployment, RR was aimed at a similarly circumscribed target audience. Stingl is right to say that the RR-argument would be laughed at by persons outside the intended ambit. But Pascal was well aware of this; and so was I. It is exceedingly difficult to get clear about real possibilities. Cer-

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tainly they are not modal entities in the manner of S5-possibilities, nor need they be attended by high degrees of subjective possibility. As I suggested in ‘Privatized Death,’2 we may have to make do for now with an operational characterization according to which something is a real possibility for a person x to the extent that x is prepared to take seriously RR-arguments with respect to it. There is a further class of pro-abortionists for whom personhood or some near thing is a condition on the wrongfulness of abortion, if abortion is indeed wrong. Of course, the nature and conditions of acquisition of personhood make for vexed metaphysical wrangles. Jane English is deeply right in finding this a regrettable situation. Those who root their position on abortion on different answers to this question risk the charge of moral unseriousness. The reason is that in ordo cognescendi, these metaphysical problems are certainly not less easy to get clear about than the ethics of abortion itself. So it is bad case-making for either side to proceed in this way. Jane English’s position is actually stronger than mine. Her view is that even if our concept of person were metaphysically unproblematic, wrangles about abortion cannot and need not be settled by recourse to it. Still, she also holds that whether a fetus is a person cannot be settled conclusively. Another claim that is central to her position is that even if fetuses are persons, it’s all right to kill them in many cases, and that even if a fetus is not a person, abortion would be wrong in many cases. Michael Wreen is right to observe that if any of these three supporting claims is defeated, the defence of the main thesis fails. He devotes considerable energy in attacking the second of this trio, on the whole convincingly, in my view. This makes me think that English’s strategy with regard to the second of Wreen’s three sub-theses was misconceived. To make her central case, it was not necessary to establish that even if fetuses were persons, aborting them would be justified in many cases. It suffices to show that it would be justified in some cases, and this would have notably blunted Wreen’s vigorous attack. Still, there are (or were) lots of people who hold that the personhood of fetuses suffices for the wrongfulness of abortion on demand. Many pro-abortionists believe (or used to believe) that the best we can say about the personhood of fetuses is that they are potential persons. It was precisely this group that I had in mind when, in Engineered Death, I pressed the argument that if personhood precludes indiscriminate feticide, so too does potential personhood. But the argument was strategic, not metaphysical. Michael Stingl recalls with verve the infamous Morgantaler debate.

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It was a difficult night for me, since I had told the organizers that I wanted no part of what passes for debate on cable television or in Canadian general elections. The organizers assured me that this would be a cross-examination debate (of a kind resembling Aristotelian refutations). My plan was to press Morgentaler in ways that would gradually show that he is not morally serious about the abortion issue. Of course, Morgentaler is not unserious about every aspect of abortion. He is not unserious about women’s rights or civil disobedience. But Morgentaler hasn’t got so much as a nanosecond of time for any argument designed to probe indiscriminate feticide morally or to call into question the prudence of policies that abet it. For some people in the audience this would have been enough to cost Morgentaler their support and it was them that I was after. In the event, the organizers had abandoned the cross-examination format and had forgotten to tell me about it. So I had to convert my attack from a Socratic evolution, point by point, to a form of exposition that was too complex for the new format. This irritated some of the attendees who hoped that Morgentaler would be more aggressively knocked about, and it tickled the daylights of those who already know that Morgentaler was a saint. But if, as Stingl says, it is no longer true that human life has trumping value, all of this is rather academic. But, then, I am simply at a loss as to how moral philosophy is to be done. This is not to say that I fail to see how moral philosophy is being done. When good and evil were matters of what God commands, it sufficed in the general case to know what God does command. For this there are legions of experts, theological specialists, whose obiter dicta would supplement established teaching. What was unneeded was sustained, rigorous, highly specialized training in ethical thinking. The hard work that was required was theological thinking. When all this collapsed, and when moral content started leeching into the sands of change, what was then needed was indeed sustained, rigorous, highly specialized training in ethical thinking. True, ancient models existed and offered some prospect of adaptation, but none fitted the texture of modernity in convincing ways. Slightly overstated, we found the utter want of what was now required. The implosion of religiously sanctioned ethics resembled the collapse of a taboo. One of the effects of making a taboo of something is to place it beyond the reach of our standard habits of case-making and justification. But once a taboo collapses, people find that they lack the dialectical savvy to defend its moral content, never having had to do so before. So morality changes. Even so, what was lacking for ethics was vigorously present for poli-

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tics and law – high levels of theoretically sophisticated thought that cut its teeth in seeking intellectually coherent and socially stable accommodations of difference. This left a gap for the stuff of morality to flow into. In the old way, moral disagreement was something for a sound ethical understanding to eliminate; but now it was for a sound political theory to accommodate, something it could do with relish. This was one of my suggestions in ‘Privatized Death,’ and I am not presently minded to give up on it. That being the case, we have an explanation for the suddenness and sheer scope of the abandonment of fetuses. It is that fetuses leave no political footprint, that they are, in this universe of endlessly negotiated self-demand, a constituency wholly without voice and without story. In light of all this, Paul Viminitz’s approach to artificial prudence strikes me as especially important in two respects, one exemplificatory and the other methodological. It exemplifies the extraordinary technical firepower one acquires with which to theorize about ethics and politics provided one is prepared to let ethics be politics. From all this formal sophistication we gain at least the sober promise of theoretical development of heretofore unparalleled subtlety. Also, finally, theory begins to comport with moral behaviour on the ground – the very behaviuor one should expect to see once it is recognized that the transformation from theological to secular ethics cannot be expected to preserve moral content. The methodological advantage relates to this directly. It is that in its suppositions about rationality, the artificial prudence approach favours the attainment of adequacy over superiority, thus emphasizing satisficization over optimization. This is music to my ears; some of the reason why is to be found in the opening pages of Gabbay and Woods’ ‘Filtration Structures and the Cut-Down Problem for Abduction.’ Yet a third advantage of Viminitz’s interest in satisficing is that it equips artificial prudence with the wherewithal to model the evolutionary turn in ethics, which is the one other place in which there is serious prospect of producing theoretical structures that are actually weight-bearing, rather than trivial or merely decorative. Let us, all the same, not keep ourselves in the dark about the transformations of modernity. When God called the shots, there was no need of moral theory, never mind that medieval scholar-clerics were free to take a crack at it if they wished. Now that God no longer calls the shots, there is a need for a stable and comprehensive moral theory, which has yet to appear. Perhaps it will never appear. Meanwhile, to great clamour, we make do with the pretence that ethics is politics.

