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 Part One: Respondeo john woods
peter alward
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) X, X (X Y), (X Y) (Y X), (X Y) ((Z X) (Z Y)), Fy FHxFx, x = y Fx { Fy, (x)(Fx { Gx) 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 y = x).Gx) is logically equivalent to (x)(Fx.(y)(Fy y = x)).Ga, where a = Hx(Fx.(y)(Fy y = x)). This arises because the latter is equivalent to Fa.(y)(Fy y = a).Ga, which entails the former. But the former is Fb.(y)(Fy y = b).Gb, with b = Hx(Fx.(y)(Fy y = x).Gx), and so entails (x)(Fx.(y)(Fy y = x)), and Fa.(y)(Fy 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
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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 y = c)) c = Hx(Fx.(y)(Fy 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 y = x)), and so also its epsilon equivalent FHx(Fx.(y)(Fy y = x)).(z)(Fz z = Hx(Fx.(y)(Fy 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 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 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-
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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) 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 (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) 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 FHxFx” is true,’ but the formula as standardly interpreted, as in ‘It is true that Fy FHxFx’, then that is necessarily true, since the first axiom is just a specification of the provable predicate calculus thesis, (x)(Fy 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
The Epsilon Logic of Fictions 45
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
Resolving the Skolem Paradox 69
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
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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.
The de Finetti Lottery and Equiprobability 165
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 <1, ...,
T1(C)
<j(C),
where both unions are disjoint. The condition (M) rules out any case where m disjoint copies of C could be placed inside n copies of C even though m > n. In particular, (M) implies that if C can be written as the disjoint union of two or more copies of itself under symmetry mappings, then the set of symmetries S does not define an equiprobability relation on the space X. If both conditions (M) and (G) are satisfied by S, then S induces a well-defined relative probability function with the formal features (R1) – (R6).21 The rough idea is that the relative probability of B to A is n (written R(B, A) = n) just in case B can be written as the disjoint union of n copies of A under symmetry mappings; this is then extended to non-integer values. In particular, A and B are equiprobable if B = T(A) for some symmetry T. In the case of the de Finetti lottery, the outcome space can be identified with the set of positive integers. If we let our set of symmetries S be all translations by a fixed integer, V(x) = x + k, then S satisfies the properties (G) and (M).22 This set of symmetries induces well-defined relations of relative probability and equiprobability. We get the same relative probabilities as we do using the fair-betting-quotient approach of section 3.3.
The de Finetti Lottery and Equiprobability 169
This analysis explains why label independence fails for a countably infinite set. A relabelling is the same as a bijective mapping. Label independence is equivalent to the proposition that any bijection on our population (or equivalently on the natural numbers) is an acceptable symmetry, that is, a member of the set S of symmetries used to define the relative probability relation. But if the bijection used in Example 1 is acceptable, then there is a symmetry that takes ODD to ONE, and (obviously) another that takes ONE to THREE. Since ONE = ODD THREE, the condition (M) is violated. The upshot is that this particular way of relabelling our population is unacceptable. Let us review. Taking equiprobability and relative probability as basic seems to lead to inconsistent assignments of probability or relative probability. We have traced the inconsistency to misleading intuitions about relabelling: the assumption that relabelling has no bearing on probabilistic relations is not generally correct. That assumption is equivalent to the view that we can define relative probability relations using the widest possible set of symmetries, namely, all bijections on an outcome space. That assumption is false except in finite outcome spaces, because the (M) condition will be violated. There is no way to define a label-independent uniform probability distribution on a countably infinite population. How can we explain our intuition that label independence is valid for countably infinite spaces? Consider two closely analogous cases: Example 2: In any finite set of outcomes, label independence is valid. Relabelling the elements does not disturb an initial uniform probability distribution. Example 3: In the outcome space of real numbers in the interval [0, 1], label independence is not valid. An initial uniform probability distribution assigns probability ½ to the intervals [0, ½] and [½, 1]. We can relabel by compressing [0, ½] into [0, ¼], and expanding [½, 1] into [¼, 1]. That is, define f(x) =
x, H½(3/2)x – ½,
0dxd½ ½dxd1
and relabel each point x as f(x). If relabelling did not matter, we would have to conclude that PrR([0, ½]) = PrR([0, ¼]), which is false. We take metric relationships, or inter-point distances, to be relevant to probability assignments in the closed interval [0, 1]. The mapping
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discussed in Example 3 is not an acceptable symmetry because it does not preserve these metric relationships. We now see that a similar point applies to a countably infinite set A. Mappings that distort interpoint distances are not acceptable symmetries. A probability measure on A, or more generally a relationship of relative probability, depends not just on the population A but also on a specific ordering or arrangement of that population.23 Comparing the three cases – a finite outcome space, a countably infinite outcome space, and the space [0, 1] – we have uncovered numerous disanalogies. Cardinality and measure coincide in the finite case, but not in either of the infinite spaces. Uniform probability is unproblematic and can be reconciled with countable additivity in a finite space and in [0, 1], but not in a countably infinite space. Symmetrybased relative probabilities allow us to retain a common framework. In each case, an underlying set S of symmetries is responsible for the fundamental relationships of relative probability and equiprobability. 4. Conclusion We began with two contrary positions. The Dutch Book argument purports to show that if we ground subjective probabilities in a betting formalism, countable additivity is inevitable. Kelly argues that the axiom is plainly inapplicable to the de Finetti lottery and more generally too powerful to be taken for granted in subjective approaches to the confirmation of scientific theories. The foregoing analysis removes one bone of contention: the de Finetti lottery. We can still work within a slightly more general betting framework while accepting that countable additivity does not apply to cases such as the de Finetti lottery where we have a principled reason to adopt only relative probabilities. The mathematical constraints imposed by countable additivity can be waived in certain very special circumstances. But ought we to limit the application of countable additivity outside such cases? I believe that there is no good reason to think so, but I will not take up that question now. notes Extracts with revisions from ‘Countable Additivity and the de Finetti Lottery,’ British Journal for the Philosophy of Science 55 (2004): 301–21. By permission. 1 K. Kelly, The Logic of Reliable Inquiry (Oxford: Oxford University Press, 1996).
The de Finetti Lottery and Equiprobability 171 2 B. de Finetti, Theory of Probability, vols. 1 and 2, trans. A. Machí and A. Smith (New York: Wiley, 1974). 3 Standard sources are L. Savage, The Foundations of Statistics (New York: Dover, 1972), and W. Edwards, H. Lindman, and L.J. Savage, ‘Bayesian Statistical Inference for Psychological Research,’ Psychological Review 70 (1963): 193–242. 4 C. Howson and P. Urbach, Scientific Reasoning: The Bayesian Approach, 2nd ed. (La Salle: Open Court Press, 1993). 5 J. Williamson, ‘Countable Additivity and Subjective Probability,’ British Journal for the Philosophy of Science 50 (1999): 401–16. 6 See P. Maher, ‘Depragmatized Dutch Book Arguments,’ Philosophy of Science 64 (1997): 291–305. 7 B. de Finetti, Probability, Induction and Statistics (New York: Wiley, 1972), 91–2. 8 De Finetti, Theory of Probability, 122. 9 S. Spielman, ‘Physical Probability and Bayesian Statistics,’ Synthese 36 (1977): 236–69. 10 See S. McCall and D. Armstrong, ‘God’s Lottery,’ Analysis 49 (1989): 223–4. McCall and Armstrong propose a mechanism of sorts: God is conducting the lottery – presumably in a fair manner! 11 D. Lewis, ‘A Subjectivist’s Guide to Objective Chance,’ in Studies in Inductive Logic and Probability, R. Jeffrey, ed. (Berkeley: University of California Press, 1980). 12 Here I draw on both Howson and Urbach’s book and Williamson’s work, but such arguments go back to R. Jeffrey, Contribution to the Theory of Inductive Probability (PhD dissertation, Princeton University, 1957). See B. Skyrms, Pragmatics and Empiricism (New Haven: Yale University Press, 1984), for a review of early examples. 13 This sum can never exceed 1, by finite additivity. 14 P. Bartha and C. Hitchcock, ‘The Shooting-Room Paradox and Conditionalizing on “Measurably Challenged” Sets,’ Synthese 118 (1999): 403–37. 15 Another option is to use nilpotent infinitesimals, as described in J. Bell, A Primer of Infinitesimal Analysis (Cambridge: Cambridge University Press, 1998). One strongly counter-intuitive consequence, however, is that on such an approach, the probability of the same ticket winning twice (in two independent draws) is literally 0. 16 Or between propositions expressing two outcomes. 17 The discussion here parallels that of P. Bartha and R. Johns, ‘Probability and Symmetry,’ Philosophy of Science 68 (Proceedings, 2001): S109–22. 18 Compare ‘Probability and Symmetry,’ where the formalism is slightly different.
172 Paul Bartha 19 The example is due to John Norton (in correspondence). 20 The idea is developed in P. Bartha and R. Johns, ‘Probability and Symmetry.’ 21 Caveat: Not all of these features have been verified in general, though there is no difficulty for the special case of the de Finetti lottery. 22 There is a slight complication: everything has to be defined first on the set of all integers and later restricted to the positive integers. 23 This result is much more readily accepted for limiting frequency interpretations of probability. Obviously, permuting an actual sequence of observations can change the limiting frequency.
10 The Lottery Paradox JARE TT WE INTRAUB
Gilbert Harman, in Change in View, twice discusses the problem of ‘the lottery paradox.’1 The first time, he uses it to show how beliefs cannot explicitly be probability-based, and he returns to it when discussing inference from high statistical probability in order to show that such inferences do not yield the same paradoxical conclusion. While it may well be that humans do not explicitly reason using subjective degrees of belief, and that we may often safely infer from statistical likelihoods, it seems to me that the paradox as presented is not a good support for these conclusions. In fact, Harman’s entire discussion of the lottery paradox (which admittedly is taken from Kyburg,2 so at least some of the blame may belong with him) appears to be based on a misunderstanding of what it is that drives the paradox's supposed contradiction. The lottery paradox itself is as follows. Given a lottery with n tickets, one believes the proposition ‘one of these n tickets will win.’3 Yet there is also a high probability for believing the proposition ‘ticket i will not win’ for each ticket i between 1 and n. So one may simultaneously believe (to a high degree, if belief is a matter of degree) both that one ticket will win and that no ticket will win, despite the logical inconsistency. When Harman revisits the paradox, he concludes that statistical inference does not lead to the lottery paradox because ‘given that one has inferred ticket number 1 will not win, then one must suppose the odds against ticket number 2 are no longer 999,999 to 1, but only 999,998 to 1 … If one could get to ticket number 999,999, one would have to suppose the odds were even, 1 to 1, so … one could infer no further.’4 He notes that in this case, the order of inference would be
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crucial, because whichever tickets were considered last would be believed most likely to win, while given a different order of consideration those same tickets would instead be discarded. My intent is to show that the paradox is not specifically generated by ‘degrees of belief’ reasoning, nor is it necessarily avoided by using statistical inference. Instead, it is merely the phrasing of the problem which leads to the paradoxical result in the first instance, and the quite curious result of our reasoned conclusion depending upon inference order in the second. In order to show this, I will first consider a number of possible factors within the paradox as laid out to see if they are in fact the culprits. By removing each factor separately from the paradox, and seeing if the paradox remains, I will test each factor as the source of the contradiction. Having shown that each is not in fact to blame, I will then attempt to make clear what I believe actually underwrites the apparent paradox5 and what an approach that more accurately reflects our reasoning looks like. The possible sources of paradox which I will examine are: (1) the negligibility of the probability of success for each case; (2) the large number of possible cases to be examined; (3) the equality of probability in each case; and (4) the presumption that one of the tickets will win. 1. The Negligibility of the Probability in Each Case One reason to suspect the negligible probability as the source of the lottery paradox is that, given how minuscule the odds are of winning for each ticket, it may be too easy to equate the odds of winning with a 0 per cent chance. In reasoning about the tickets then, we might conflate an extremely low probability with no possibility at all. The probability of each ticket winning is tied closely to the number of tickets in the lottery (assuming each ticket has an equal probability of winning). At first glance, it appears there is no way to eliminate the small probability of winning without also reducing the large number of tickets in the lottery. But all that needs to be done to affect this separation is to allow for more than one winner. So, suppose we have 100,000 tickets, each printed with a number between 1 and 10. The number of tickets would remain the same, as would the equal probability of each ticket winning. The presumption that one ticket will win is altered, but only in quantity, as we now believe that 10,000 tickets will win. While a 10 per cent chance of winning is clearly not negligible or easily conflated with a 0 per cent chance, Harman still implicitly endorses (in the quote above) the idea of counting any ticket with a sufficiently
The Lottery Paradox 175
high probability of losing as one we can reasonably conclude will not win, and a 90 per cent chance of failure seems reasonably high.6 We can easily see that even with each ticket holding a 10 per cent (or higher) chance of success, the proposition ‘ticket i will not win’ will still be true for each ticket, and so the paradox will still hold using the degrees of belief approach. As for statistical inference, Harman’s conclusion only becomes more odd than it originally was. Given that same-numbered tickets will either all win or all lose, it seems certain that one will have inferred that thousands of tickets will lose that should in fact be thought likely winners given one’s ultimate belief that one of only two tickets are the set of possible winners. In addition, his conclusion will vary depending not only upon the order of inference but also upon how many of each number remain. If the last ten tickets are numbered consecutively 1, 2, 3 ..., 10 then his conclusion will be as originally stated. But if they are all the same number, and ticket number 99,990 is a different number, then he can infer statistically that the odds against that ticket are 10 to 1, and discard it. What then of the remaining ten tickets? We can infer either that they will all win, generating a certainty that is clearly unjustified, or that they will all lose, which brings about the lottery paradox Harman had supposedly eliminated! Perhaps the problem here is caused by the fact that all ten remaining tickets are considered as a unit, whereas Harman only considers tickets serially. Even considered one at a time though, statistical inference will still be faced with the dilemma just presented when the last two tickets are the same number. In any case, as we will next see, the source of the paradox is not the number of cases to be considered or the concomitant physical requirement of considering them serially. 2. The Large Number of Cases Given the number of cases, and the limitations of the human brain, we are simply incapable of holding all of the possible situations in our minds simultaneously in the Lottery Paradox. This forces us to consider the cases either as a unit or as an aggregate of serial considerations. The first underwrites the conclusion that ‘one ticket will win,’ the second that ‘ticket i will lose’ for all tickets i through N. But perhaps there is a third way to consider the possibilities. The large number of cases from i to N means that in order to reason about all of them, we must discard entirely the possibility of each ticket winning after it has been considered, instead of weighing them all against one another. What happens if we alter the paradox to
176 Jarett Weintraub
include a small enough number of cases that we can consider them all simultaneously? Reducing the number of tickets also increases the odds of each ticket winning, and this connection is at least in part ineliminable. But as we have just shown, so long as the probability of losing remains sufficiently high for each ticket, the problem is still as Harman describes it. Let us assume, then, that there are only three differently numbered tickets in our lottery. Each has a 33 per cent chance of winning, and a 66 per cent chance of losing. Examined serially, the Lottery Paradox holds. But now, instead of discarding each ticket after examining it, we can retain its chance at winning while examining the other two. This would be a relational approach to the problem, so that instead of examining ticket 1 vs. tickets 2&3, and then examining ticket 2 vs. ticket 3 (or considering ticket 1 vs. 2&3, then 2 vs. 1&3, then 3 vs. 1&2, and adding the results), we could consider 1 vs. 2&3, 2 vs. 1&3, and 3 vs. 1&2 all at once. In each case, we would acknowledge that the individual ticket under examination is unlikely to win but that its chance of winning is still to be considered in weighing the odds of the other two tickets’ success and that its chance of winning depends in part upon the conclusions we reach about the other tickets. Unfortunately this approach does not seem to give us any real traction on the problem but only generates a variation on the paradox. Either we will conclude, as in the case of 100,000 tickets, that no ticket has a chance to win (only now with the added caveat that each ticket will not win because one of the other two will) or we may be able to conclude only that there is no conclusion to be made and that our reasoning will simply cycle endlessly among the possibilities with no reason at all to prefer one over the others. Here we may use Harman’s ‘satisficing’ approach to pick the ticket which we want to win as the one we believe will win, since there is no reason to select one over the other beyond simple whim or desire (presumably, the interest in favouring the ticket that is one’s own would tip the balance here). This is not a reasoned view, however, based as it is solely on desire and not on principles of reasoning. It thus does not solve the question of how we are able to reach a reasoned conclusion, but instead is merely wishing or hoping. 3. Equality of Probability In the last case, we were unable to arrive at a conclusion because each case, even when considered relationally, was equally probable as the
The Lottery Paradox 177
other; there was no reasoned basis for concluding that any of them was more likely than the other, and so either they must all win or all lose (neither of which is acceptable), or else there is no reasoned conclusion to be made at all. Perhaps this is the source of the Lottery Paradox, that it is an attempt to reach a conclusion where no reasoned conclusion is to be found. The lack of context in the problem may lead to a confusion of reason’s capacities. Given no basis for choosing any one ticket over the others, there is no reason to select any of the tickets, and so it appears from that alone that no ticket can win. Certainly this is very much not the way reason typically functions. The ordinary circumstances surrounding reasoned selection involve some contextual variation, where options and outcomes have distinct and relevant differences in their likelihood. This allows us to pick a likely ‘winner’ simply on the basis of its receiving a higher relative ‘score’ than all its competitors, even if it does not have a majority statistical likelihood. Thus we may perhaps conclude that, despite a preponderance of evidence to the contrary, Lee Harvey Oswald was the lone gunman, given the greater likelihood of this view than of any other individual theory put forth to date.7 While it is not certain how much variation is ‘enough’ (would ten tickets with a 9 per cent chance and one with a 10 per cent chance be enough?), it seems safe to conclude that four tickets with an 18 per cent chance each and one ticket with a 28 per cent chance would be sufficient. Our gut instinct here would be to conclude that the last ticket is the most likely to win; but this is not the same as believing that it will win. As was the case when we considered the factor of negligible probability, serial consideration of the tickets still generates the Lottery Paradox, and examination by statistical inference generates conflicting conclusions that depend upon the order of examination. If the highestprobability ticket is the last or second to last considered, statistical inference will bring about the belief that that ticket is the winner, and while it may be reasonable to conclude that it is the most likely to win, it may not be reasonable to conclude that it will be the winner. When the 28 per cent ticket is considered first, second, or even third, it is discarded as unlikely, relative to the remaining odds against it, and we are left with the conclusion that the last two tickets are the equally likely winners. This conclusion may be more surprising than the one in which other factors were removed, given that the shape of the paradox may appear to rest upon the combined knowledge that there is a correct answer as to which ticket will win, having the set of all possible answers avail-
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able, and there being no way to conclude which answer is the correct one. But this only results in indecision, not paradox. The last factor should not be ‘having no way to conclude which is correct’ but ‘having no way not to conclude that all of them are wrong.’ This is the factor which we have been attempting to counter so far, and each attempt has led inexorably back to paradox. The second piece of knowledge, having the set of all answers available, is logically ineliminable, since we cannot reason about all possible solutions unless we can know what they are. However, the first fact, that there will be a winner, is not entirely unassailable. 4. Certainty of a Winner The knowledge that there will be a winning ticket is necessary for the paradox to the degree that one leg of the paradox simply is the fact that we know for certain that one ticket will win (combined with the other leg, that taken serially, we believe that none will win). But instead of being certain either that none will win or that one will win, suppose we are considering both on an equal probabilistic standing. Imagine a lottery with 1,000 tickets, and only 999 of them are sold. The probability of any given ticket winning is then equal to the chance that no ticket will win, and this latter can be reasoned about in just the same way. If we examine this problem using Harman’s statistical inference approach, the order of inference again matters tremendously and leads to two distinct conclusions. In every situation where the chance of no ticket winning is considered other than last or next to last, the conclusion will be that some ticket will win, and it will end as he originally outlined. But in the other two cases, it seems that we will conclude with certainty that if any ticket wins, it will be the one we examine last, and that there are even odds of this occurring. As for the degree of belief approach, it is, if anything, worse off than ever. Not only is our reasoning implicated in concluding that no ticket will win, but it is also now the culprit in our conclusion that some ticket will win. Essentially, this is the paradox as originally laid out by Harman; we have a high degree of belief that no individual ticket will win and a high degree of belief that some ticket will win. 5. Picking through the Wreckage So far, every approach attempted has resulted in a trail of logical carnage. The closest we have come to avoiding paradox altogether has
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been when we reduced the number of cases to a set which could be reasoned about simultaneously. In this case, we were able to conclude that perhaps the best answer was that there was no reasoned conclusion to be had, and it is this possibility that I wish to explore further, as I believe that it is the correct one and will allow for a better way of thinking about how we think about cases like these. The major problem in the Lottery Paradox is that it is a case where we know that there is an answer yet we cannot determine what it is. This is not at all a new or uncommon situation. Aristotle’s famous ‘sea battle’ problem was concerned with the truth-value of statements concerning non-necessitated future events;8 given that there would ultimately be a definite value applicable to such statements, but that it was unknown at the time of their utterance, what was their truth-value at that time? Were they already true or false, and if so how? And if they were neither true nor false, what were they? Ultimately, it seems, such claims have no truth-value at all – they are logical ‘holes’ filled in only when the event occurs. Unfortunately, we are not concerned with truth-value so much as belief, be it true or false. And while we can claim a statement has no real truth-value, we cannot so easily claim that it is not really believed – this will not solve our dilemma. But there is an appropriately similar tack, and that is to conclude that a reasoned approach will not actually reach the same conclusions as Harman puts forth in either of his descriptions. In one sense, this seems obvious, since it is doubtful that anyone would really be tempted to think, as Harman describes, either that no ticket will win or that only one of the last two tickets considered will win. But what then do we think, and why? When approached relationally, the problem seems no longer to be one which is paradoxical, but only insoluble, a situation which we are much better prepared to handle. The relational approach pictures the odds of each ticket winning or losing as (at least implicitly) dependent upon the odds of all other tickets winning, as with the degree of belief approach, but resists aggregating those beliefs into one unitary belief, which the statistical-inference approach also avoids. But each of the approaches Harman considers has just the drawback that the other does not. What both these other approaches have in common is a mistaken understanding of how beliefs are fixed in working towards a conclusion. The statistical-inference approach requires that, having inferred that ticket 1 will lose, we discard it entirely from all future consideration. People just do not think this way. The degree of belief approach requires that having decided that a ticket will likely lose, we keep that
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particular belief about the ticket when aggregating it with our beliefs about every other ticket. But this is not how we think either. In thinking that ticket 1 will lose, I include the belief that some ticket 2 through n will win. When I reason about ticket 2, and decide that it will lose, I include the belief that the winning ticket is among tickets 3 through n and ticket 1. In each case, when I believe that another ticket will win, I believe that any other ticket may win. My reasoning about tickets 2 through n revives my belief in the possibility of ticket 1 being a winner, and so forth for all other cases. In short, whenever I reason about any particular ticket, I am forced to reconsider the possibility of every other ticket as in the class of possible winners, even those previously discarded. Because of this, one simply cannot aggregate these beliefs in the way required in order to generate the Lottery Paradox according to the ‘degrees of belief’ approach. They are not logically compatible, in that each is dependent upon not believing what is implied by all of the others. Even if we cannot hold each distinct case in our heads simultaneously, as when there are a large number of cases, we still do not reason about the cases in a non-relational matter. Instead, we reason that we cannot come to a reasoned conclusion about them as a whole, or at least that our reasoning about them individually is not the same as our reasoning about them as a whole. Saying that we are thinking about the cases in aggregate is very different indeed from saying that we are aggregating our thinking about each individually. That being said, the error in Harman’s second approach is so blatant as to almost beg mentioning. We simply do not, when reasoning about the second ticket, take the first ticket no longer to exist. Because of this, the odds of each ticket losing do not decrease as we move from inference to inference but remain the same as they did with the first ticket. Perhaps because reasoning as we do seems to violate both his Get Back Principle and his claim that ‘full acceptance ends inquiry,’ Harman is reluctant to acknowledge this. But neither seems as irreparably harmed by the relational view as it may at first blush appear. Even when full acceptance ends inquiry, the acceptance is still understood to be provisional. My firm belief that Elvis is dead is certain, but not therefore unshakable. Only the pathologically fixated would be forever unwilling to reconsider their beliefs if sufficient counter-evidence was presented. Evidence that challenges the original belief requires that that belief be reopened to investigation, and I believe that the consideration of the odds of any other ticket winning or losing qualifies as just such a challenge.
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Given that the belief in ticket 2 losing inherently implies a chance of ticket 1 winning, holding such a belief should serve to return the belief that ticket 1 is a loser to only tentative acceptance. Thus we may fully accept ticket 1 as a loser while considering it, but as soon as we consider any other ticket, that belief is undermined. This is where concerns about the Get Back Principle come into play. The Get Back Principle states that ‘One should not give up a belief one can easily (and rationally) get right back.’9 When considering ticket 2, why give up the belief that ticket 1 will lose, since considering ticket 1 again will allow us to get it right back? When considering a problem such as a lottery, all possible outcomes are seen as merely provisional for the period of inquiry. One does not simply eliminate one possibility and move on, as one might in investigating a murder or a plane crash, since here the possibility of each winning or losing is logically dependent upon the odds for each and every other.10 So the belief about each ticket is not a belief of the sort to which the Get Back Principle would apply, because of both their provisional status and their logical interdependence. The fact that we do not just eliminate tickets from consideration is where the statistical inference approach goes awry. Once we have discarded ticket 1 as a loser, the odds for ticket 2 do not, in the real world, change from 999,999 to 1 to 999,998 to 1 but remain the same. In a sense, then, such cases are indeed like Aristotle’s sea-battle problem. Until such time as the winning ticket is selected, and inquiry can definitively end, there is no reasoned conclusion to be reached, and no explicitly reasoned belief about any given ticket can be made on a basis other than satisficing. Otherwise, we risk seriously violating the Principle of Clutter Avoidance, filling our heads with beliefs about any and every unknown potentiality. While the solution to the question of which ticket will win remains unknown and unknowable, conclusions about each possibility remain open, and the only reasonable conclusion is simply that there is no conclusion to be reached. notes 1 Gilbert Harman, Change in View (Cambridge, MA: MIT Press, 1986). 2 Henry Kyburg, Probability and the Logic of Rational Belief (Middleton, CT: Wesleyan University Press, 1961). 3 For the purposes of most of this paper I will assume that this is not merely probable but certain. It simplifies the discussion, and none of the paradox’s bite is dulled.
182 Jarett Weintraub 4 Harman, Change in View, 71. 5 While it may not always be possible to entirely remove a factor from the equation, each in turn should be able to be sufficiently diluted in influence to insulate the argument from the charge that it is all the factors working in concert that are the actual source of contradiction. 6 Although the parenthetical remark Harman makes regarding the question of when one can safely discard a ticket as a loser is vague, I will (in the next section) take it as given for my purposes that even a ticket with a 1-in-3 chance of winning might be discarded as likely to lose. 7 On the other hand, lumping a group of theories into the general category of conspiracy may produce a higher overall likelihood. 8 De Interpretatione, 19a23–19b3. 9 Harman, Change in View, 58. 10 By contrast, the likelihood of the butler having committed a murder isn’t logically dependent upon the likelihood of the maid having done it. Discarding (or even confirming) one possibility need not raise (or lower) the chances of the other having done it at all. Perhaps they both did it, or the maid was herself dead at the time of the murder – in either instance, one’s belief in the butler’s guilt or innocence may have no effect on the likelihood of the maid’s.
11 Reliabilism and Inference to the Best Explanation S A M UE L R U HMK O R F F
Two of Bas van Fraassen’s significant challenges to the claim that inference to the best explanation (IBE) is a rationally required rule are the problem of the bad lot and the argument from indifference. These challenges rely on the observation that, even if the explanatory virtues (e.g., consilience, simplicity) are assumed to be evidential, the best explanation available to us might very well not be the best explanation overall. If we don’t have evidence that the best explanation available to us is the best explanation overall, then IBE can hardly be a rationally required rule. Due to his permissive voluntaristic epistemology, van Fraassen does not conclude that it is a mistake of reason to use IBE tout court, but rather that it is irrational to use it as a rule, and even more irrational to regard it as a rationally required rule.1 Some proponents of IBE may find comfort in van Fraassen’s permission of piecemeal IBE. However, if IBE has the defects alleged by van Fraassen, scientific realists who pin their hopes of solving the problem of underdetermination on IBE and who do not share van Fraassen’s permissive epistemology will be greatly distressed; for the defects alleged by van Fraassen are serious indeed. In this paper, I first discuss the problem of the bad lot and the argument from indifference, and argue that they are best answered by invoking a reliabilist epistemology (or some other version of externalism). I then discuss the general nature of reliabilist defences of rules of inference and propose a list of criteria which they must meet. If I am correct, a successful response to the problem of indifference must give a reliabilist defence of IBE by appeal to the success of science. I argue that the problem of indifference cannot be easily dismissed; yet the realist can hope to address it successfully.