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notes 1 Engineered Death: Abortion, Suicide, Euthanasia, Senecide (Ottawa: University of Ottawa Press/Éditions de l’Université d’Ottawa, 1978). 2 ‘Privatized Death: Metaphysical Discouragements of Ethical Thinking,’ in Midwest Studies in Philosophy 24, Peter A. French and Howard K. Wettstein, eds. (Boston, MA: Blackwell Publishers, 2000), 199–217.

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Contributors

Peter Alward is Assistant Professor of Philosophy at the University of Lethbridge. His research is in the philosophy of language, the philosophy of mind, and metaphysics. He is currently working on mental causation and the semantics of fictional names. Paul Bartha is Associate Professor of Philosophy at the University of British Columbia. His research interests include general philosophy of science, foundations of probability and decision theory, and inductive inference. George Boger is Professor of Philosophy at Canisius College, Buffalo, New York. Recent publications treat the metalogical sophistication of Aristotle’s logic, his treatment of fallacious reasoning, and Aristotle’s and Plato’s political philosophy. Recent research treats the formal logic underlying fallacy theory. Leslie Burkholder is Senior Instructor in the Department of Philosophy, University of British Columbia. His interests include logic, critical thinking, and computer applications in instruction. He has published ‘Computing’ in A Companion to the Philosophy of Science, W.H. NewtonSmith, ed. (Blackwell, 2001). Jim Cunningham is a Reader in Computer Science at Imperial College London, where he leads a research group in Communicating Agents. He has published on rationality in machines. In recent years he has also taught Human Computer Interaction and Natural Language Processing, and has been engaged in several European projects developing Software Agent Technology.

512 Contributors

Darcy A. Cutler currently teaches at the University of British Columbia and Douglas College. His research interests include history and philosophy of logic, foundations of mathematics, and philosophy of physics. David DeVidi is Associate Professor of Philosophy at the University of Waterloo. His recent research is mostly in philosophical logic, philosophy of mathematics, and analytical metaphysics. He is co-author, with J.L. Bell and the late Graham Solomon, of Logical Options (Broadview, 2001), and has articles in various journals, including Journal of Philosophical Logic, Australasian Journal of Philosophy, Synthese, and Mathematical Logic Quarterly. Lisa Lehrer Dive recently completed her doctorate at the University of Sydney, Australia. Her research interests are in epistemology and metaphysics, with a focus on mathematical knowledge and the ontology of mathematics. Her thesis project was the development of an epistemically driven physicalist philosophy of mathematics. James B. Freeman is Professor of Philosophy at Hunter College of The City University of New York. His research is in informal logic and argumentation theory. He is the author of Thinking Logically (Prentice Hall, 1988, 1993), Dialectics and the Macrostructure of Arguments (Foris, 1991), and Acceptable Premises: An Epistemic Approach to an Informal Logic Problem (Cambridge University Press, 2005). Dov M. Gabbay is Augustus de Morgan Professor of Logic and Professor of Philosophy and Computing Science at King’s College, London. He is author of over two hundred papers and monographs. Recently he has published The Reach of Abduction (North-Holland, 2005) in collaboration with John Woods. Trudy Govier is Associate Professor of Philosophy at the University of Lethbridge. She is the author of many articles and books including A Practical Study of Argument (Wadsworth, six editions), The Philosophy of Argument (Vale Press, 1999), and Forgiveness and Revenge (Routledge, 2002). Nicholas Griffin is Director of the Bertrand Russell Centre at McMaster University, Hamilton, Ontario, where he holds a Canada Research

Contributors 513

Chair in Philosophy. He has written widely on Russell and is the author of Russell’s Idealist Apprenticeship (Clarendon, 1991), the editor of Russell’s Selected Letters (Routledge, 2002), and general editor of The Collected Papers of Bertrand Russell (Allen & Unwin, 1983). David Hitchcock is Professor of Philosophy at McMaster University in Hamilton, Ontario. He is the author of Critical Thinking (Methuen, 1983) and of articles in informal logic, the theory of argumentation, ancient Greek philosophy, and the history of logic. He is co-author (with M. Jenicek) of Evidence-Based Practice: Logic and Critical Thinking in Medicine (AMA Press, 2005). Andrew D. Irvine is Professor of Philosophy at the University of British Columbia. His edited and authored books include Bertrand Russell: Critical Assessments (Routledge, 1999), Argument: Critical Thinking, Logic and the Fallacies with John Woods and Douglas Walton (Prentice-Hall, 2000), and David Stove’s On Enlightenment (Transaction, 2003). Irvine is a founding member of the editorial board of the online Stanford Encyclopedia of Philosophy. Dale Jacquette is Professor of Philosophy at the Pennsylvania State University. He is the author of articles on logic, metaphysics, philosophy of mind, and Wittgenstein. He has published David Hume’s Critique of Infinity (Brill, 2001), Ontology (McGill-Queen’s University Press, 2002), and On Boole (Wadsworth/Thomson Learning, 2002). He has edited The Blackwell Companion to Philosophical Logic (2002) and The Cambridge Companion to Brentano (2004). R.E. Jennings is Professor of Philosophy at Simon Fraser University. His publications on logic, philosophy of language, and philosophy of mind include The Genealogy of Disjunction (Oxford University Press, 1994). Matthew McKeon is Assistant Professor of Philosophy at Michigan State University. His major research interests are validity in intensional languages, the role that the concept of necessity plays in grounding logical consequence, the relation of set theory to logic, and theories of what counts as a logical constant. Kent A. Peacock is Associate Professor of Philosophy at the University

514 Contributors

of Lethbridge. His interests include philosophy of physics and the environment, and he has published Living with the Earth: An Introduction to Environmental Philosophy (Harcourt Brace Canada, 1996). Victor Rodych is Associate Professor of Philosophy at the University of Lethbridge. He has published widely on Wittgenstein’s philosophy of mathematics and the philosophy of Karl Popper, including ‘Popper versus Wittgenstein on Truth, Necessity, and Scientific Hypotheses,’ Journal for General Philosophy of Science (2003). Samuel Ruhmkorff is Assistant Professor of Philosophy at Simon’s Rock College of Bard in Great Barrington, Massachusetts. His research focuses on inference to the best explanation, probabilism, and reliabilism. Barry Hartley Slater, a graduate of Cambridge and Kent Universities, is now Honorary Senior Research Fellow in Philosophy at the University of Western Australia. His research interests are philosophical logic and aesthetics, with special reference to the epsilon calculus, paradoxes, and fictions. He has published over one hundred journal articles and four books, the latest being Logic Reformed (Peter Lang, 2002). Michael Stingl has been a member of the University of Lethbridge philosophy department since 1989. His main research is in bioethics and evolutionary ethics. He is at work on a book on evolution and ethics with John Collier, and is also the principal investigator with Alberta’s Provincial Health Ethics Network (PHEN) on a project on the just allocation of health resources within a regionalized health system. Jonathan Strand is Associate Professor of Philosophy at Concordia University College of Alberta. His primary research interests are philosophy of religion and philosophical logic, in particular the semantics and logic of English conditionals. Bas C. van Fraassen is McCosh Professor of Philosophy, Princeton University. Recent research interests include empiricism in the philosophy of science; epistemology compatible with empiricist scruples; and relations between the sciences, the arts, and literature. Publications include The Scientific Image (Oxford University Press, 1980), Laws and Symmetry (Oxford University Press, 1989), Quantum Mechanics: An