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Conclusions of the form ‘Those who wish to avoid argument A ought to )’ are difficult to defend. The worry, indeed, is that the alternative responses considered by the author compose a bad lot. There may be a better response to van Fraassen than the course outlined below. But I hope that I have at least shifted the burden of proof onto those who think that a satisfactory response to van Fraassen’s challenges to inference to the best explanation need not involve appeal to the success of science and an externalist epistemology. 1. Inference to the Best Explanation and Bad Lots Some terminology will be useful. Available theories are theories which are available for consideration, that is, theories which have been proposed and are known to the relevant agent. Following Shimony, I will call the hypothesis that an unavailable theory is true the pessimistic hypothesis (PH). The optimistic hypothesis (OH) is the hypothesis that an available theory is true. Van Fraassen distinguishes between the comparative step and ampliative step of IBE.2 In the comparative step, available theories are compared with each other using the explanatory and empirical virtues as criteria. In the ampliative step, the available theory richest in the explanatory and empirical virtues (Hbest) is inferred to be the theory richest in the explanatory and empirical virtues, and hence true. Thus, inference to the best explanation involves a commitment to the optimistic hypothesis and a denial of the pessimistic hypothesis. The problem of the bad lot grants the defender of IBE the claim that the explanatory virtues are evidential.3 It then purports to show that, even in the face of such generosity, IBE is deeply flawed. The problem of the bad lot is based on the observation that IBE is only reasonable if P(OH) is high enough. Van Fraassen sets the threshold of full belief at >.5 and assumes that P(Hbest / OH) = 1. Given these liberal assumptions, the defender of IBE needs only to show that P(OH) > .5. But how is she to do this? There are many more unavailable explanations than available explanations. It is possible that the available explanations constitute a bad lot: they do not contain the true explanation. In other words, for all we know, P(OH) d .5. I think that IBE should be construed probabilistically.4 On my view, the result of the comparative step is the assignment of prior probabilities to available theories conditional on the optimistic hypothesis based on both explanatory and empirical virtues. The ampliative step
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involves the determination of the probability of the optimistic hypothesis and the subsequent updating of the probabilities of available theories through Jeffrey conditionalization. When IBE is construed probabilistically, the problem of the bad lot does not apply. Non-probabilistic IBE demands that we have reason to think that P(OH) > .5 – a weighty requirement. But probabilistic IBE demands only that we have evidence that P(OH) > 0; for the explanatory virtues make a difference to our posterior probabilities for available theories just in case P(OH) > 0. The bar has been lowered considerably. It seems quite reasonable to assume that the P(OH) > 0; and so it seems quite reasonable to hold that the explanatory virtues, if evidential, should play a role in determining the posterior probabilities of available theories in question. 2. The Argument from Indifference In response to probabilistic IBE, van Fraassen modifies the problem of the bad lot. Instead of asking ‘What evidence do we have that P(OH) > .5?’ van Fraassen presents a positive reason to think that P(OH) is too low for IBE to have any standing as a rule of inference. Van Fraassen calls this the argument from indifference: I believe, and so do you, that there are many theories, perhaps never yet formulated but in accordance with all evidence so far, which explain at least as well as the best we have now. Since these theories can disagree in so many ways about statements that go beyond our evidence to date, it is clear that most of them by far must be false. I know nothing about our best explanation, relevant to its truth-value, except that it belongs to this class. So I must treat it as a random member of this class, most of which is false.5 Hence it must seem very improbable to me that it is true.6
Strictly speaking, if the argument from indifference is going to apply to probabilistic IBE, then the number of explanations at least as good as the best available explanation must be unbounded. If this is true, the probability of OH will be 0, and the explanatory virtues will not figure in the probabilities of available explanations. If the number of explanations at least as good as the best available explanation is not unbounded, and is merely very large, then P(OH) > 0, and the explanatory virtues will make a difference to our probabilities for available explanations. Even if available explanations are assigned low posterior probabilities,
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the posterior probabilities they are assigned will be based in part on the explanatory virtues. Although probabilistic IBE is off the hook if the number of explanations at least as good as the best available explanation is large but not unbounded, this is a mere technicality which offers no real comfort to the scientific realist. For example, one thing the scientific realist wants to do with IBE is to use it to solve the problem of underdetermination. This is not going to work if the posterior probability of the best available explanation is very low! The scientific realist can muck around with small bits of probability all he wants, but any significant reply to the argument from indifference must show that it is sometimes warranted to assign relatively high probabilities to the best available explanation. Van Fraassen has another consideration in his arsenal which he does not mention in the passage quoted above. The scientific realist must not only refute the claim that ‘there are many theories, perhaps never yet formulated but in accordance with all evidence so far, which explain at least as well as the best we have now,’ she must show that we are sometimes capable of the non-arbitrary assignment of probabilities to the optimistic hypothesis. Van Fraassen’s challenge can be construed as having two components: (1) The scientific realist cannot refute the claim that there is an indefinite number of unavailable theories at least as rich in the empirical and explanatory virtues as the best available theory. (2) The scientific realist cannot show that we are ever justified in assigning non-arbitrary, non-zero probabilities to OH. 3. Direct Responses to the Argument from Indifference The argument from indifference can be directly and indirectly challenged. The argument from indifference seems to rely on the claim that we can only consider a small, finite number of theories, whereas there are an infinite number of unavailable theories. The direct challenge responds to the argument from indifference by challenging this claim. There are several strategies for doing this, all of which fail. I will discuss two of note. The first strategy for enlarging the class of available explanations invokes cases in which an agent considers a series of recursively generated theories. Take van Fraassen’s example of empirically equivalent
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theories. He starts with Newton’s theory TN and then generates an infinite number of theories by appending to TN the claim that the universe has constant velocity v relative to absolute space for all real numbers v.7 Newton may have considered this infinite range of theories and chosen TN(0) because it is simpler than the other theories. If he did this, it is fair to say that he implicitly considered all of the theories TN(v), even though he did not explicitly consider each one. This strategy succeeds in allowing the number of available theories to be unbounded. However, it does not refute the argument from indifference because there are always going to be an indefinite number of unavailable theories, no matter how many theories and their recursively generated empirical equivalents we consider. The second strategy for claiming that available theories can constitute a significant portion of possible theories centres on the subsumption of vast numbers of theories under broader claims. For example, Peter Lipton points out that the range of competing explanations can be narrowed simply by considering the best available explanation and its contradictory.8 (Alternatively, we might compare available explanations and the pessimistic hypothesis.) By considering the contradictory of the best available hypothesis, Lipton argues, we are implicitly considering all of the explanations which are subsumed under it. If the best available explanation is a better explanation than its contradictory, then we are warranted in inferring it. This strategy has the advantage that, if a theory’s contradictory or the pessimistic hypothesis is taken to be available, then available theories can exhaust logical space. However, it is at bottom a trick. The challenge presented by the argument from indifference centres on the possibility of there being an indefinite number of possible theories at least as good as the best available explanation. It should not console us that these theories can all be subsumed under a single, more general claim. Another way to put this point is as follows: the explanatory virtues are intended to be used to evaluate fine-grained scientific theories, not general claims such as ‘All available theories are false.’ Although scientists sometimes have informed opinions on the truth or falsity of the pessimistic hypothesis or the contradictories of available theories, a scientist does not get epistemic credit for evaluating the indefinite number of theories that entail the pessimistic hypothesis just by reflecting in a general sort of way on the pessimistic hypothesis. In conclusion, direct responses to the argument from indifference fail. The fact that unavailable explanations are unbounded, and the fact
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that they cannot be considered simply by lumping them together under a general claim, make a direct response impossible. 4. The Thesis of Privilege Indirect responses assume the thesis of ‘privilege,’ as van Fraassen labels it. According to this thesis, available theories are not a statistically normal sample of potential theories. Rather, they are a privileged sample, one which is disposed to contain the best overall explanations. If the thesis of privilege is correct, then P(OH) should be relatively high in some cases (assuming as usual that these virtues are evidential). In these cases, the explanatory virtues will make a significant and warranted contribution to the posterior probabilities of available theories. One way to support the existence of a privileging mechanism is to appeal to evolutionary concerns:9 we evolved the ability to generate good lots because possession of true theories about the world increases fitness. It is tempting to tell this kind of story. But appeals to evolution are tricky. It is hard to show that a trait increases fitness in the way posited by the theory. In addition, not all fitness-enhancing traits are selected for. I do think something like this evolutionary story is true. But I wouldn’t want to stake the fate of scientific realism on it. For one, there is the general – and deserved – skepticism of armchair evolutionary psychology. For another, the constructive empiricist will be able to propose a similar evolutionary story: humans evolved the ability to generate empirically adequate theories because possession of empirically adequate theories increases fitness. It is hard to conceive of evidence that would favour the defender of IBE’s account over the constructive empiricist’s. Another way to support the thesis of privilege is to argue that the success of science demonstrates that we are disposed to generate good lots. But arguing that the best explanation for the success of science is that scientists are disposed to generate good lots seems to be viciously circular in that it uses IBE to vindicate IBE. More on this later. Yet another strategy for defending the thesis of privilege invokes the possibility that the explanatory virtues are such that available theories are much more likely than unavailable theories to participate in them. Van Fraassen claims that the problem of the bad lot and the argument from indifference are neutral among accounts of the explanatory virtues.10 But if (say) the scientific realist considers a theory to be rich in the explanatory virtues just in case it is simple, consilient, and
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inscribed in a leading scientific journal, the argument from indifference does not apply to his view. Only a limited number of theories are inscribed in leading scientific journals; so there is no chance that there are a large number of unavailable explanations as good as or better than the best available explanation. Now, it is silly to think being inscribed in a leading scientific journal is an explanatory virtue. But, as Paul Thagard has argued, the extent to which a theory participates in the explanatory virtues depends in part on the background beliefs of the evaluator.11 For example, Thagard defines the consilience of a theory in terms of the number of classes of facts it explains. The individuation of classes of facts is dependent on the evaluator’s background beliefs. This relativity to background beliefs promises to narrow the class of exceptional explanations such that available explanations are privileged. This strategy for defending privilege can be combined with the other strategies. If we adopt it, the constructive empiricist will no doubt question the evidentiality of explanatory virtues that are relative to background beliefs. Van Fraassen does this by taking simplicity as a stalking-horse: ‘it is surely absurd to think that the world is more likely to be simple than complicated (unless one has certain metaphysical or theological views not usually accepted as legitimate factors in scientific inference).’12 On the face of it, this is a compelling intuition. Why think that the world is likely to be such that simplicity, or consilience, or analogy to accepted theory, are marks of truth? But there is something slippery going on. A priori, there is no reason to think that the world is more likely to be simple than not. However, it is possible that we have evidence that the world is simple. Our current scientific picture of the world – even the observable world – is of a relatively orderly place. These observations point towards a scientific justification of the evidentiality of the explanatory virtues. Again, the scientific realist’s response to the argument from indifference turns upon an argument based on the success of science. If it is illicit to use IBE to justify IBE, then the success argument is viciously circular and the scientific realist is left with no defence of her position. 5. Circularity and Reliabilism I have suggested that defending the thesis of privilege requires an argument based on the instrumental success of science. Putnam’s ‘mir-
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acle’ argument is the best-known success argument. It runs as follows: the instrumental success of science would be a miracle unless scientific method (assumed to include IBE) gets at the truth; therefore scientific method (including IBE) reliably gets at the truth.13 In other words, the best explanation of the instrumental success of science is that scientific methodology reliably gets at the truth. Many philosophers have objected that empiricism can explain the success of science just as well as, if not better than, realism.14 This objection is important, but it lies outside of the scope of this paper. The objection to success arguments with which I am concerned contends that they are viciously circular.15 Success arguments are inferences to the best explanation; but they are arguments intending to vindicate IBE. This circularity seems vicious, for those who do not think that IBE is justified in the first place should not be convinced by an IBE with the conclusion that IBE is justified. Boyd meets this argument for vicious circularity with the following line of thought. To arbitrate between the empiricist and the realist, we need to compare the overall plausibility of the philosophical packages they offer us. Thus, for example, the fact that it is difficult for the empiricist to say how we can be justified in believing in the empirical adequacy of scientific theories counts as a mark against the empiricist package. By examining the advantages and disadvantages of the empiricist and realist packages, we can decide which package is better overall. Since realism offers a better package, we are justified in embracing realism.16 I find this response inadequate. First, as Fine points out, weighing the various merits of philosophical packages seems itself to be a type of IBE.17 Second, if empiricism has trouble saying how we are justified in our judgments of empirical adequacy, that fact is a demerit for empiricism. But it has nothing to do with the question of the circularity of the success argument. Either the success argument is viciously circular or it is not. If it is, then it does not count as evidence for realism, no matter what other considerations support realism. The package argument is simply a change of topic. Scientific realism is no longer being defended by the success argument. Instead, other considerations, such as the intuitive implausibility of empiricism, are in play. This constitutes a retreat from the success argument rather than a defence of it against the circularity objection. Boyd’s idea may be that the realist’s defence of IBE, even if circular, increases the overall coherence of the realist package.18 But if the argu-
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ment really is viciously circular, it is hard to see how it can improve the value of the realist package. As Fine points out, the realist who finds comfort in the coherence produced by the success argument is in the same position as the proponent of an inconsistent logical system who can produce a consistency proof in that system.19 I suppose it is nice to have a demonstration that one’s view is coherent. An internally coherent view is preferable over an incoherent view. But surely the realist wants more out of the success argument than the conclusion that her view is (at least in this respect) coherent. There is no denying that the success argument uses IBE to justify IBE. If this circularity is vicious, realism accrues little benefit from the success argument. Thus any successful response to the circularity objection will involve the claim that the circle is not vicious.20 The most natural way to do this is to invoke an externalist epistemology. For example, reliabilism defuses the circularity of the success argument in the following way. According to the simplest version of reliabilism, all it takes for conclusions drawn on the basis of IBE to be justified is for IBE to be a reliable rule of inference. Thus, if IBE is in fact reliable, and part of the best explanation of the instrumental success of science is the reliability of IBE, then the belief that IBE is reliable is justified. The issue of circularity simply does not arise. Of the scientific realists who attempt to avoid the circularity objection to the success argument, only Stathis Psillos discusses reliabilist solutions to this objection.21 This is surprising given that reliabilism provides for the vindication of self-endorsing rules of inference.22 This vindication, however, is not as easily attained as one might think. I distinguish between the requirements for justification according to reliabilism and requirements for a reliabilist defence of a rule of inference. A reliabilist defence of a rule of inference occurs in the context of a philosophical or scientific debate regarding the appropriateness of that rule of inference. In providing a reliabilist defence, the supporter of a rule of inference is attempting to change the mind of those who are skeptical of that rule of inference. For this reason, the requirements for a reliabilist defence of a rule of inference may be more stringent than the requirements for justification according to reliabilism. Consider the problem of the bad lot as it applies to chess. Let’s say that the chess equivalents of explanations are sequences. A sequence is a series of moves. The shortest sequences consist of one half-move, or ply; the longest are of indefinite length. Now, in order to discover which move is best, attention must be paid to the various sequences which
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result from every legal move. Typically, the longer the sequences considered, the better the chance of determining which move is best. Suppose we had a computer, Deeper Thought, which was capable of evaluating a billion positions per second. The average chess position presents 38 legal moves. Deeper Thought would require thirty seconds to perform a brute force evaluation of such a position through six ply, and 2854 years through twelve ply. (The latter calculation would encompass 9 u 1018 positions!) Grandmasters can sometimes evaluate their moves through twelve ply (or more). But the sequences they consider constitute only a small fraction of the possible sequences. To make things more concrete, let’s suppose that a grandmaster decides that move F is better than move G through twelve ply. One version of the problem of the bad lot for chess is this: how does the grandmaster know that there are not twelve-ply sequences resulting from F and G that she has not considered and that have the consequence that G is a better move than F?23 We might answer the question posed by saying that grandmasters have some sort of process which reliably overlooks unpromising sequences of moves, generating statistically abnormal good lots. We might even go so far as to say that the sequences they consider are those most relevant to determining which move is best. This kind of story constitutes a reliabilist defence of grandmasters’ inferences regarding the desirability of chess moves. The naive way to give a reliabilist defence of a rule of inference is to claim: (1) reliabilism is true; and hence (2) if the rule is reliable, beliefs generated by it are justified. Unfortunately, it is not this easy. The most obvious sign that something has gone wrong is that I can use the naive reliabilist defence to defend myself against the chess version of the problem of the bad lot. I am a rank-and-file player, yet I can blithely assert that if I am reliable in my beliefs about which move is best, I am justified in these beliefs. Clearly, this is not a sufficient reliabilist defence of my process of selecting chess moves. No one is going to be convinced by this simple statement that my beliefs about the desirability of chess moves are justified. Since I should not be able to avoid the problem of the bad lot in chess, the naive reliabilist defence is too simplistic. Consideration of the failure of naive reliabilist defences of rules of inference suggests that the conditions for giving a reliabilist defence of a form of inference may be more stringent than the conditions placed by reliabilism on the justification of beliefs generated by that form of
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inference. This makes intuitive sense: a reliabilist defence of a form of inference is an attempt to convince others of its epistemic credentials, while a main theme of externalism is that a belief can be justified when the believer does not have access to its epistemic credentials. I propose that one requirement for a reliabilist defence of a process R is an argument that there is not a reliable process which recommends the belief that R is unreliable. This requirement succeeds in eliminating the possibility that I could give a reliabilist defence of my ability to generate good lots in chess, because there is a reliable process, induction, which recommends the belief that I am unreliable at generating good lots in chess. Note that this requirement is different from the requirement for justification cited by comparative reliabilism. The latter is that, in order for belief B to be justified, there must not be a reliable, competing process which undermines B. The requirement I am proposing is the demand for an argument that there is not a reliable, competing process which undermines B. A reliabilist defence of a process must do more than show that there are no reliable, undermining processes. Imagine a shy clairvoyant. She has only made ten predictions, nine of which have turned out to be accurate. There is not available to her the process of induction from past failures of clairvoyance to the unreliability of clairvoyance. If her clairvoyance is reliable, she may or may not be justified in her beliefs based upon clairvoyance.24 In either case, she certainly is not going to be able to convince us that her beliefs generated by clairvoyance are justified! The record of success of her clairvoyance is simply not substantial enough. In order to convince us that the beliefs she generates by clairvoyance are justified, she must give us strong, presumptive reason to think that her process is reliable. To give strong, presumptive reason to think that a process is reliable involves the appeal to a robust track record of apparent reliability. I say ‘apparent reliability’ because the demand for conclusive, non-circular evidence for the reliability of a rule is an internalist one (one that is in fact often impossible to satisfy). Another part of giving strong, presumptive reason in favour of the reliability of a process is to sketch a plausible mechanism by which the process reliably gets at the truth. This helps eliminate the possibility that the apparent success of the process is an accident. To summarize, I propose that the following are two conditions for a reliabilist defence of a process R: (a) An argument that there is no reliable process which undermines R.
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(b) Strong, presumptive reason to think R is reliable. i. A robust track record of apparent success. ii. A sketch of a plausible mechanism by which R reliably gets at the truth. I hold that (a) and (b) are necessary conditions for a reliabilist defence of R. I am agnostic regarding the sufficiency of these conditions. I think that (b) is fulfilled when (i) and (ii) are fulfilled to a sufficient degree. I think (i) and (ii) can be traded off: the more robust the track record, the less of a need for a sketch of a mechanism, and vice versa. 6. Conclusion I have argued that plausible attempts to address the problem of the bad lot and the argument from indifference will defend IBE based on the success of science and will embrace reliabilism (or some other form of externalism) in order to defuse the threat of vicious circularity. I have also distinguished between reliabilist conditions for justification and reliabilist defences of a rule of inference and proposed criteria that the latter ought to meet. If I am correct, a successful response to the problem of indifference must provide the detailed arguments required by these criteria. The problem of indifference cannot be easily solved; but neither should the scientific realist despair of solving it. notes 1 Bas van Fraassen, Laws and Symmetry (Oxford: Clarendon Press, 1989), 142. 2 Ibid., 142–3. 3 Ibid., 143. 4 Van Fraassen refers to this as ‘retrenched’ IBE (ibid., 145). 5 Van Fraassen appears to be relying upon the principle of indifference here. The principle of indifference is associated with the classical theory of probability and has fallen into disrepute. This might suggest a response to the argument from indifference: since it is based on a false principle, it can be safely ignored. However, this response is inadequate. It is true that some Bayesians think the assignment of priors is entirely subjective. According to them, we can choose to assign a high probability to the best available explanation and low probabilities to unavailable explanations. But this defence avoids the argument from indifference only by making the assign-
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6 7 8 9 10 11 12 13 14
15
16
17 18 19 20
21 22
ment of priors arbitrary. It does not treat the explanatory virtues as evidence, so it is not what the scientific realist is after. Ibid., 146. Bas van Fraassen, The Scientific Image (Oxford: Clarendon Press, 1980), 46. Peter Lipton, ‘Is the Best Good Enough?’ Proceedings of the Aristotelian Society 93 (1993): 94–5. See van Fraassen, Laws and Symmetry, 143–4. Ibid., 143. Paul Thagard, ‘The Best Explanation: Criteria for Theory Choice,’ Journal of Philosophy 75 (1978): 76–92. Van Fraassen, The Scientific Image, 90. Hilary Putnam, Mathematics, Matter and Method, vol. 1 of Philosophical Papers (Cambridge: Cambridge University Press, 1975), 73. Van Fraassen, The Scientific Image, 70–96; Peter Lipton, Inference to the Best Explanation (London: Routledge, 1991), 168–74; Arthur Fine, ‘The Natural Ontological Attitude,’ reprinted in The Philosophy of Science, Richard Boyd, Philip Gasper, and J.D. Trout, eds. (Cambridge, MA: MIT Press, 1991), 261– 5; P. Kyle Stanford, ‘An Antirealist Explanation for the Success of Science,’ Philosophy of Science 67 (2000): 266–84. Larry Laudan, ‘A Confutation of Convergent Realism,’ reprinted in Philosophy of Science, Boyd, Gasper, and Trout, eds., 240–1; Fine, ‘The Natural Ontological Attitude,’ 262–3; Lipton, Inference to the Best Explanation, 158– 68; Bas van Fraassen, ‘Empiricism in the Philosophy of Science,’ in Images of Science: Essays on Realism and Empiricism, Paul Churchland and Clifford Hooker, eds. (Chicago: University of Chicago Press, 1985), 259–60. Richard Boyd, ‘Realism, Approximate Truth, and Philosophical Method,’ in Scientific Theories, C. Wade Savage, ed., Minnesota Studies in the Philosophy of Science, vol. 14 (Minneapolis: University of Minnesota Press, 1990), 385–9; Richard Boyd, ‘On the Current Status of Scientific Realism,’ reprinted in Philosophy of Science, Boyd, Gasper, and Trout, eds., 219. Arthur Fine, ‘Unnatural Attitudes: Realist and Instrumentalist Attachments to Science,’ Mind 95 (1986): 163. See Lipton, Inference to the Best Explanation, 161. Fine, ‘The Natural Ontological Attitude,’ 263. See Boyd, ‘Approximate Truth’; Lipton, Inference to the Best Explanation; Stathis Psillos, Scientific Realism: How Science Tracks Truth (London: Routledge, 1999). Psillos, Scientific Realism, 81–90. See William Alston, The Reliability of Sense Perception (Ithaca: Cornell University Press, 1993), 15–17, 115–19.
196 Samuel Ruhmkorff 23 There are more versions of the problem of the bad lot for grandmasters. For example, how does a grandmaster know that there are not longer sequences which would reverse the results of her calculations? 24 According to comparative reliabilism, the shy clairvoyant’s belief is not justified because there is a reliable process (epistemic caution) which endorses agnosticism concerning the reliability of her clairvoyance. According to ordinary reliabilism, she would be justified simply because her clairvoyant process is reliable.
Part Two: Respondeo JOHN WOODS
James Franklin sees in probability an interesting parallel with continuity and perspective.1 All three of these things took a long time before yielding to mathematical formulation, and, before that happened, judgements of them tended to be unconscious and mistaken. I have a somewhat different version of this story. Sometimes a conceptually inchoate idea is cleaned up by a subsequent explication of it. Sometimes these clarifications are achieved by modelling the target notion mathematically. Sometimes the clarification could not have been achieved save for the mathematics. We may suppose that something like this has proved to be the case with perspective and continuity. To the extent that this is so, anything we used to think of these things which didn’t make its way into the mathematical model could be considered inessential if not just mistaken. This is one particular approach to what, in the comments on Part One, I called the residue problem. It is interesting to reflect on how well this line of thought fits the case of probability. In raising the matter, we are calling attention to two questions: (1) What was probability like before Pascal? (2) How do we now find it to be? Concerning the first of this pair of questions, I think that we may suppose that, in their judgements under conditions of uncertainty, people routinely smudged such distinctions as may have obtained between and among ‘it is probable that,’ ‘it is plausible that,’ and ‘it is possible that.’ If we run a strict version of the line we have been considering over this trio, then not making it into the calculus of probability leaves all that is left of these blurred idioms in a probabilistically defective state. There is a sense in which this is not the wrong thing to conclude, but it is a trivial one. For if what I sometimes intend by ‘probability’ fails to find a safe harbour in the probability calculus,
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then it is not a fact about probability that the probability calculus honours. But, unlike what may have been the case with perspective and continuity, we must take care not to say without further ado that those reasonings that don’t make the Pascalian cut are mistakes of reason or even mistakes of probabilistic reason. In this I cast my lot with Jonathan Cohen2 and Stephen Toulmin,3 albeit for somewhat different reasons. With Cohen I agree that some of the Kahneman-Tversky4 experimental results which show their subjects to have been bad Pascalians do so only if they had undertaken to be good Pascalians. The alternative, of course, is that, even though they were invited to be Pascalians and primed to make a workmanlike job of it, their sole mistake is that they slid unaware into a non-Pascalian disposition toward reasoning under conditions of uncertainty. Had they been drawn to the task of compounding plausibilities, it is far from clear that the Kahneman-Tversky results would show their efforts in a bad light. Here, too, I am with Toulmin when he says that not all judgements of probability, even when made by working scientists, express or attempt to express the concept of aleatory probability or to comport with its theorems. A similar moral can be drawn from the sheer semantic sprawl of the idioms of possibility. In my comments on Part Five I try to make something of the notion of something being a ‘real possibility’ for a cognitive agent. It is a difficult notion to pin down. But two things are clear. One is that it is not to be identified with the ‘possibility’ of any known modal logic. The other is that this does not show that my orphaned notion is untenable or philosophically impotent. Let us take it that, unlike perspective and continuity, idioms of probability (or probability/plausibility/possibility) that don’t cut the Pascalian mustard leave residues of philosophically interesting usage. If this were so, there might well be philosophically important issues, the successful handling of which requires the wherewithal of this conceptual residue. Again, standard answers to Kahneman-Tversky questions don’t cut the mustard of aleatory probability, but they do comport with conditions on plausible reasoning. What, then, are we to say? That these bright, well-educated subjects are Pascalian misfits or that they are more comfortably at home (though unconsciously) with a plausibility construal of their proffered tasks? If we say the second, we take on an onus we might be unable to discharge, or anyhow discharge at will. It is the task of certifying the conditions under which these non-Pascalian manoeuvres are well justified. In lots of cases, we won’t have much of a clue as to how to achieve these elucidations. (For years I have tried with-
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out success to get a deep grip on the notion of real possibilities.) Small wonder, then, that what Gabbay and I call the Can Do Principle beckons so attractively. This is the principle that bids the theorist who is trying to solve a problem P to stick with what he knows and to make a real effort to adapt what he knows to the requirements of P. One of the great attractions of Pascalian probability is that we know how to axiomatize it. Can Do is right to say that it would be advantageous if we could somehow bend the probability calculus to the task to hand. But sometimes, the connection just can’t be made. Bas van Fraassen is spot on in pointing out that 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 self-indulgence.’ This calls to mind Gabbay’s and my Make Do Principle, which is the degenerate case of Can Do. Make Do is just Can Do in circumstances in which the fit with P cannot be achieved satisfactorily. If P is the problem of avoiding ‘the naivety and oversimplification inherent in much of traditional epistemology,’ then a decision to deploy the theory of probability by brute force would be a case of Make Do. It would capture the mood of the tasker who, not knowing what to do, does what he does know how to do, and wholly ignores that it is all beside the point. A point on which ‘The Day of the Dolphin’ and ‘Cognitive Yearning’ are at one is their recognition of the naivety and oversimplification that do indeed inhere in much of traditional epistemology. A case in point is the question of what is observable, a matter at once fundamental to van Fraassen’s constructive empiricism and yet not much elucidated by traditional epistemology. Van Fraassen seeks to repair this omission via the artful device of epistemic marriages. For this he needs, among other things, degrees of belief and indexicality, and hence constraints on iterations of belief-expression. And for this in turn he needs, or wants, subjective probability adapted to the requirements of a theory of changes of view. The Can Do Principle bids van Fraassen to do what in part he does do. It bids him to try to accommodate changes of view in a Bayesian interpretation of the probability calculus. But it doesn’t work. This presents the theorist with alternatives. One is to interpret subjective probability afresh. The other is further to adapt what already lies to hand. A third (although discredited) option is to brazen it out with Make Do. Van Fraassen picks the option that comports best with the spirit of Can Do. He replaces Conditionalization (which doesn’t work) with the General Reflection Principle. This is a signifi-
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cant and principled deviation from Bayesianism, which leaves as much of Bayesianism standing as is compatible with the epistemological objectives at hand. That, together with the policy of envaguement, arguably puts the idea on a feasible and load-bearing footing, wherewith we have a substantial and methodologically circumspect contribution to epistemology. Subjective probability is also at the centre of the essays of Paul Bartha and Jarett Weintraub, and each plays interesting variations on Can Do and other methodological issues. Suppose, with Bartha, that we want to ground subjective probability in a betting formalism. For this we must have, by the Dutch Book argument, the countable additivity axiom. The de Finetti lottery appears to discomport with the axiom. So what are we to do? With considerable ingenuity, Bartha shows that in special cases we can replace countable addivity with relative probabilities. This solves the problem in the spirit, if not the letter, of Can Do. For, like van Fraassen, Bartha retains as much of what we already know how to do as is compatible with the obligation to make some changes to what went before. The solution is a conservative one, which is what Can Do in general calls for. Both van Fraassen and Bartha accommodate a problem by changing the structure within which the problem exists. This is done in ways that might attract Quine’s approval, with his interest in minimal mutilation. Weintraub pursues, also with considerable elegance, a different strategy. It is the strategy of direct attack, that of showing that accommodation is unnecessary since there is nothing to accommodate. In this case, the issue is (Kyburg’s) Lottery Paradox and Harman’s proposed accommodation of it in a structure of degrees of belief. The force of Weintraub’s argument is to show that the ‘problem’ is not one that requires this kind of accommodation. I would go him one further: It is not a problem at all. Paradoxes are interesting things. They attract different kinds of response. One is to cut and run, as Tarski did in the face of the Liar. The other is to stare them down, as Russell did with the Barber. In cutting and running from the Liar, Tarski gave up on the English predicate ‘true’ and stipulated its replacement by a transfinite number of artificial look-a-likes. But of the barber who shaves himself if and only if he doesn’t, Russell’s face-down was to retain the English predicate ‘barber’ and dismiss the Paradox’s alleged instantiation of it. Weintraub’s response to the Lottery is thus in the spirit of facingdown. I myself am inclined to go whole hog. At the heart of the issue are certain facts about lotteries which all
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concerned are rightly expected to agree on. Let L be a million-ticket lottery. Then, for each ticket, the odds against it being the winner are astronomical. But this does not change the fact that it is certain or astronomically probable that one of them is indeed the winner (assuming no monkey business, or ticket-printing errors, and the like). Notice how these probability idioms behave. Notice in particular that they don’t respect the standard-closure operations of classical logic. This being so, no one is in the slightest inclined to conclude from the fact that, with respect to each ticket, there is a massive probability against it being the winner, that there is a massive probability that none at all is the winner. Now we may also suppose that when confronted with the example of any given ticket, and asked whether it is the winning ticket, a wholly reasonable answer could be, ‘No, it isn’t.’ People who see a paradox looming take liberties with this fact. They observe (or anyhow claim) that there is an interpretation of the response ‘No, it isn’t,’ such that the totality of such responses does indeed honour the classical closure conditions. So, from concessions in the form ‘not-P,’ for all P, it is concluded ‘Never P.’ There is no doubt that ‘No, it isn’t’ sometimes does bear such an interpretation. But if it bears it in the case of the Lottery, it is notable that the response in each case is classically stronger than the evidence for it. The probability claims in which, in turn, each answer ‘No, it isn’t’ is embedded, don’t close classically. Why, then, would we think that the answers to which they lend their respective support would do the opposite, especially when assuming so leads to a contradiction? Clearly, ‘No, it isn’t’ also bears interpretations for which classical closure fails. Such would be the case if, for example, ‘No, it isn’t’ were taken as expressing a high implausibility. So construed, we would have two agreeable facts to take comfort in. One is that such a response is both reasonable and proportionate to the probabilistic evidence in each case. The other is that the idiom of plausibility, like the idioms of probability, does not close classically. The Lottery Paradox is a Barber. Samuel Ruhmkorff makes an attractive appropriation of reliabilism to defend inference to the best explanation (IBE). This gives me pause on two counts. One is, What are we to make of the claim that science (anyhow our science) has been a success? The other is, What are we to make of the idea of explanation in anomalous contexts? Consider these in order. It is wholly natural to assume that what constitutes the success of science is that it gets at the real truth of things as they actually are. In