Contributors 515

Empiricist View (Clarendon, 1991), and The Empirical Stance (Yale University Press, 2002). Paul Viminitz teaches philosophy at the University of Lethbridge, specializing in game theory, philosophy of war, political philosophy, and theodicy. Jarett Weintraub is a PhD candidate in Philosophy at the University of California, Riverside. Currently, he is an instructor at Crafton Hills College. He is working on his dissertation on the role of the Formula of Universal Law in Kant’s Groundwork. John Woods is Director of the Abductive Systems Group at the University of British Columbia and Adjunct Professor at the Universities of Lethbridge and British Columbia. He was formerly Professor and Chair of Philosophy at the University of Lethbridge and President of that university. He is also the Charles M. Peirce Professor of Logic in the Logic and Computation Group at King’s College London. He is the author of numerous papers and books on logic, argumentation theory, and philosophy. Michael Wreen is Professor of Philosophy at Marquette University. His research interests include argumentation theory, the philosophy of logic, aesthetics, and death-related ethical issues. He has published about eighty articles in a variety of books and journals.

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Books by John Woods

Necessary Truth (New York: Random House, 1969). (Co-edited with L.W. Sumner.) Proof and Truth (Toronto: Peter Martin Associates, 1974). The Logic of Fiction: A Philosophical Sounding of Deviant Logic (The Hague and Paris: Mouton and Co., 1974). Engineered Death: Abortion, Suicide, Euthanasia, Senecide (Ottawa: University of Ottawa Press/Éditions de l’Université d’Ottawa, 1978). Formal Semantics and Literary Theory (Amsterdam: North-Holland, 1979). (Coedited with Thomas Pavel.) The Importance and Relevance of the Humanities in the Present Day (Waterloo: Wilfrid Laurier University Press, 1979). (Co-edited with Harold Coward.) Argument: The Logic of Fallacies (Toronto and New York: McGraw-Hill, 1982). (With Douglas Walton.) Fallacies (Dordrecht: Reidel, 1987). (Edited, with a Preface.) Fallacies: Selected Papers 1972–1982 (Dordrecht and Providence, RI: Foris Publications, 1989). (With Douglas Walton.) Critique de l’Argumentation (Paris: Editions Kimé, 1992). (With Douglas Walton.) Fundamentals of Argumentation Theory: A Handbook of Classical Backgrounds and Contemporary Developments (Hillsdale, NJ, and London: Erlbaum, 1996). (With Frans H. van Eemeren, et al.) Handboek Argumentatietheorie (Groningen: Martinus Nijhoff uitgevers, 1997). (With Frans H. van Eemeren, et al.) Human Survivability in the Twenty-First Century (Toronto: University of Toronto Press, 1999). (Edited with David Hayne.) Argument: Critical Thinking Logic and The Fallacies (Toronto: Prentice-Hall, 2000). (With Andrew Irvine and Douglas Walton; 2nd ed., 2004.) Aristotle’s Earlier Logic (Oxford: Hermes Science Publications, 2001).

518 Books by John Woods Logical Consequence: Rival Approaches (Oxford: Hermes Science Publications, 2001). (Edited with Bryson Brown.) New Essays in Exact Philosophy: Logic, Mathematics and Science (Oxford: Hermes Science Publications, 2001). (Edited with Bryson Brown.) Handbook of the Logic of Argument and Inference: The Turn toward the Practical, volume 1 in the series Studies in Logic and Practical Reasoning (Amsterdam: North-Holland, 2002). (Edited with Dov M. Gabbay, Ralph H. Johnson, and Hans Jürgen Ohlbach.) Agenda Relevance: An Essay in Formal Pragmatics, volume 1 of the series The Practical Logic of Cognitive Systems (Amsterdam: North-Holland, 2003). (With Dov M. Gabbay.) Handbook of the History of Logic, vol. 1: Greek, Indian and Arabic Logic (Amsterdam: North-Holland, 2003). (Edited with Dov M. Gabbay.) Handbook of the History of Logic, vol. 3: The Rise of Modern Logic I: Leibniz to Frege (Amsterdam: North-Holland, 2003). (Edited with Dov M. Gabbay.) Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences (Cambridge: Cambridge University Press, 2003). The Death of Argument: Fallacies in Agent-Based Reasoning (Dordrecht and Boston: Kluwer, 2004). The Reach of Abduction: Insight and Trial, volume 2 of the series The Practical Logic of Cognitive Systems (Amsterdam: North-Holland, 2005). (With Dov M. Gabbay.) Handbook of the History of Logic, vol. 6: Logic and the Modalities in the Twentieth Century (Amsterdam: North-Holland, 2005). (Edited with Dov M. Gabbay.) Handbook of the History of Logic, vol. 5: Logic from Russell to Gödel (Amsterdam: North-Holland, 2006). (Edited with Dov M. Gabbay.) Handbook of the History of Logic, vol. 7: The Many Valued and Non-Monotonic Turn in Logic (Amsterdam: North-Holland, 2006). (Edited with Dov M. Gabbay.) Handbook of the History of Logic, vol. 10: Topics in Classical 20th Century Logic: Sets, Recursion and Complexity (Amsterdam: North-Holland, 2006). (Edited with Dov M. Gabbay.) Seductions and Shortcuts: Fallacies in the Cognitive Economy, volume 3 of the series The Practical Logic of Cognitive Systems (Amsterdam: North-Holland, 2006). (With Dov M. Gabbay.) Handbook of the History of Logic, vol. 8: Logic: A History of Its Central Concepts (Amsterdam: North-Holland, forthcoming). (Edited with Dov M. Gabbay.) Handbook of the History of Logic, vol. 11: Topics in Classical 20th Century Logic: Models, Categories & Proof Systems (Amsterdam: North-Holland, forthcoming). (Edited with Dov M. Gabbay.) The Handbook of the Philosophy of Science, 16 volumes (Amsterdam: North-Hol-

Books by John Woods 519 land, to appear beginning in 2005) (Edited with Dov M. Gabbay and Paul Thagard.) Handbook of the History of Logic, vol. 9: Inductive Logic (Amsterdam: North-Holland, forthcoming). (Edited with Dov M. Gabbay.) Handbook of the History of Logic, vol. 2: Mediaeval and Renaissance Logic (Amsterdam: North-Holland, forthcoming). (Edited with Dov M. Gabbay.) Handbook of the History of Logic, vol. 4: British Logic in the Nineteenth Century (Amsterdam: North-Holland, forthcoming). (Edited with Dov M. Gabbay).