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‘Cognitive Yearning and the Fugitivity of Truth,’ I’ve tried to make explicable the naturalness of this assumption. Our desire for knowledge is fulfilled by our being in the requisite states of belief. So if I desire to know whether P, believing that P is in the general case thinking that you know it. This fulfills the desire to know whether P, even if the belief that P is mistaken. Bearing on this is the quotationality of belief. Believing that P is believing that P is true. And because beings like us take the ‘realist stance,’ we are realists about truth, which is just to say that we have a natural inclination to think that realism is true. This is all grist for the mill of the success of science. The beliefs that science prompts us to have do not by and large do us much damage. On the contrary, we survive, we prosper, we do particle physics, and occasionally we build splendid civilizations. It is in these respects that the success of science (and most of our other cognitive practices too) consists. To say as well that our survival, prosperity, scientific acuity, and civilization are the resultant of beliefs that satisfy realist conditions is to venture a philosophical explanation of what our success consists in. But, as we all very well know, there is no respectable anti-realist who doubts our success – our survival, our prosperity, and so on. What he or she doubts is that particular philosophical interpretation of what our success amounts to. This seems an appropriate place for the reliabilist to play his cards. What explains the fact, it might be asked, that these are the beliefs upon which our survival, prosperity, and so on, rest, rather than (say) their negations? Surely it is that these beliefs are true, or some near thing, and that their negations are false. Is it that way? I have my doubts. Consider a different context. Suppose we are trying to fashion an answer to someone who denies (or seriously calls into question) the existence of the external world. The answer proposed is an IBE answer. The best explanation of the coherence, stability, and so on of our experiences is that they are caused by an external world of the kind they purport to be experiences of. I am not minded to accept this argument. Yes, the external world is the best explanation of external world experiences, but that is insufficient to the task at hand. The reason is that explanation is already an external world-presupposing concept. The IBE is therefore circular. Suppose that the actual fact of the matter were that the external world did not exist. What, then, would explain our external world experiences? The answer is, I think, that nothing would, not even that our experiences are implanted by God (since God is conceived as external to us). I am inclined to think, therefore, that there is no successful reliabialist-IBE
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answer to van Fraassen and his ilk, notwithstanding Sam Ruhmkorff’s adroit handling of the self-referential factor in the answer he proposes. notes 1 James Franklin, The Science of Conjecture: Evidence and Probability before Pascal (Baltimore, MD: Johns Hopkins University Press, 2001). 2 L. Jonathan Cohen, An Essay on Belief and Acceptance (Oxford: Oxford University Press, 1992). 3 Stephen Toulmin, The Philosophy of Science (London: Hutchinson University Library, 1960). 4 David Kahneman, Paul Slovic, and Amos Tversky, eds., Judgments under Uncertainty: Heuristics and Biases (Cambridge and New York: Cambridge University Press, 1982).
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Part III Logic and Language
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12 Aristotle and Modern Logic D.A. CUTLER
1. Introduction In general, a completeness theorem has the following form: If a conclusion is a semantic consequence of a set of premises, then the conclusion is deducible from the premises. In modern treatments, the deducibility relation is completely syntactic: it is defined on sequences of strings of uninterpreted symbols. Semantic consequence is defined in terms of truth in a structure and structures are usually regarded as set-theoretic constructs. The statement of the completeness theorem thus might appear to depend on concepts that are particular to modern mathematical logic. Can the question of completeness be raised in a form that does not rely on these concepts? Could Aristotle, for example, have raised it? Jonathan Lear has argued that it would be ‘anachronistic’ to suppose that Aristotle could have posed the problem of completeness for his syllogistic logic.1 Lear supposes that a completeness problem can be raised only for a logical system whose underlying language is ‘formal’ in the sense of having none but model-theoretic semantics. Languages of this sort are particular to modern mathematical logic and have no place in Aristotle’s conception of logic. In response to Lear, John Corcoran and Michael Scanlan argue that Aristotle could indeed have posed a form of completeness problem.2 They correctly point out that Lear’s construal of the problem is, in certain respects, too narrow. Corcoran and Scanlan attempt to bridge a conceptual gap between ancient and modern logic by showing that Aristotle and the modern logician share a common conception of certain logical problems – among them, completeness – in spite of a variety of accidental differences in presen-
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tation. Timothy Smiley has gone further in arguing that Aristotle actually raised the completeness problem and attempted to settle it by means of a proof.3 In spite of my agreement with Corcoran and Scanlan’s criticisms of Lear, I think that their discussion of completeness neglects precisely the features of the notion that are most important from the point of view of modern logic. These features, which I discuss in subsection 3.4 below, give the completeness theorem for first-order logic, for example, a significance that their version of the completeness theorem for Aristotle’s syllogistic logic does not share. It follows, I think, that Corcoran and Scanlan have been overly hasty in assimilating Aristotle’s problematic to our own. In my view, an examination of Smiley’s reconstruction of Aristotle’s ‘completeness proof’ only supports my contention. Did Aristotle formulate the problem of proving the completeness of his syllogistic logic? Did he attempt a proof of completeness? As interesting as these historical questions are, my primary interest for now is the notion of completeness itself. Proper discussion of the historical questions presupposes an analysis of the meaning and significance of completeness, and this, rather than the historical questions per se, will be my concern. 2. Informal Logical Notions and Their Formal Counterparts4 In modern mathematical logic, we understand a completeness theorem as stating a relationship between a formal notion of semantic consequence and a formal notion of deducibility. In order to comment on the philosophical significance of this relationship, it will be necessary to describe the formal notions of consequence and deducibility that the theorem refers to. I will present the view that the formal notions of mathematical logic represent theoretical reconstructions of an intuitive notion of logical consequence that has been used – or at least presupposed – by logicians since Aristotle’s time. 2.1 Logical Consequence and Model-Theoretic Consequence Let us look at a claim that involves the intuitive notion of logical consequence: the claim that Pythagoras’ theorem is a logical consequence of the axioms of Euclidean geometry. Logical consequence is a relation that can hold between a set of premises – the Euclidean axioms – and a conclusion – Pythagoras’ theorem. A pair composed of a set of
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premises and a conclusion is a premise-conclusion argument. A premiseconclusion argument is valid just in case its conclusion is a logical consequence of its premises.5 If we give the axioms and theorems of Euclidean geometry their traditional interpretation, they concern a particular set of objects, points, lines, and figures in space and a particular set of relations that can hold between these various objects. Points can lie on lines, lines can intersect, and so on. In mathematical logic, we represent the notions of validity and logical consequence as they apply to formal languages by presenting model-theoretic semantics. A formal language is an inductively defined set of strings over an arbitrary set of symbols. As such, the language is just a set of strings of uninterpreted symbols until we specify a class of structures that interpret it and assign truth-values to its sentences. We call a structure that interprets the Euclidean axioms a Euclidean space. If a conclusion is a logical consequence of a set of premises, then for any way that things could be, which includes the truth of the premises, if things were that way, the conclusion would have to be true as well. A sentence is true in a structure if certain well-defined objectual relations hold between the parts of the structure named by the sentence’s terms. In this sense a structure which satisfies a sentence represents a way that things could be, such that the truth-conditions of the sentence are realized. If Pythagoras’ theorem is true in all Euclidean spaces then it is a logical consequence of the Euclidean axioms. 2.2 Informal and Formal Deduction Attempts to establish the claim that Pythagoras’ theorem is a logical consequence of the Euclidean axioms have traditionally taken the form of chains of simpler premise-conclusion arguments. The conclusion in the last argument in such a chain is Pythagoras’ theorem, and each premise of each argument in such a chain purports to be either the conclusion of an argument coming before it in the chain or else one of the Euclidean axioms. A chain of arguments of this sort establishes the claim that Pythagoras’ theorem is a logical consequence of the Euclidean axioms for anyone able to recognize that each of the arguments in the chain is valid and that the final conclusion of a sequential chain of valid arguments is entailed by the ‘ultimate premises’ of the sequence: that is, by those sentences that occur in the sequence but which are not the conclusions of any of the arguments in the sequence. An argumentation or deduction is a three-part structure composed of a set of premises,
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a conclusion, and a discourse called a chain of reasoning which purports to show that the conclusion is a logical consequence of the premises. An argumentation is cogent if and only if its chain of reasoning is sufficient to show that its conclusion is a logical consequence of its premises. If an argumentation is cogent, then it is capable (at least in principle) of conferring knowledge of validity. That is, if an argumentation is cogent, then one could come to know that its premises logically entail its conclusion by following its chain of reasoning. Thus, an attempt to establish the claim that Pythagoras’ theorem is a logical consequence of the Euclidean axioms traditionally takes the form of an argumentation whose premises are the Euclidean axioms and whose conclusion is Pythagoras’ theorem. A deductive system is a set of deductions in a formal language. It is supposed to model the process of reasoning in natural language. One way of specifying a deductive system is to present a set of formal inference rules. This gives rise to a relation of deducibility on the formulas of the language: A formula I is deducible from a set of formulas * if and only if there exists a finite sequence of formulas I1, ..., In such that (1) each Ii is a member of *, or results from an application of one of the inference rules to earlier members of the sequence; and (2) In is I. The sequence I1, ..., In is called a formal deduction of I from *. A deductive system is effective if its set of inference rules is effective: that is, if we can decide by a finitary procedure whether any particular premiseconclusion argument is an instance of one of the rules. Since a proof is a finite sequence of formulas, it follows that if the rules are effective, there is an effective procedure for deciding whether or not a given sequence of formulas is a deduction. 2.3 Logical Theories Let us say that a logical theory or logic is a triple composed of a formal language, a model-theoretic semantics, and a formal system of deduction. These notions are characteristic of the modern approach to logic, and it is in terms of them that we understand questions of completeness: for a given logic, is there a formal deduction associated with every case of the relation of model-theoretic consequence? For first-order logic the question is answered in the positive sense by Gödel’s completeness theorem. For second-order logic, with the standard relation of secondorder model-theoretic consequence, and any effective deductive system, it is answered in the negative sense by Gödel’s incompleteness
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theorem.6 The completeness theorem says something about the relationship between a sentence’s being a logical consequence of other sentences and the possibility of knowing that it is a consequence of other sentences. If a logic is complete then whenever the relation of logical consequence holds, it can, in principle, be known to hold by someone who can follow the appropriate deduction. 3. Completeness and Exhaustiveness 3.1 Lear on Aristotle Jonathan Lear claims that ‘[i]t would be anachronistic to attribute to Aristotle the ability to raise the question of completeness’ because the question ‘depends on an awareness of the syntax/semantics distinction.’7 By ‘the syntax/semantics distinction,’ Lear means the distinction made in modern systems of logic between the syntactic notion of deducibility and the notion of semantic consequence. In these systems, recall, the rules of inference may be regarded as purely syntactic operations on uninterpreted strings. What allows the relation of deducibility to be regarded as a consequence relation at all is that it can be shown to preserve a particular model-theoretic consequence relation. By contrast, Aristotle takes the notion of ‘following of necessity’ – his analogue to the intuitive notion of logical consequence – as primitive and attempts to define rules of inference, for interpreted sentences, such that any sentence deduced from others by means of a rule obviously ‘follows of necessity’ from them. So instead of defining a particular relation of semantic consequence and then showing that his rules of deduction are sound relative to it, Aristotle chooses a set of simple, obviously valid, inferences and ‘invites one to agree that these are cases in which the conclusion follows of necessity from the premises.’8 Our intuition that the rules really are valid modes of inference makes a proof of soundness unnecessary. On Lear’s view, Aristotle also makes a claim about his system that is in some respects analogous to the assertion that it is complete. According to Lear, Aristotle argues that every deduction that we intuitively recognize as cogent (in the sense of section 2.2 above) can be represented as a series of syllogistic inferences.9 On Lear’s view, Aristotle’s attitude to syllogistic inference is comparable to a modern logician’s attitude to the notion of a computable function: a precisely defined mathematical notion – that is, that of a recursive function – is presented
212 D.A. Cutler
as an analysis of an informal notion – that is, that of a computable function – but the very informality of the latter notion precludes a rigorous proof that the two notions are coextensive. Nevertheless, there are various informal arguments, falling short of rigorous proof, which at least indicate that the two notions are coextensive. According to Lear, we should understand Aristotle’s presentation of the theory of the syllogism in a similar light: that is, as a mathematical analysis of a more or less imprecise intuitive notion. Making the obvious analogy to Church’s thesis, Lear refers to the claim that every informally cogent deduction can be represented as a chain of syllogisms as ‘Aristotle’s Thesis.’ Lear interprets A23 of Prior Analytics as an argument in support of Aristotle’s Thesis and hence as the closest kin to a modern completeness proof that Aristotle’s conception of logic will allow. 3.2 Corcoran and Scanlan on Lear and Aristotle In their discussion of Lear’s book, Corcoran and Scanlan argue that Lear has tied the question of completeness too closely to what they call ‘the particular artifacts of modern logic.’10 They see as one of these artifacts the view that we must regard systems of deduction as purely formal: Lear ... says ‘from the perspective of modern logic the point of a completeness theorem is to establish the extensional equivalence of two distinct ... relations.’ The two relations are, of course, deducibility and having no counter-interpretations. But Lear fails to see that the possibility of this kind of question transcends the particular artifacts of modern logic and can be raised in any situation involving a positive and negative criterion [for validity]. He seems to think that the possibility of raising such a question of exhaustiveness (or completeness) depends on the syntactical character of deducibility and the model-theoretic character of ‘no counterinterpretations.’11
Corcoran and Scanlan give the following account of completeness questions. Suppose that, within some well-defined class of premiseconclusion arguments, we have a sufficient criterion for validity and a sufficient criterion for invalidity. The criteria are exhaustive if every argument satisfies one criterion or the other. If a sufficient criterion for validity and a sufficient criterion for invalidity jointly exhaust a class of arguments then the two criteria together constitute a necessary and sufficient condition for validity within that class of arguments. In gen-
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eral, according to Corcoran and Scanlan, a completeness question is a question about the exhaustiveness of a criterion for validity together with a criterion for invalidity. One way of attacking the problem of capturing the extension of logical validity in general is to attempt to isolate a proper subclass of the all premise-conclusion arguments and show that there are criteria that are exhaustive of that subset. This would constitute a partial solution of the problem of characterizing logical validity. One might then hope to extend one’s criteria so as to make them exhaustive of a more comprehensive class of arguments.12 Aristotle was interested in the class of arguments called ‘categorical syllogisms.’ These are composed of the four types of categorical sentences: A: ‘All A is B9 ; E: ‘No A is B9 ; I: ‘Some A is B9 ; and O: ‘Some A is not B9 (where A and B are one-place predicates). However, a categorical syllogism is more than just a premise-conclusion argument constructed from categorical sentences. In order to be a syllogism, a categorical argument must satisfy four additional conditions.13 First, it must be valid: the assertion of the argument’s premises necessarily involves the assertion of its conclusion. Second, the premises must be minimal: the removal of any premise would destroy the argument’s validity. Third, the argument must be non-circular: the argument’s conclusion cannot repeat any of the premises, either in exactly the same words or as a syntactical variant that is semantically equivalent. Finally, the argument must be multi-premised: it must have more than one premise. These four features of the syllogism make it rather more special than an arbitrary premise-conclusion argument. John Woods and Andrew Irvine provide an illuminating discussion of these conditions and Aristotle’s probable reasons for concentrating his interest on arguments that satisfy them.14 With the exception of validity, these conditions mark a difference in focus between Aristotle’s logic and modern logic. The latter has most typically been concerned with the more general conception of a premise-conclusion in conveniently restricted languages and with appropriate notions of validity and deducibility for such languages. However, this difference in focus is unimportant from the point of view of the present discussion of completeness. Aristotle had a sufficient criterion for validity of a categorical syllogism in the form of his notion of ‘perfectibility’: a syllogism is perfect if it is obviously valid. A syllogism that is not obviously valid can be perfected – that is, shown to be valid – by chaining together perfect syllogisms. According to Corcoran and Scanlan, Aristotle also had a suffi-