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Index

abduction. See logic abortion, 6–7, 453–69, 474–89, 504–8 actualism. See realism Adams, Ernest, 275 additivity, axiom of. See probability ad hominem. See fallacy Agenda Relevance. See Gabbay, Dov M.; Woods, John Alexander of Aphrodistas, 227–8, 242 Algra, Kempe, 237 Aliseda-Lera, Atocha, 415 Allen, Derek, 352–3, 357, 360–1, 437 Allen, J.F., 383–4, 395, 396 Alston, William, 195, 355, 362 Alward, Peter, 102, 104–5, 107–8, 511 Amato, Joseph A., 460, 471 Amis, Kingsley, 33 anaphora, 35, 43 ancient logic. See logic Anderson, Alan Ross, 409 Anderson, Ardis, 470 Anderson, Marjorie C., 470 anti-realism. See realism Apollonius Dyscolus, 238 Apuleius, 240 Argumentation, Communication and

Fallacies. See Grootendorst, Rob; van Eemeren, Frans argument theory, 418–37; WoodsWalton approach, 421–6, 430, 449– 50. See also fallacy; logic, formal vs informal; pragma-dialectics Aristotle, 3, 137, 321, 409, 430–1, 441, 483; Organon, 321; Prior Analytics, 212, 215, 322, 440; sea battle, 179– 81; Sophistical Refutations, 227, 237, 436–7; Topics, 227, 436. See also logic Armstrong, David M., 171 Arrington, Robert L., 471 Arrow, Kenneth J., 469 Artificial Morality. See Danielson, Peter Atlantis, 27 Aulus Gellius, 238 Austen, Jane, 310, 319 Austin, J.L., 32 authority, reasoning from, 334–47, 435 Baars, B.J., 392, 396 Bach, E., 396 Bachman, James, 157

522 Index barber paradox. See Russell, Bertrand Barnes, Jonathan, 237 Barrett, Robert B., 302, 319 Bartha, Paul, 163, 171–2, 200, 511 Bayesianism/Bayes’ theorem, 125–8, 199–200, 331–47, 444–5 Beaney, Michael, 91 begging the question. See fallacy behaviourism, 23, 461–3, 465 being, theory of, 52 belief, degrees of, 122–30, 173–4 Bell, J., 171 Belnap, Nuel, 409 Benacerraf, Paul (dilemma), 64–6, 68, 76, 91–2 Bentham, Jeremy, 477 Berlinski, D., 258, 262–3 Bernays, Paul, 262 Birbaum, M.H., 417 bivalence, law of. See logic, laws of Blackburn, P., 388, 395–6 Blair, J. Anthony, 157, 348, 359–61, 437–9, 450 Bobzien, Susanne, 224, 237–8, 241–2 Boger, George, 326, 441, 449–50, 511 Boghossian, P., 258 Boole, George, 224, 237 Boolos, George, 222–3, 247, 258–9, 262 Bourbaki, Nicholas: Éléments de Mathématique, 34, 47 Boyd, Richard, 190, 195 Bozic, M., 295 Bradshaw, G.L., 416 Brady, R.T., 275 Brandom, Robert, 102 Bratman, Michael, 381, 395, 397 Brown, Bryson, 277, 503 Burkholder, Leslie, 444, 511

Bury, R.G., 238 Calvin, William C., 300, 319 Canfield, John V., 471 Cantor, Georg, 64, 67, 86 Carnegie, Andrew, 465 Cavender, Nancy, 347, 438 Change in View. See Harman, Gilbert Chatterton, Thomas, 28 Chellas, Brian F., 62 Chihara, Charles, 62–3 choice, axiom of. See set theory, axioms of Church, Alonzo, 219, 440; Church’s thesis, 212 Churchland, Paul M., 132, 195, 415 Chrysippus, 224–6, 231–6, 238, 321– 3 Cicero, Marcus Tullius, 238–9, 366 cognitive science, 112, 401, 420 Cohen, Jonathan, 198, 203 Cohen, Paul J., 89 Coleman, Athena V., 361 Colver, A. Wayne, 471 completeness, 207–21, 322–3 composition, fallacy of. See fallacy computability, 211–12 connective, 301–18, 326 consequence relation 324, 432–3; logical vs model-theoretic, 207–21; substitutional vs model-theoretic, 243–57; transitivity vs non-transitivity of, 264–75, 433 consolation argument, 364–78 constructive empiricism. See empiricism continuum hypothesis, 69 contractarianism, 492 contradiction, law of. See logic, laws of

Index 523 Copernicus, Nicolaus, 382, 463 Copps, Sheila, 374 Corcoran, John, 207–8, 212–16, 219, 221–2, 420, 430, 438, 440 counterfactuals, 6, 264–75 Cresswell, M.J., 38, 47–8, 61 Crews, Frederick, 471–2 Critias. See Plato critical thinking theory, 418 Crossley, J.M., 295 Cunningham, Jim, 383, 395–6, 447–9, 511 cut down problem, 404, 406–11 Cutler, Darcy, 321–2, 512 Czermak, J., 47 Dafoe, Daniel: Journal of the Plague Year, 313, 319 Danielson, Peter, 497, 499; Artificial Morality, 493, 502–3 D’Arms, Justin, 368–70, 378 Darwin, Charles, 463–4; Origin of Species, The, 491 Darwinism. See evolution; social darwinism Davidson, D., 385, 396 De Bono, Edward, 336–8, 347 deduction theorem, 225, 233–4 de Finetti, B., 158–60, 162, 166, 171 de Finetti lottery, 158–70, 200 de Lacy, Phillip 239 Democritus, 365 Dennett, Daniel, 392, 397, 491 descriptions. See Russell, Bertrand Desmond, Adrian, 472 DeSousa, Ronald, 319 determinacy, axiom of. See set theory, axioms of DeVidi, David, 294, 324–5, 512 de Vincentis, Mauro Nasti, 239