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cient criterion for the invalidity of a categorical syllogism: say that two categorical syllogisms have the same narrow form if and only if ‘there is a one to one correspondence between their respective sets of terms which transforms one ... [syllogism] into the other.’ A syllogism (*, I) is invalid if there is another syllogism (*9 , I9 ) that has the same narrow form as (*, I) and has true premises and a false conclusion.15 (In this case, we say that (*9 , I9 ) is a counter-interpretation to (*, I)). So, according to Corcoran and Scanlan, Aristotle could raise the question whether these criteria are exhaustive of the class of categorical syllogisms. Thus, according to Corcoran and Scanlan, the truth of Lear’s claim that Aristotle lacks a ‘syntax/semantics’ distinction is consistent with allowing that Aristotle could pose a completeness problem. The fact that we can consider the terms that occur in a syllogism as fully interpreted does not preclude us from testing for validity by considering other syllogisms of the same form and trying to find one with true premises and a false conclusion. In fact, this kind of test presupposes that we are working in a fully interpreted language since we are taking the truth or falsity of the component sentences of syllogisms into consideration. Aristotle’s method of ‘perfecting’ syllogisms is sound just in case no syllogism perfected by this method has a counter-interpretation. It is complete just in case the method allows for the ‘perfection’ of every syllogism that has no counter-interpretation. In fairness, it should be noted that Lear himself is very close to this interpretation of the completeness problem.16 The reason he rejects it seems to be that he supposes that in order for Aristotle to pose the completeness problem, he would have had to regard ‘having no counter-interpretation’ as an analysis of ‘following of necessity.’ For, to be an analysis of the latter notion, ‘having no counter-interpretations’ would have to be both a necessary and a sufficient condition of ‘following of necessity.’ The observation that, in a language with a small enough vocabulary, an argument that is intuitively invalid will satisfy the criterion is commonplace.17 Aristotle presumably understood this and hence did not offer ‘having no counter-interpretations’ as an analysis of ‘following of necessity.’ The possible insufficiency of the ‘no counter-interpretation’ criterion is irrelevant on Corcoran and Scanlan’s construal of the completeness problem because they see completeness as the joint exhaustiveness of two criteria, neither of which need be both necessary and sufficient.
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3.3 Smiley: Aristotle’s Attempt at a Completeness Proof Although Corcoran and Scanlan argue that Aristotle could have raised the completeness problem, they note that ‘whether he did or not is another question,’18 a question that they do not attempt to answer.19 By contrast, Smiley has argued that Aristotle not only raised the problem of completeness, but also attempted to settle it by means of a proof.20 Smiley locates Aristotle’s completeness proof at A23 of Prior Analytics. Recall that, on Lear’s view, this is the chapter that contains the argument for ‘Aristotle’s Thesis.’ Smiley regards the grounds for Lear’s view that it would be ‘anachronistic to attribute to Aristotle the ability to raise the question of completeness’ as ‘undeniable.’21 However, he takes the view that in emphasizing the difference between Aristotle’s project and the modern one, there is a danger of overlooking their similarity: a similarity, which seems to me to be more significant than their difference. To the objection that Aristotle was ‘not conscious of the distinction between syntactic and semantic consequence,’ I would rejoin that Aristotle was conscious of the difference between what follows and what can be shown to follow – and therefore of the need to prove completeness.22
It will be instructive to look at Smiley’s reconstruction of Aristotle’s argument. Smiley points out that in the syllogistic case, valid arguments satisfy what I will refer to as the chain condition. The chain condition can be stated as follows: Let AB denote a sentence of any of the categorical forms A, E, I, or O, regardless of whether A is the first or second predicate (i.e., in more traditional terms, regardless of whether A is the subject term or predicate term). A syllogism satisfies the chain condition if it is of the form: AC, CD, DE, EF, ..., GH, HB; therefore, AB. According to Smiley, Aristotle attempts to prove completeness by attempting to prove that every valid syllogism satisfies the chain condition.23 A valid argument is one whose conclusion follows of necessity from its premises. The conclusion is shown to follow from the premises if it is deducible from them. If the chain condition holds then it is clear that a valid syllogism must be one that can be shown to be
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valid because it is actually a very trivial matter to construct a deduction of the conclusion from the premises of a valid argument – as one can see from a few moments of experimentation.24 3.4 Completeness versus Exhaustiveness Corcoran and Scanlan are certainly correct to point out a sense in which the formulation of the completeness question for a logic does not require that the logic’s underlying language have no interpretation beyond that given it by a formal semantics. They are also correct to point out that Aristotle’s criterion of invalidity – having a counterargument – is analogous in some ways to the criterion of invalidity in modern logic – having a counter-model. Thus, there is indeed a sense in which Aristotle could raise a question that is analogous to the completeness question. But this being said, it is important to point out that the completeness problem, as it arises in twentieth-century logic, has a significance that goes beyond the exhaustiveness of two criteria. The completeness theorem for first-order logic shows that a non-finitary notion, first-order consequence, has a finitary characterization in terms of first-order deduction. The possibility of this kind of characterization and the mathematical and conceptual problems that it raises are not even hinted at in the case of syllogistic logic. First-order consequence is a non-finitary notion because there are first-order sentences that are satisfiable only in infinite structures. Consider, for example, the following sentence:
xy(x z sy) & xy(x z y sx z sy). Thus there are first-order arguments with finitely many premises that can be shown to be invalid only by considering the infinite models of some set of sentences. Even if we restrict our attention to arguments that have a finite number of premises, the formulation of the completeness problem for first-order logic requires the notion of satisfaction in an infinite structure. By contrast, there is no sentence in the language of syllogistic logic that is satisfied only in an infinite model.25 Every sentence in the language of syllogistic logic may be expressed in the language of the monadic predicate calculus.26 It is a meta-theorem of monadic logic that if I is a pure monadic sentence, then if I is satisfiable, I is true in a structure whose domain contains at most 2k members, where k is the number of predicate letters in I .27
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It is interesting to note that Herbrand came very close to proving the completeness theorem for first-order logic.28 What seems to have kept him from proving completeness – and even from thinking that the problem can be formulated in a meaningful way – was his unwillingness to accept the notion of satisfaction in an infinite domain of objects. This observation might profitably be applied to Aristotle: how would his views on the actual infinite affect his view of the use of infinite domains to capture logical consequence? The infinitary nature of first-order consequence might be considered an accidental difference between it and syllogistic consequence. After all, the truth-functional consequence relation that operates in sentential logic is finitary in much the same way – and for much the same reason – as the consequence relation of syllogistic logic. Yet the same concept of ‘completeness’ seems to operate in both the completeness theorem for sentential logic and the completeness theorem for first-order logic: in each case we show that for every valid argument there is a deduction. But an important consequence follows from the fact that, from the modern point of view, deduction can be regarded as an effective procedure: namely, that the first-order consequence relation is effectively enumerable in spite of the fact that it involves the notion of truth in an infinite domain. We can effectively enumerate the first-order deductions, and since every valid argument is associated with a deduction, we can effectively enumerate the valid arguments. More important, perhaps, we can effectively enumerate all of the logical consequences of a given premise set. Syllogistic consequence and sentential consequence are both complete and hence share this feature. But they are also both very restricted cases of the general notion of logical consequence and in both cases the formal semantics includes only finite ‘models.’ In neither case is it particularly surprising or particularly hard to prove that every case of logical consequence is associated with a formal deduction. So, in these cases it would be easy to miss the significance of the restriction of deduction to effective procedures. Let us consider a case of exhaustiveness in which the positive criterion does not give rise to an effective enumeration of the consequence relation; it is difficult – for me, at least – to see how, in this kind of case, we could call an exhaustiveness result a completeness result. Consider, for example, the case of first-order Peano arithmetic. For the sentences of first-order Peano arithmetic, we have a negative criterion for arithmetical truth: A sentence is not an arithmetical truth if it is not satisfied in some structure that is isomorphic to the standard model of first-
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order arithmetic. We also have a positive criterion for arithmetical truth: A sentence is an arithmetical truth if it is provable from the firstorder Peano axioms, using a first-order system of deduction. Gödel’s incompleteness theorem shows that these criteria are not exhaustive. If they were, then every true sentence of first-order arithmetic would be provable. We could attempt to ‘remedy’ the situation by adding rules of inference to the first-order Peano axioms. Shoenfield describes a system of second-order arithmetic that has as its theorems all true sentences of first-order arithmetic.29 But this system includes the Z-rule: From
I[1], I[2], I[3],..., we infer,
xIx (where I is an open formula with one free variable, I[t] is the result of substituting the term t for the free variable in I, and ‘1,’ ‘2,’ ‘3’ ... are terms). The Z-rule is not effective because, in order to apply it, one must evaluate an infinite number of formulas. It can be shown that every true sentence of arithmetic is either ‘provable’ in the new system or false in the standard model of arithmetic. But we would not regard this as a completeness result even if it were an exhaustiveness result. This example – which admittedly is not an example of an exhaustive set of criteria for logical truth – illustrates a distinction between two kinds of problem. There is no way of turning the method of proof in the system into an effective enumeration of the true sentences of first-order arithmetic even though it gives rise – in some sense – to a criterion of truth in arithmetic. In order to consider a question of completeness as a special case of exhaustiveness we need to have made clear, to some extent, the notion of an effective procedure. We need at least some informal criteria that are sufficient for a procedure to be effective and we must view deductive systems in such a way that they can be shown to satisfy these criteria. This implies that we must be able to regard the steps in a deduction as finite syntactic manipulations. In this sense, rather than the sense that Lear intends, the completeness question does depend on the ‘syntactic character of deducibility.’ In order for Aristotle to have raised a completeness question in the modern sense as opposed to an
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exhaustiveness question, he would have to have realized that his method of perfecting syllogisms has this character. Corcoran and Scanlan might well object that the distinction I have made is purely verbal. Perhaps it depends as much on an ‘artifact’ of modern logic as the ‘syntax/semantics’ distinction that Lear points to. After all, the modern concern with effective methods of deduction is associated with Hilbert’s formalist programme in foundations of mathematics. Formalism flourished, say, from 1905 to 1931. The span of modern formalism is a mere tick of the clock compared with the span of logical theory since Aristotle. Moreover, most would say that formalism was decisively refuted by Gödel’s second incompleteness theorem and hence that it has little relevance for logic even now. There are two replies that can be made. The first reply, and the one that I tend to favour since I am trying to highlight differences between modern logic and Aristotle’s logic, is that concerns with effectiveness are still part and parcel of modern logic, regardless of what we say about the success of the formalist programme. Formalism motivated Gödel, Turing, and Church to clarify the notion of an effective procedure and to construct new mathematical concepts with which to study this notion. The fact that Gödel and others showed that certain things cannot be captured by effective procedures is no reason to refrain from considering what can be captured by them. One of the things that characterize logic post-1931 is that we have a robust mathematical conception of ‘effective procedure’ with which to pursue these investigations. There is little doubt that Aristotle, and many other logicians and mathematicians since his time, have thought about something akin to the notion of an effective procedure. What they lacked was a mathematical theory of the notion. This, I think is something characteristic of twentieth-century logic. The second possible reply to Corcoran and Scanlan, and the one that I think they ought to regard as more telling, is that the notion of effectiveness is tied up with one of the notions that seems important to their own conception of logic: namely, that the purpose of a deduction is to convey knowledge of logical consequence. One who can follow the deduction can come to know that the conclusion is a logical consequence of the premises. On this view, a completeness theorem shows that whenever the relation of logical consequence holds, it can in principle, be known to hold. There is a venerable conception of knowledge – closely associated with Aristotle – according to which anything that can be known must
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be known directly or else on the basis of other things that are known directly and this chain of justification cannot go back to infinity. It seems that on this conception of knowledge, anything that is known must be known through the application of a finite number of effective rules of inference to an effective set of things that we know directly. The idea behind the Aristotelian conception of knowledge is that, as finite beings, we cannot ever make use of infinitely many pieces of information in deducing a new piece of information nor can we store an infinite number of pieces of information without having them finitely coded. This is perhaps why we balk at thinking of the Z-rule as a rule of inference. The only way to apply it is to go through the infinite process of justifying the rule’s premises. Whatever one thinks of the Aristotelian conception of knowledge, two things about it seem undeniable: it relies implicitly on an intuitive notion of effective procedure and it is anything but marginal in traditional conceptions of logical inference. I think that the difference between the first-order logic and syllogistic logic is further borne out by Smiley’s completeness proof. Whether or not he is correct in interpreting Aristotle as attempting a completeness proof at 40b90–41a20, the fact that a proof of the kind Smiley envisages is possible for syllogistic logic shows just how close the relation between consequence and knowledge of consequence is when we restrict ourselves to the syllogism. Compare, for example, any valid argument that can be expressed as a multi-premise syllogism with the valid first-order argument whose premises are the axioms of ZFC and whose conclusion is the well-ordering theorem. The point of the comparison is not that the only valid syllogisms are trivially recognizable as such. The point is rather that once one has identified a syllogism as valid, the premises that must be included to make it valid are tantamount to the steps in a deduction of the conclusion from them. There are further steps one must go through to produce a rigorous deduction but it is a trivial matter to fill them in. This is what follows from the observation that every valid syllogism satisfies the chain condition. By contrast, if I tell you that the well-ordering theorem is a consequence of the axioms of ZFC, I have told you nothing about how to construct the relevant deduction. Inspection of the premises gives no indication of how to proceed. Given this feature of the relationship between valid first-order arguments and first-order deductions, it really might be regarded as surprising that every valid argument is associated with a deduction.