Dewey, John, 462, 464–5, 471; Ethics, 456, 469 diagonal argument, 64, 67 Dickens, Bernard M., 472 Diels, Hermann, 239 Diocles of Magnesia, 238–42 Diogenes Laertius, 238 Dive, Lisa Lehrer, 106–7, 325, 512 Donnellan, Keith, 35–6, 38, 47 Dosen, K., 295 Dowty, D.R., 395 Doyle, Arthur Conan: Sherlock Holmes, The Adventures of, 16–19, 21, 24–28, 33, 45, 104–5 Duhem, Pierre, 79 Dummett, Michael, 279, 295 Dunn, M., 133 Durant, Will, 459, 470 Dutch Book Argument, 158, 162–5, 170, 200 Dyscolus, 239 Earman, John, 133 Ebbinghaus, Heinz-Dieter, 70–1, 77 Edgington, Dorothy, 282 Edwards, Paul, 470 Edwards, W., 171 Einstein, Albert, 72 Éléments de Mathématique. See Bourbaki, Nicholas emotivism, 461, 463, 465 empirical adequacy. See empiricism, constructive empiricism, 135–6, 140, 155, 460; constructive, 111–19, 188–90, 199 Engineered Death. See Woods, John English, Jane, 474–89, 506 entailment. See consequence relation Epictetus, 366 epistemic logic. See logic

524 Index epistemology, 8–10, 65–6, 111–131, 135–56, 183, 199–200, 220, 349–50, 352, 356–9, 420, 432; naturalized, 131 epsilon logic, of fiction, 33–47, 104–5 Etchemendy, John, 222, 263 euthanasia, 6–7, 458–9 evolution, 188, 300, 303, 463–4 excluded middle, law of. See logic, laws of exhaustiveness, 216–20 explanation, inference to the best, 138, 183–94, 201–2 externalism, vs internalism, 142, 183, 191, 193–4 Fainsinger, Robin L., 470 fallacy, 3, 8–10, 150–3, 321–2, 351, 367, 380, 402, 418, 422–3, 430, 432– 6; ad hominem, 435; begging the question, 424, 434; composition, 108; hasty generalization, 152; heuristic, 6, 326; moralistic, 369; problematic premiss, 348; two wrongs, 373. See also argument theory, Woods-Walton approach Farr, Daniel, 450 Faust, 28–30, 36 Feinberg, Joel, 490 Feldman, Richard, 349–50, 359 Ferguson, G., 396 fiction: genealogy of, 26–30; logic of, 6–7, 33–47, 49–60, 103–6; paradox of, 46; semantics of, 15–30. See also semantics, say-so Field, H., 247, 255, 258, 262–3 Fillmore, C.J., 396 Fine, Arthur, 132, 190–1, 195 Finocchiaro, Maurice A., 437–9 first-order logic. See logic

Fisher, R.A., 417 Fitch, Frederic B., 229, 241 Forbes, Graeme, 62 formalism, 89, 219, 301, 461 formal system, 61, 64, 72–6, 82–3, 106–7, 208–11, 225–31 Franklin, James, 197, 203 Frede, Michael, 224, 237 free logic. See logic Freeman, James B., 362–3, 437, 445, 512 Frege, Gottlob, 23, 82, 83–4, 91, 438; intensions, theory of/names, theory of, 36, 43, 52 French, Peter A., 469, 509 Freud, Sigmund, 459, 462, 470 Gabbay, Dov M., 3, 11–12, 108, 134, 151, 157, 199, 222, 327, 380, 395, 415, 439, 443, 448, 450, 508, 512; Agenda Relevance, 11, 447, 449; Practical Logic of Cognitive Systems, A, 10, 399, 449–50; Reach of Abduction, The, 11, 78, 90, 92, 156, 398–9, 414, 416, 449 Galen, 238–9, 241 Gallin, D., 258, 262–3 game theory, 491–501 Gasper, Philip, 195 Gauthier, David: Logic of Leviathan, The, 492, 501; Morals by Agreement, 492–3, 501–2 Gay, Peter, 470 Gentzen, Gerhard, 240 geometry, 84, 89, 208–9; parallel postulate, 89 Georgeff, M.P., 397 Gergen, Kenneth J., 416 Gettier problem, 141 Gigerenzer, Gerd, 412–13, 416–17

Index 525 Gillies, Donald, 416 Goble, Lou, 275 Goddard, L., 39–40, 43–4, 47–8 Gödel, Kurt: completeness theorem, 210–11, 217; incompleteness theorems, 70, 75, 107, 210–11, 218–19, 498; pragmatism, 65, 68, 76, 78, 81, 85–92 Goethe, Johann Wolfgang von, 29 Goldfarb, Warren, 223 Goldman, Alvin I., 353–4, 362 Govier, Trudy, 349–50, 352, 359, 438, 445–7, 512; Practical Study of Argument, A, 375, 379 Gradiva. See Jensen, Wilhelm Graham, George, 369, 378 Graumann, Carl, 416 Greenbaum, S., 396 Grice, Paul, 302, 304, 306–9, 315–16, 318–19 Griffin, Nicholas, 103–5, 512 Grootendorst, Rob, 157, 360–1, 438; Argumentation, Communication and Fallacies, 372–3, 379 Guenthner, F., 108, 450 Gulliver’s Travels. See Swift, Jonathan Gupta, A., 133 Haack, Susan, 73, 77 Hacking, Ian, 132 Hájek, Alan, 275 Halonen, Ilpo, 157 Halpern, Joseph Y., 133, 383, 395 Hamblin, C.L., 420, 437 Hamlet. See Shakespeare, William Hansen, Hans V., 361, 438–9, 450 Hanson, N.R., 413, 417 Hanson, W., 247, 258–9, 261, 263 Hansson, Sven Ove, 415 Harel, D., 396

Harman, Gilbert, 200, 471; Change in View, 173–6, 178–82, 406, 416 Harms, R.T., 396 Harper, W.L., 276 Harsanyi, John C., 469 Hart, W.D., 76 hasty generalization. See fallacy Heiberg, Johan L., 240 Heidegger, Martin, 459–60, 470 Herbrand, Jacques, 217, 223 heuristic fallacy. See fallacy Heyting, Arend, 291 Hilbert, David, 34, 82, 104, 219, 262 Hinman, P., 247, 253, 257–9, 262 Hintikka, Jaakko, 49–50, 61, 102, 136, 157; Knowledge and Belief, 120, 132 Hintikka’s problem, 120 Hitchcock, C., 163, 171 Hitchcock, David, 321–2, 364, 438, 513 Hobbes, Thomas, 477, 493, 499, 503 Hodkinson, I., 395 holism, 250, 257 Hooker, Clifford A., 132, 195 Horgan, T., 415 Howson, Colin, 158, 160, 171 Hughes, G.E., 38, 47–8, 61 Hughs, R., 222 Hülser, Karlheinz, 224, 237 Hume, David, 150, 155, 370, 464, 471 Husserl, E., 438 Huxley, Henry, 464–5 Hyde, Dominic, 294 hypothetico-deductive method, 433–4 identity, law of. See logic, laws of Ierodiakonou, Katerina, 241 impossible worlds, 267–72 indispensability argument, 78–90

526 Index inference to the best explanation. See explanation infinitary logic. See logic infinity, actual vs potential, 217, 322 infinity, axiom of. See set theory, axioms of internalism. See externalism interrogative logic. See logic intuitionistic logic. See logic Irvine, Andrew D., 77, 92, 213, 222, 347, 513 Jacobson, Daniel, 368–70, 378 Jacquette, Dale, 62, 105–6, 323, 325, 513 Jané, Ignacio, 67, 76 Jeffrey, Richard, 133, 171, 185, 222–3 Jennings, Ray, 275, 319, 325–6, 513 Jensen, Wilhelm: Gradiva, 33, 46 Johns, Richard, 171–2 Johnson, Ralph H., 11, 157, 348, 351– 3, 359–61, 363, 437–9, 450 Johnstone, Henry, 9–10 Kadane, Joseph, 133 Kahane, Howard, 347, 438 Kahneman, David, 198, 203 Kalbfleisch, Karl, 238 Kant, Immanuel, 84, 121, 155, 382, 395, 503 Kasparov, Gary, 492–3 Keefe, Rosanna, 294 Kelly, H.H., 412, 417 Kelly, Kevin, 158, 166, 170 Kempson, Ruth M., 157 Kepler, Johannes, 382 Keynes, John Maynard, 415 Kim, J., 257–9 knowledge, theory of. See epistemology

Koch, Sigmund, 416 Koetschau, Paul, 241 Kolenda, Konstantin, 471 Kozen, D., 396 Kripke, Saul A., 49–50, 54, 58–9, 61–2, 93; Naming and Necessity, 51, 97, 102 Kripke semantics. See semantics Kripke’s puzzle, 93–101, 107–8 Kruger, L., 416 Kuipers, Theo, 157 Kyburg, Henry, 173, 181, 200 Laan, David Vander, 276 Lackey, Douglas, 92 Langford, C.H., 276 Langley, P., 416 language: formal vs natural, 73–5, 326; object language vs meta-language, 249, 256. See also semantics Laudan, Larry, 195 laws of thought. See logic, laws of Lear, Jonathan, 207–8, 211–15, 218– 19, 221–2 Leary, David E., 416 Leblanc, Hughes, 62 Leech, G., 396 Leibniz’ Law, 34 Leisenring, A.C., 47 Leith, M., 383, 386, 395–6 Leplin, J., 132 Levine, D., 417 Lewis, C.I., 49, 61, 272–4, 276 Lewis, David K., 61, 171, 276–7 Lewis’ Principal Principle, 161 liar paradox, 41, 200 libertarianism, 463, 465–7 Lilly, Reginald, 471 Lindman, H., 171 Linsky, L., 102

Index 527 Lipton, Peter, 187, 195 Locke, John, 9–10, 125, 503 logic, 8, 381, 394–5, 418; abductive, 134–5, 380, 398–414, 448; ancient, 207–21, 224–37; Aristotelian, 9, 207–21, 224–5, 228, 231, 235, 321–4, 434–6; of belief, 120; connectionist, 400–1; of discovery, 405–6, 411–14; dynamic, 389; epistemic, 143; firstorder, 73, 208, 216–18, 220, 243–57, 388; formal vs informal, 348, 418– 37; free, 15–16, 22, 33, 37, 104; infinitary, 71; interrogative, 136; intuitionistic, 281, 283–94; laws of, 22–3, 34, 89, 231, 250, 283–4, 290, 432, 447; many-valued, 33, 44, 59– 60, 104; modal, 6, 49–60, 105, 120, 143–4, 198, 283, 288–94, 324, 383–8, 393; plausibility, 134–5, 137, 139, 197–8, 201, 342–7, 407–11; propositional, 422; of relations, 24–5, 105; relevant, 8, 324, 407–10; secondorder, 210, 218; sentential, 217; Stoic, 224–37; tense, 380–95. See also fallacy; fiction, logic of logical consequence. See consequence relation logical positivism, 460, 463 logical truth, 243–4, 249–50, 253–7 Logic of Fiction, The. See Woods, John logicism, 83–4 Logic of Leviathan, The. See Gauthier, David Long, H.S., 238 Lopes, L.L., 417 lottery paradox, 173–81, 200–1, 344–5 Löwenheim-Skolem theorem. See Skolem-Löwenheim theorem Luce, R.D., 501 Sukasiewicz, Jan, 59, 438

Lyons, David, 469 Mackie, John L., 471 Maddy, Penelope, 65, 76, 81, 87, 89, 91–2 Maher, P., 171 Malory, Thomas: Morte d’Arthur, Le, 27 Mansfeld, Jaap, 237 many-valued logic. See logic Marcus, Ruth Barcan, 47 Mares, Edwin, 276 Marlowe, Christopher, 29–30 Marshall, P.K., 238 Martinez, C., 440 Massey, Gerald J., 438 materialism, 463 Mates, Benson, 224, 237 McCall, S., 171 McDonnell, Kathleen, 469 McKeon, Matthew, 323, 325, 513 McTaggart, J.M.E., 389 Meinong, Alexius von: theory of objects, 15–16, 20–2, 24, 30–31, 33, 36, 103 Mellor, D.H., 389, 396 Menand, Louis, 470 Mendelson, E., 259 Menzel, Christopher, 62 Meyer, Nicholas: Seven Per-Cent Solution, The, 25, 27 Meyer, R., 39–40, 43–4, 47–8, 275 Mill, John Stewart, 464–5; theory of names, 36 Milne, Peter, 232, 241–2 minimum mutilation, principle of, 80 modal actualism. See realism, vs actualism modality, alethic, 58–60 modal logic. See logic

528 Index modal realism. See realism, vs actualism modal semantics. See semantics model-theoretic consequence. See consequence relation model-theoretic semantics. See semantics, formal Moens, M., 385, 387, 396 Moggridge, D.E., 415 Mohr, Heinrich, 471 Mongin, Philippe, 133 Montague, R. 41, 48 Moore, G.E., 8 Moore’s paradox, 120, 124, 127 moralistic fallacy. See fallacy Morals by Agreement. See Gauthier, David Moreschini, Claudio, 240 Morgan, M.S., 416 Morgenstern, O., 501 Morgentaler, Henry, 453–6, 468, 506– 7 Morscher, E., 47 Morte d’Arthur, Le. See Malory, Thomas; Tennyson, Alfred Lord Muggleton, S., 396 Mulroney, Brian, 374 multiplicative axiom. See set theory, axioms of Murcock, B.B., 417 Murray, Malcolm, 497 Mutanen, Arto, 157 Myers, Frederic W.H., 471 Naming and Necessity. See Kripke, Saul A. Newell, Alan, 415 Newton, Isaac, 187 Nickel, Dawn D., 470 Nietzsche, Friedrich, 134