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4. Conclusion I have not dealt conclusively with the question of whether or not Aristotle did realize that his method of deduction could be captured by a set of effective rules, that is, by a deductive system in the modern sense. Corcoran and Scanlan, I think, ignore this question and give no reason to suppose that he did realize this. Some of the things I have said might be taken to indicate that he could have and I have certainly given no reasons to conclude that he did not. At any rate, that question is not relevant to the problem that has been my main concern: namely, the problem of what conceptual resources are required to raise the problem of completeness. It seems to me that Corcoran and Scanlan, in their desire to bridge the conceptual gulf between ancient and modern logic, have neglected an important aspect of the completeness problem as we understand it today. Thus I have attempted to correct a deficiency in their analysis. In analysing completeness as the exhaustiveness of a criterion for validity and a criterion for invalidity, Corcoran and Scanlan neglect to include the requirement that the criterion of validity be effective among the features of a completeness problem. In my view, this is a necessary feature of completeness problems and one that distinguishes them from the more general class of exhaustiveness problems. I also think that the prominence given to concerns about effectiveness in twentieth-century logic is one of the features that differentiate it from ancient logic. notes 1 Jonathan Lear, Aristotle and Logical Theory (Cambridge: Cambridge University Press, 1980). 2 John Corcoran and Michael Scanlan, ‘The Contemporary Relevance of Ancient Logical Theory,’ Philosophical Quarterly 32, no. 126 (1982): 76–86. 3 Timothy Smiley, ‘Aristotle’s Completeness Proof,’ Ancient Philosophy 14 (1994): 25–37. 4 Much of the terminology in this section comes from John Corcoran, ‘Argumentations and Logic,’ Argumentation 3 (1989): 17–43, and John Corcoran, ‘The Conceptual Structure of Classical Logic,’ Philosophy and Phenomonological Research 33, no. 1 (1972): 25–47. 5 I have side-stepped the issue of whether logical consequence is a relation that can hold between a set of propositions and a proposition or a relation
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6
7 8 9 10 11 12 13
14 15 16 17
18 19
between a set of sentences and a sentence. Nothing that I will have to say depends very much on choosing sentences rather than propositions as the domain of intuitive logical consequence. For the completeness theorem for first-order logic, see any intermediate logic text, for example, George Boolos and Richard Jeffrey, Computability and Logic, 2nd ed. (Cambridge, and New York: Cambridge University Press, 1980). For a description of the standard semantics for second-order logic and an account of the differences between it and other second-order semantic theories, see chap. 4 of Stewart Shapiro, Foundations without Foundationalism: The Case for Second-Order Logic (Oxford: Clarendon Press 1991); see p. 87 for the uncompletability of second-order logic with standard semantics. Lear, Aristotle and Logical Theory, 15–16. Ibid., 2. Ibid., 16. Corcoran and Scanlan, ‘The Contemporary Relevance of Ancient Logical Theory,’ 85. Ibid. Ibid. Cf. John Corcoran, ‘The Conceptual Structure of Classical Logic.’ See John Woods and Andrew Irvine, ‘Aristotle’s Early Logic’ in The Handbook of the History and Philosophy of Logic, vol. 1: Greek, Indian and Arabic Logic, Dov Gabbay and John Woods, eds. (Amsterdam: North-Holland, 2003): 5–81. Ibid., 27–42. Corcoran and Scanlan, ‘Contemporary Relevance of Ancient Logical Theory,’ 82. Lear, Aristotle and Logical Theory, 7. See John Etchemendy, The Concept of Logical Consequence (Cambridge: Harvard University Press, 1990), 28ff., for a recent instance of this observation. But see also Alfred Tarski, ‘On the Concept of Logical Consequence,’ in Logic, Semantics, Metamathematics: Papers from 1929 to 1998, 2nd ed., J.H. Woodger, trans., J. Corcoran, ed. (Indianapolis: Hackett, 1983), 409–20, and John Corcoran, ‘Meanings of Implication’ in A Philosophical Companion to First-Order Logic, R. Hughs, ed. (Indianapolis: Hackett, 1993), 85–100. Corcoran and Scanlan, ‘The Contemporary Relevance of Ancient Logical Theory’ 85. Corcoran, however, has presented a completeness proof for syllogistic logic based on a rigorous formal reconstruction of Aristotle’s system of syllogistic logic. See ‘Completeness of an Ancient Logic,’ Journal of Symbolic Logic 37 (1972): 696–702.
Aristotle and Modern Logic 223 20 21 22 23
24 25
26
27 28
29
Timothy Smiley, ‘Aristotle’s Completeness Proof,’ 25. Ibid., 28. Ibid. The main paragraph of the proof is 40b30 to 41a20. It should be noted that Smiley’s use of ‘chain condition’ is different from mine. The ‘chain condition’ in his sense is the assertion that every valid syllogism satisfies the ‘chain condition’ in my sense. It is a consequence of the fact that every valid syllogism satisfies the chain condition that every valid syllogism must have a finite premise-set. Of course, there are infinite sets of categorical sentences that are satisfiable only in structures with infinite domains. For example: All A1 is A2, Some A2 is not A1, All A2 is A3, Some A3 is not A2, All A3 is A4, Some A4 is not A3, ... Of course the representation in first-order logic must take account of the existential import of categorical sentences. For a discussion of this issue, see Timothy Smiley, ‘Syllogism and Quantification,’ Journal of Symbolic Logic 27 (1962): 1. For example, ‘All A is B’ ought to imply ‘Some A is B.’ And, of course, this is not the case if these sentences are represented by, respectively, ‘x(Ax Bx)’ and ‘x(Ax & Bx)’ because ‘x(Ax Bx)’ is true if the extension of ‘A’ is empty. Thus ‘All A is B’ must be represented as ‘xAx & x(Ax Bx)’ while ‘no A is B’ must be represented as xAx & x(Ax Bx). This is one solution suggested in Richard Jeffrey, Formal Logic: Its Scope and Limits. (New York: McGraw-Hill, 1967). Perhaps a more natural solution to this problem is to use many-sorted first-order logic, assigning a different sort of variable to each predicate. This takes care of the existential import because the domain of each sort of variable must be non-empty. See Jeffrey, 116ff.; Smiley, 58ff. Boolos and Jeffrey, Computability and Logic, 254. Jacques Herbrand, ‘On the Fundamental Problem of Mathematical Logic,’ Logical Writings, Warren Goldfarb, ed. (Cambridge: Harvard University Press, and Dordrecht: D. Reidel Press, 1971). For an interesting and useful account of Herbrand’s understanding of the completeness problem, see also Timothy Scanlon, ‘Review of Herbrand: Logical Writings,’ Synthese 27 (1974): 271–84. J. Shoenfield, Mathematical Logic (Don Mills, ON: Addison-Wesley, 1967), 277ff.
13 The Peculiarities of Stoic Propositional Logic DAVI D H ITCH CO CK
Aristotle, the founder of logic, nowhere defines the concepts of argument and of validity. He simply uses them in his definition of a syllogism as ‘an argument in which, certain things being posited, something other than those things laid down results of necessity through the things laid down.’1 In reconstructing Aristotle’s early theory of syllogisms, John Woods2 uses Aristotle’s reticence to interpret the basic concepts of argument and validity very liberally: arguments may have any number of premisses, even zero, and validity is the absence of a counter-model, constrained only by a requirement that premiss(es) and conclusion belong to the same discipline. Thus Woods finds in Aristotle’s earliest logical writings considerable resources for his on-going sophisticated defence of classical validity against contemporary relevantist objections. The properties of Aristotelian syllogisms which relevantists find so congenial – exclusion of redundant premisses, nonidentity of the conclusion with any premiss, multiplicity of premisses – turn out to be constraints over and above those imposed by the requirement that a syllogism be a valid argument. Between Aristotle, writing in the fourth century bce, and Boole, writing more than two millennia later,3 only one logician published a system of logic. That was Chrysippus (c. 280–207 bce), the third head of the Stoic school. Chrysippus’s system of propositional logic was dominant for 400 years, until bits of it were eventually absorbed into a confused amalgamation with Aristotle’s categorical logic, a bowdlerization nicely described by Speca.4 For centuries the system from which these surviving bits were extracted was forgotten. Only the careful work of such scholars as Mates,5 Frede,6 Hülser,7 and Bobzien8 has allowed us to appreciate once again the achievement of Chrysippus. Despite its rigour and soundness, the system is oddly incomplete.
The Peculiarities of Stoic Propositional Logic 225
One can show that a conjunction follows from its conjuncts, but not that either conjunct follows from the conjunction. One can detach the antecedent from a conditional, but not put it back on; in other words, there is no deduction theorem, no rule of ‘conditional proof’ or ‘if introduction.’ One can show what follows from an exclusive disjunction and the affirmation or denial of one of its disjuncts, but not what the exclusive disjunction follows from. Further, there is no evidence that anybody ever tried to extend the system. Why was Stoic propositional logic so incomplete? I shall argue that many of its peculiarities can be explained by the rather restrictive accounts of argument and of validity that Chrysippus adopted as the foundation of his system. The omissions from the system were not accidental oversights, or not just accidental oversights, but were dictated by the requirement that everything demonstrable in the system be a valid argument. With much more complex and restrictive accounts of argument and of validity than those adopted by Woods in his reconstruction of Aristotle’s earlier logic, Chrysippus was forced into a much more restrictive formal system than contemporary classical propositional logic. 1. The System 1.1 Its Language The language of the system is a punctuation-free, regimented Greek, whose syntax supposedly corresponds to the structure of the incorporeal propositions (axiômata) signified by its sentences. In contemporary symbolism: 1. If p is a proposition, then so is p. (Read: not p.)9 2. If p and q are propositions, then so is o p o q. (Read: if p then q.)10 3. If p1, ..., pn (n > 1) are propositions, then so are & p1 & ... & pn (read: both p1 and ... and pn) and ¢ p1 ¢ ... ¢ pn (read: either p1 or ... or pn).11 Only basic propositions (not defined here) and propositions formed by a finite number of applications of the above three rules are propositions in the system. Every Stoic proposition is either true or false.12 Negation and conjunction are classically truth-functional: the negation of a true proposition is false, and of a false proposition true;13 a conjunction is true if all
226 David Hitchcock
its conjuncts are true and false if a conjunct is false.14 The conditional connective if indicates that the consequent follows from the antecedent;15 hence a conditional is true if the contradictory of the consequent ‘conflicts with’ (machetai) the antecedent, and false otherwise.16 Disjunction is exclusive; according to the quasi-connexionist version of its truth-conditions,17 a disjunction is true if and only if one of its disjuncts is true and each disjunct conflicts with each other disjunct.18 An argument is ‘a system ácomposedñ of premisses and a conclusion.’19 The plural of ‘premisses’ is quite intentional: Chrysippus denied that there are one-premissed arguments,20 for unknown reasons. An argument is valid if and only if the contradictory of its conclusion conflicts with the conjunction of its premisses.21 Thus validity is the same as truth of the conditional whose antecedent is the conjunction of the argument’s premisses and whose consequent is the argument’s conclusion.22 The concept of conflict is thus basic to the truth-conditions for conditionals and disjunctions, and to the concept of argument validity. Unfortunately our sources do not preserve a complete account of this concept. Conflicting propositions cannot be simultaneously true.23 The incompatibility need not be logical; we are told24 that Not it is light conflicts with It is day. Some pairs of conflicting propositions can be simultaneously false, as in the example, at night by lamplight. A proposition cannot conflict with itself.25 Conflict implies some sort of connection (sunartêsis);26 hence the mere fact that a proposition is always false, or even necessarily false, is not sufficient for it to conflict with any arbitrarily chosen proposition. Any proposition conflicts with its contradictory;27 hence it cannot be a requirement that each conflicting proposition is at some time true, or even possibly true. To sum up, one proposition conflicts with another only if (1) they are distinct, (2) they cannot both be simultaneously true, and (3) this impossibility is not due to the necessary falsity of one of them. These necessary conditions may not be jointly sufficient; the difficulty is to specify when an always false, or necessarily false, proposition conflicts with another proposition.28 1.2 Its Primitives The primitives of Stoic propositional logic are ‘undemonstrated29 arguments’ (anapodeiktoi logoi) of five ‘moods’ (tropoi): A1. o p o q, p £ q (modus ponendo ponens)30
The Peculiarities of Stoic Propositional Logic 227
A2. o p o q, q £ p (modus tollendo tollens)31 A3. & p & q, p £ q (modus ponendo tollens)32 A4. ¢ p ¢ q, p £ q33 A5. ¢ p ¢ q, p £ q (modus tollendo ponens)34 For present purposes, I ignore extended descriptions of third, fourth and fifth undemonstrated arguments to cover conjunctions with multiple conjuncts and disjunctions with multiple disjuncts.35 1.3 Its Rules of Inference There are rules, called themata, for generating new valid arguments from arguments already known to be valid, apparently four in number.36 The first thema is a contraposition rule for arguments, of the same sort as we find already in Aristotle’s Topics37 and Sophistical Refutations:38 T1. If from two [or more]39 a third follows, from either one [or all but one] of them together with the contradictory of the conclusion there follows the contradictory of the remaining one.40
The term contradictory is more general than the term negation. A proposition and its negation are said to be contradictories (antikeimena);41 hence the contradictory of a negated proposition may fall short of it by a negative. The first thema thus allows us to demonstrate variants of the moods in which we have such a contradictory rather than a negation, or in which the other conjunct or disjunct of the ‘leading premiss’ (hêgemonikon) occurs in the ‘added premiss’ (proslêgon).42 It also allows us to ‘reduce’ (anagein) second undemonstrated arguments to corresponding first undemonstrated arguments, and conjunction introductions to third undemonstrated arguments, as in the following example: (A3)
& F & G, F £ G F, G £ & F & G
(T1)
The second, third, and fourth themata are taken by Alexander of Aphrodisias43 to be equivalent to the following ‘synthetic theorem’ of the Peripatetics:
228 David Hitchcock Whenever from some something follows, and that which follows along with one or more yields something, then those from which it follows, along with the one or more with which it yields that something, will themselves also yield the same thing.44
Symbolically, if p1, ..., pn £ qi (n > 1) and q1, ..., qi, ..., qm £ r, (m > 1), then p1, ..., pn, q1, ..., qi – 1, qi + 1, ..., qn £ r (1 d i d m). The synthetic theorem is thus a rule for chaining arguments together, or in contemporary language a cut rule.45 It is perfectly general, except that it requires both the subordinate argument (i.e., the one whose conclusion qi is a premiss of the other argument) and the superordinate argument (i.e., the one whose premiss qi is a conclusion of the other argument) to have at least two premisses, in accordance with Aristotle’s definition of a syllogism. The third thema has survived in two versions, one reported by Alexander (fl. c. 200), the other by Simplicius (writing after 532). Following Bobzien,46 I shall use Simplicius’ version:47 T3. If from two a third follows, and that which follows along with another [or others]48 from outside yields something, then also from the first two and the one [or ones] assumed in addition from outside there will follow the same one.49
Symbolically, if p1, p2 £ qi and q1, ..., qi, ..., qn £ r, then p1, p2, q1, ..., qi – 1, qi + 1, ..., qn £ r (1 < n, 1 d i d n; p1 z qj and p2 z qj for each j such that 1 d j d n). The third thema differs from the synthetic theorem in two respects. First, the subordinate argument has exactly two premisses rather than any number greater than one; this limitation is not substantive, since all chains of reasoning must begin with undemonstrated arguments with two premisses. Second, the superordinate argument cannot have a premiss of the subordinate argument as its premiss, as the words ‘from outside’ indicate; this latter difference is substantive, and provides the key to the reconstruction of the second and fourth themata, which are not extant. The following proof illustrates the use of the third thema: (A1)
o D o L, D £ L
(A3)
& L & N, L £ N
(T3)
& L & N, o D o L, D £ N50 Bobzien51 has proposed as the second thema the special case where
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the superordinate argument uses no external premisses:52 T2. If from two a third follows, and that which follows along with one or both from which it follows yields something, then also from the first two there will follow the same one.