Nightmares of Eminent Persons. See Russell, Bertrand Nixon, Richard, 311 non-contradiction, law of. See logic, laws of non-monotonicity, 232–3 nonsuches, 16–21, 24, 26–7, 103–4 Norman, D.A., 417 Northcott, Herbert C., 470 Norton, John, 172 Nunberg, G.: indexicality, theory of, 99, 102 Occam’s razor, 304 Ohlbach, Hans Jürgen, 11, 157 Ono, H., 295 ontological commitment, criterion of, 78, 80–1 Organon. See Aristotle Origen, 241 Origin of Species, The. See Darwin, Charles Our Gang. See Roth, Philip Pap, A., 247, 258 Pappas, George, 361 paradox. See fiction; liar paradox; lottery paradox; Moore’s paradox; Putnam’s paradox; relabelling paradox; Russell, Bertrand, barber paradox; Russell’s paradox; Simpson’s paradox; Skolem paradox; sorites paradox parallel postulate. See geometry Pareto conditions, 128, 130 Parsons, Terence, 22, 24–25, 31–32, 388, 396 Pascal, Blaise, 197; wager, 505 Peacock, Kent, 157, 513 Peacocke, C., 258

Index 529 Peano arithmetic, 81–2, 217–18 Pearce, G., 276 Peirce, Charles Sanders, 141, 362, 398 Pepys, Samuel: Diary, 311, 313 perfectibility, 213–14, 322 phenomenology, Husserlian, 87–90 Philoponus, Ioannes, 239 Pigozzi, Gabriella, 415 Pinto, Robert C., 437–9 Planck, M., 403 Plantinga, Alvin, 61–2, 276, 358, 362– 3 Plato: Critias, 27; Theaetetus, 141; Timaeus, 27 Platonism. See realism, vs anti-realism plausibility logic. See logic Plumwood, V., 275 Poincaré, Pierre, 79 Polish notation, 307 Popper, Karl, 413, 417 possible-world semantics. See semantics postmodernism, 23 Practical Logic of Cognitive Systems, A. See Gabbay, Dov M.; Woods, John Practical Study of Argument, A. See Govier, Trudy pragma-dialectics, 372–3, 418–19 Pragmatic Theory of Fallacy. See Walton, Douglas pragmatism, 78–90 Presocratics, 9 Price, John Valdimir, 471 Priest, Graham, 31, 107, 440 Principia Mathematica. See Russell, Bertrand; Whitehead, Alfred North Principles of Mathematics. See Russell, Bertrand

Prior, A.N., 48, 395 Prior Analytics. See Aristotle prisoner’s dilemma, 493–4, 499 probability: axioms of, 123, 158–62, 165–6, 200; calculus of, 197–9; conditionalization, 125, 130, 184–6, 445; non-standard, 162–3; subjective, 111, 122–8, 131, 158, 160–70, 199–200 problematic premiss. See fallacy proof theory, 41, 447 Proof and Truth. See Woods, John Psillos, Stathis, 191, 195 psychologism, 380, 419 Pucella, Riccardo, 133 Putnam, Hilary, 64–9, 73, 78–83, 90– 2, 189–90, 195, 281–3, 294, 325; Realism and Reason, 67, 76–7 Putnam’s paradox, 117 Pyrrhonism, 155–6 Pythagoras’ theorem, 207–10 quantum mechanics, 72 Quine, W.V.O., 41–2, 78–83, 90–1, 131, 200, 243–63, 323, 326 Quirk, R., 387, 396 Rachels, James, 474, 490 Rackham, H., 238 Radford, C., 48 Raiffa, H., 501 Rao, A.S., 397 rationalism, 135–6 rationality. See reason Rawls, John, 442, 457, 505; Theory of Justice, A, 456, 469, 472 Reach of Abduction, The. See Gabbay, Dov M.; Woods, John Read, Stephen, 247, 258, 261, 281 realism: vs actualism, 49–50, 53,

530 Index 55–7, 106; vs anti-realism, 8, 46, 64–76, 78–90, 130–1, 142, 147–8, 151–2, 155–6, 202–3, 463, 491; scientific, 112–5, 183–6, 188–91, 194 reason, 3, 8–10, 442–3, 447, 491–3; bounded vs unbounded, 380 reducibility, axiom of. See set theory, axioms of reflective equilibrium, 442–3 Reichenbach, Hans, 302, 385, 395, 405, 415 relabelling paradox, 166–70 relations. See logic relativity theory, 72 relevant logic. See logic reliabilism, 142, 183, 189–94, 201–2, 354–5 Rescher, Nicholas, 134–5, 137, 276, 342, 344, 347, 356, 363, 409, 416 Restall, Greg, 276 Reynolds, M., 395 Richard II. See Shakespeare, William Richards, Robert J., 472 Rivas, U., 440 Rodych, Victor, xii, 92, 107, 325, 514 Roth, Philip: Our Gang, 467, 472 Rott, Hans, 415 Routley, Richard, 24, 31–2, 39–41, 43– 4, 47–8, 105, 275 Routley’s formula, 39–40 Royal Society of Canada, 7 Ruddick, Sara, 472 Ruhmkorff, Samuel, 201, 203, 514 Russell, Bertrand, 30, 458, 461, 465, 470–1; barber paradox, 24, 200–1; descriptions, theory of, 15–17, 19– 22, 24, 33, 35–8, 50, 103–4; Nightmares of Eminent Persons, 20–21; pragmatism, 78, 83–8, 90; Principia

Mathematica, 84, 91–2; Principles of Mathematics, 83, 91 Russell’s paradox, 83–4 Ryan, Alan, 472 Sainsbury, R.M., 285 Salisbury, Lord, 134 Salmon, Merrilee H., 347 Salmon, Nathan, 96, 101–2 Salmon, Wesley C., 127, 389, 396 Santayana, George, 458, 470 Savage, C. Wade, 195 Savage, L.J., 171 say-so semantics. See semantics Scanlan, Michael, 207–8, 212–16, 219, 221–2 Scanlon, Timothy, 223 Schilpp, P.A., 90 Schmitt, F.F., 360 Schneewind, J.B., 471 Schneider, L.N., 396 Schneider, Richard, 238 Schofield, Malcolm, 237 Schotch, Peter, 275 Schrödinger, Erwin, 115 Schwartz, Stephen, 294 Scriven, Michael, 437 Seager, William, 115–17, 132 second-order logic. See logic Seidenfeld, Teddy, 133 semantic ascent, 249–50 semantic closure, 41 semantic consequence. See consequence relation semantic kinds, 6, 105 semantics 65–6, 93–101, 296–319, 325–6, 433; algebraic, 284; biological model of, 298–301; formal, 207– 21, 447; Kripke/modal/possible worlds, 37–8, 43, 49–60, 266–75,

Index 531 289–90; say-so, 17–18, 23–25, 29– 30, 45, 139. See also fiction, semantics of; impossible worlds; supervaluation semantics sentential logic. See logic set theory 50, 59, 66–9, 72, 75–6, 447; axioms of, 69–70, 75, 84, 86, 88–90, 220 Seven Per-Cent Solution, The. See Meyer, Nicholas Sextus Empiricus, 238–9, 241–2 Shakespeare, William: Hamlet, 28; Richard II, 16–17 Shapiro, Stewart, 222, 258 Sherlock Holmes, The Adventures of. See Doyle, Arthur Conan Sherwin, Susan, 472 Shiffrin, R.M., 415 Shimony, A., 184 Shoenfield, J., 218, 223 Shoham, Y., 383, 395 Simon, Herbert, 415, 416 Simplicius, 228, 240–1 Simpson’s paradox, 133 skepticism, 8, 144–5 Skolem-Löwenheim theorem, 64, 67, 106–7 Skolem paradox, 64–76, 106–7 Skyrms, Brian, 171 Slater, B.H., 47–8, 104, 514 Slovic, Paul, 203 Smiley, Timothy, 208, 215–16, 220–1, 223 Smith, Holly, 497, 503 Smith, Norman Kemp, 395 Smith, Peter, 294 Smith, Susan L., 470 Soames, Scott, 101 social darwinism, 465 Socrates, 454, 489

Solomon, Graham, 294 Sophistical Refutations. See Aristotle Sorabji, Richard, 365–6, 378 Sorenson, Roy, 139, 157 sorites paradox, 279–83, 286, 325 Sosa, E., 396 Speca, Anthony, 224, 237 Spencer, Herbert, 464–5 Spielman, S., 160, 171 Stalnaker, R.C., 39, 44, 47, 62, 276 Stanford, Kyle, 195 Stanford University, 6 Steedman, M., 385, 387, 396 Stenner, Alfred J., 302, 319 Stewart, James, 31 Stich, S., 257–9 Stingl, Michael J., 470–1, 505–7, 514 Stoic logic. See logic Strand, Jonathan, 277, 323–4, 514 Strawson, Peter F., 15–16, 36, 73 strict implication, 264–8, 272–3 structuralism, mathematical, 65 Sumner, William Graham, 465 supervaluation semantics, 36, 44, 104, 287 Svartvik, J., 396 Swift, Jonathan: Gulliver’s Travels, 36 Swanson, Carolyn, 31 Swets, J.A., 412, 416 syllogistic logic. See logic, Aristotelian Szabo, M.E., 240 Tanner, W.P., 412, 416 Tarski, Alfred: truth, theory of, 41, 200, 222, 430 temporal logic. See logic, tense Tennyson, Alfred Lord: Morte d’Arthur, 27 tense logic. See logic

532 Index Thagard, Paul, 189, 195, 416 Thalberg, Irving, 368, 378 Theaetetus. See Plato Theophrastus, 236 Theory of Justice, A. See Rawls, John Thomason, R., 39, 41, 44, 47–8 Thompson, Paul, 472 Thomson, Judith Jarvis, 465–7, 472, 474, 480, 490 Throop, William, 294 Tienson, J., 415 Timaeus. See Plato Tindale, Christopher W., 361, 437 Tooley, M., 396 Topics. See Aristotle Toulmin, Stephen, 198, 203, 416, 437 transworld identity, problem of, 54 Trout, J.D., 195 Truman, Corrine D., 470 truth, 138–9, 148–9, 155–6, 348–50 Tufts, James: Ethics, 456, 469 Turing, Alan, 219 Tversky, Amos, 198, 203 two wrongs fallacy. See fallacy Unger, Peter, 280, 325 University of Calgary, 6 University of Lethbridge, 7, 9 University of Michigan, 5 University of Toronto, 5 University of Victoria, 6 Urbach, Peter, 158, 160, 171 Urquhart, Alasdair, 30 utilitarianism, 460, 464 vagueness, 279–94, 324–5 van Dalen, Dirk, 70–1, 77 van Eemeren, Frans H., 157, 360–1, 438; Argumentation, Communication and Fallacies, 372–3, 379

van Fraassen, Bas C., 36, 132–3, 183– 9, 194–5, 199–200, 203, 514 Vendler, Z., 385, 395–6 Venema, Y., 383, 395 verificationism, 65, 68 verisimilitude, 141 Verkuyl, H., 387, 396 Villegas-Forero, L., 440 Viminitz, Paul, 503, 508, 515 Viol, W.P.M. Meyer, 47 von Neumann, J., 501 Wallies, Maximilian, 239–41 Walton, Douglas, 8, 347, 367, 370–1, 374–5, 379, 438–9; Pragmatic Theory of Fallacy, 373, 379. See also argument theory, Woods-Walton approach Warren, Mary Ann, 474, 490 Watson, John B., 471 Weingartner, P., 47 Weintraub, Jarett, 200, 515 well-ordering theorem, 220 Wettstein, Howard K., 469, 509 Wheatley, Henry B., 319 White, Nicholas, 378 Whitehead, Alfred North, 88; Principia Mathematica, 84, 91–2 Wicklegreen, W.A., 417 Wilkins, Augustus S., 239 Willard, Charles A., 157, 360–1 Williamson, J., 158, 160, 171 Williamson, Timothy, 294 Wilson, Donna M., 470 Wittgenstein, Ludwig, 301 Wong, Jan, 336 Woodger, J.H., 222 Woods, John, 3–12, 32–3, 47–8, 151, 157, 213, 222, 277, 331, 338–47, 380, 395, 415, 436, 439, 441, 443, 448–9,

Index 533 453–69, 472, 503, 508, 515; Agenda Relevance, 11, 447, 449; Aristotle’s Earlier Logic, 224–5, 237, 240, 321–2, 326; Death of Argument, 326–7; Engineered Death, 6, 9, 11, 460, 465, 468–9, 504, 506; fiction, semantics of, 15–30, 33, 36, 44–7; Logic of Fiction, The, 6, 15, 30, 49, 63, 103, 105, 108, 323; Paradox and Paraconsistency, 9, 11, 107–8; Practical Logic of Cognitive Systems, A, 10, 399, 449– 50; Proof and Truth, 6; Reach of Abduction, The, 11, 78, 90, 92, 156,

398–9, 414, 416, 449. See also argument theory, Woods-Walton approach Wreen, Michael, 437, 506, 515 Wright, Crispin, 279, 281–3, 286, 293, 325 Yablo, Stephen, 63 Zalta, Ed, 276 Zermelo, Ernst, 64, 67, 70–1, 74 Zytkow, J.M., 416