Symbolically, if p1, p2 £ q and either q, pi £ r (i = 1 or i = 2) or q, p1, p2 £ r, then p1, p2 £ r. The following proof illustrates the use of the second thema: (A1) (A1)
o K o D, K £ D
o K o D, K £ D D, K £ o K o D
o K o D, K £ o K o D
(T1)
(T2)
(T1)
o K o D, o K o D £ K53 This proof illustrates how the combination of the first thema and one of the cut rules (in this case, the second thema) gives the effect of a reductio ad absurdum rule. The two first undemonstrated arguments have conclusions that contradict each other; from them, by a series of manoeuvres, one derives an argument whose conclusion is the contradictory of one premiss of the two undemonstrated arguments and whose premisses are their remaining premisses. Because the first thema is formulated in terms of contradictories rather than negations, the system has both a negation-introduction rule (illustrated here) and a negationelimination rule of the sort popularized by Fitch.54 The fourth thema55 provides for the case where the superordinate argument includes both one or two premisses of the subordinate argument and one or more premisses external to it: T4. If from two a third follows, and that which follows along with one or both from which it follows and one or more from outside yields something, then also from the first two and that or those from outside there will follow the same one.
Symbolically, if p1, p2 £ qj and either q1, ..., qj, ..., qn, pi £ r (i = 1 or i = 2) or q1, ..., qj, ..., qn, p1, p2 £ r, then p1, p2, q1, ..., qj – 1, qj + 1, ..., qn £ r (1 d n, 1 d j d n; p1 z qk, p2 z qk for 1 d k d n). The following proof of constructive
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dilemma illustrates the use of the fourth thema:
(A2)
o q o r, r £ q
o q o r, r, (A2)
o p o r, r £ p
¢ p ¢ q, p £ q
q, p £ ¢ p ¢ q p £ ¢ p ¢ q
(T1) (T3)
(T1)
¢ p ¢ q, r, p £ o q o r
¢ p ¢ q, o p o r, r £ o q o r
(A5)
(T4)
(T1)
¢ p ¢ q, o p o r, o q o r £ r Thus the system had a derived rule of disjunction elimination using conditionals. 1.4 Two Conjectures CONJECTURE 1 (cut): The second, third, and fourth themata have the same demonstrative power in the system as an unrestricted cut rule. SUPPORTING CONSIDERATIONS: The themata cover all cases in which the subordinate argument has two premisses. Hence it is sufficient to prove that it is not necessary to have a cut rule for subordinate arguments with other than two premisses. Since each undemonstrated argument has two premisses and no thema permits one to derive an argument with fewer than two premisses, there is no need for a cut rule for subordinate arguments with fewer than two premisses. In a straightforward chaining of several undemonstrated arguments together, with no application of the first thema, a reduction which generates a subordinate argument with more than two premisses can be transformed into one which generates only two-premiss subordinate arguments, simply by changing the order in which the cut rule is applied; the trick is to always find two premisses of the analysandum from which a proposition not in the analysandum follows. Applications of the first thema, which are sometimes required (as in the reduction above using the fourth thema) to produce two premisses from which something follows, do not affect this basic strategy.56 CONJECTURE 2 (negation introduction and negation elimination): If p1, ..., pn £ q (n > 1) and pn + 1, ..., pn + m £ q (m > 1), then p1, ..., pi – 1, pi + 1, ..., pn + m £ ctr(pi), where ctr(pi) is r if pi is r for some proposition r,
The Peculiarities of Stoic Propositional Logic 231
and otherwise ctr(pi) is pi, and p1, ..., pi – 1, pi + 1, ..., pn + m includes only once any premiss which occurs both in p1, ..., pn £ q and pn + 1, ..., pn + m £ q. PROOF: Apply the first thema to whichever argument has pi as a premiss, putting ctr(pi) as the resulting argument’s conclusion. By conjecture 1, p1, ..., pi – 1, pi + 1, ..., pn + m £ ctr(pi). QED In addition to a standard conditional elimination rule (A1), the system has conjunction introduction (from A3, by T1 – see the example above of the use of the first thema), disjunction elimination using conditionals (from A2 twice and A5 – see the example above of the use of the fourth thema), and the following qualified version of disjunction elimination using arguments: For any propositions p, q, r, p1, ..., pm, ..., pn (1 d m, m < n, pi z pj for any i z j, 1 d i,j d n), if p1, ..., pm, p £ r and pm + 1, ..., pn, q £ r, then ¢ p ¢ q, p1, ..., pn £ r (proved by adapting the demonstration of disjunction elimination using conditionals). 2. Peculiarities 2.1 Absence of Theorems THEOREM 3: For any proposition p, /£ p. PROOF: Only arguments with at least two premisses are demonstrable in the system. QED EXPLANATION: Anything demonstrated in the system is a syllogism, which is a species of valid argument.57 Every argument has at least two premisses. Hence there are no theorems in the system. In particular, although the Stoics held that an argument is valid if and only if the conditional whose antecedent is the conjunction of its premisses and whose consequent is its conclusion is true,58 Chrysippus did not incorporate this principle in his system; even if p1, ..., pn £ q (n > 1), /£ o & p1 & ... & pn o q. Nor does the system permit proofs of several moods of logical truths that are expressible in the system, for example, & p & p (law of non-contradiction, known to Aristotle), ¢ p ¢ p (law of excluded middle, known to Aristotle, and the syntactic counterpart of the Stoics’ principle of bivalence), o p o p (law of identity). 2.2 Absence of One-Premissed Syllogisms THEOREM 4: For any propositions p and q, p /£ q. PROOF: Every undemonstrated argument has two premisses. No
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thema permits one to derive an argument with fewer than two premisses. QED EXPLANATION: Since they are arguments, syllogisms must have more than one premiss. In particular, for any proposition p, p /£ p (irreflexivity), even though any proposition p follows from itself and there is no general ban on syllogisms with a conclusion identical with a premiss.59 Also, for any propositions p and q, & p & q /£ p and & p & q /£ q (counterconjunction-elimination), even though any conjunct follows from a conjunction. Also, there are no propositions p, q, and r such that p £ r, q £ r, and ¢ p ¢ q £ r (absence of simple disjunction elimination using arguments), even though, if a proposition follows from each disjunct in a disjunction, then it follows from the whole disjunction. The rejection of one-premissed arguments shows that the premisses of a Stoic argument did not constitute a set, contrary to Peter Milne’s suggestion.60 If they did, Stoic propositional logic would easily generate one-premissed arguments, a fact Chrysippus could hardly fail to have noticed. For example, the application of the first thema to the first undemonstrated argument If not it is day, it is day; not it is day; therefore it is day produces the argument Not it is day; not it is day; therefore if not it is day, it is day. If the two premisses of the latter argument are a set, it is a set with one member, so that the argument would have one premiss. Thus repetitions of premisses can occur. The fact that repetitions can occur in turn explains why Chrysippus needed three cut rules; he had to rule out repetitions of a premiss that occurred in both arguments being chained together. 2.3 Non-Monotonicity LEMMA: If p1, ..., pn £ q and some basic proposition r occurs in the propositions p1, ..., pn and q, then r occurs in at least two of those propositions. PROOF: Each basic proposition in an undemonstrated argument occurs either in both premisses or in both a premiss and the conclusion. If the first thema is applied to an argument in which a basic proposition occurs in at least two of its components, then that basic proposition occurs in at least two components of the resulting argument. If a basic proposition occurs in at least two components of either of two arguments chained together using the second, third, or fourth thema, then it occurs in at least two components of the resulting argument. (The proof of this last proposition requires a rather complex enu-
The Peculiarities of Stoic Propositional Logic 233
meration of cases, in order to show that any basic proposition in the conclusion of the subordinate argument must occur both in a premiss of the subordinate argument and in a component of the superordinate argument other than the premiss which drops out.) QED THEOREM 5: For some propositions p1, ..., pn, q and r, p1, ..., pn £ q but r, p1, ..., pn /£ q (n > 1). PROOF: o F o G, G £ F but, by the preceding lemma, o F o G, G, H /£ F. QED EXPLANATION: The concept of conflict was interpreted in such a way that an argument could be invalid not only through obvious disconnection (diartêsis), bad form, or deficiency of a premiss, but also through redundancy.61 Redundancy is the introduction of a superfluous premiss into an argument which would otherwise be valid, as in the argument If it is day, it is light; but it is day; and virtue benefits; therefore it is light. The explanation of redundancy as a source of invalidity62 is perhaps that, according to the Stoics’ first thema, if this argument is valid, then so is its contrapositive: If it is day, it is light; but it is day; and not it is light; therefore not virtue benefits. But this argument is clearly invalid on the connexive criterion of validity; the contradictory of the conclusion has nothing to do with the conjunction of the premisses. Chrysippean propositional logic is close to being counter-monotonic; apart from arguments like that mentioned in note 62, the only exceptions to counter-monotonicity which come readily to mind involve adding as a premiss a ‘duplicated’63 conditional whose antecedent and consequent are identical to the conclusion, as in the argument o F o F, o F o G, G £ F. 2.4 Absence of a Deduction Theorem THEOREM 6: For some propositions p1, ..., pn, q and r (1 d n), p1, ..., pn, q £ r but p1, ..., pn /£ o q o r. PROOF: If n = 1, p1, ..., pn /£ o q o r, since only arguments with at least two premisses are demonstrable in the system. More generally, the system contains no general procedure for demonstrating the validity of an argument with a conditional conclusion. For such a demonstration, the conditional must be embedded in a more complex premiss, either a disjunction or a more complex conditional. QED EXPLANATION: The deduction theorem is unsound on the connexionist account of the conditional. That is, there are cases where a proposition q follows from a combination of some proposition p with one or
234 David Hitchcock
more other premisses but where it does not follow from those other premisses that q follows from p. For example, grass is green follows from the combination of snow is white and not both snow is white and not grass is green, but it does not follow from not both snow is white and grass is not green that grass is green follows from snow is white. In other words, Not both snow is white and not grass is green; but snow is white; therefore, grass is green is valid, but if snow is white, then grass is green does not follow from not both snow is white and not grass is green. In general, in fact, if p, then q does not follow from not both p and not q. 2.5 Absence of Hypothetical Syllogism, Dilemma, and Other Apparently Valid Moods THEOREM 7: (a) o p o q, o q o r /£ o p o r (no hypothetical syllogism), (b) p, q /£ ¢ p ¢ q, (c) ¢ p ¢ q, p /£ q, and (d) o p o r, o q o s, ¢ p ¢ q /£ ¢ r ¢ s (no dilemma). PROOF AND EXPLANATION: The system is sound, as can be shown by applying the above truth-conditions for negations, conjunctions, conditionals and disjunctions, and the definition of the validity of an argument to the five types of undemonstrated argument and by working out that each of the themata preserve validity. But each of the above moods has counter-examples: (a) Put you know that you are dead for p, you are dead for q, not you know that you are dead for r; (b) Put grass is green for p and snow is pink for q;64 (c) Put grass is green for p and snow is pink for q; (d) Put it is day for p, it is light for r, it is night for q, not the sun is shining for s. All four counter-examples exploit the fact that the conclusion is true only if there is a conflict between two propositions. Non-syllogisticity of the above moods can also be shown by reinterpreting the connectives as truth-functions in a semantics of multiple truth-values, in such a way that undemonstrated arguments have no counter-model, the themata preserve absence of a counter-model, but each mood has a counter-model.65 2.6 Absence of Some Valid Moods THEOREM 8: (a) o p o q, p o q /£ ¢ p ¢ q, (b) ¢ p ¢ q, p /£ o p o q, and (c) & p & q, o p o r /£ r. PROOF: Non-syllogisticity of these moods can be shown by reinterpreting the connectives as truth-functions in a semantics of multiple truth-values.66
The Peculiarities of Stoic Propositional Logic 235
EXPLANATION: These moods are valid. Their indemonstrability must be regarded as an unintended consequence of the way Chrysippus constructed his system. He evidently included as undemonstrated moods all two-premiss valid argument schemes with a complex leading premiss with two propositional variables, an added premiss which was one of the propositional variables or its contradictory, and a conclusion that was the other propositional variable or its contradictory. To these primitives he added the rule for contraposing arguments which Aristotle had explicitly stated in his early logical writings, as well as a threepart cut rule. Special cases like the above are inadvertently not provided for, even though they are valid two-premiss arguments. To add more primitives to the system to accommodate such special cases is likely to be a never-ending task, because there is no single principle that accommodates them all. 3. Conclusion 3.1 The Requirement of at Least Two Premisses Chrysippus assumed that an argument must have at least two premisses. Hence each primitive of his system has two premisses, and the rules for demonstrating arguments on the basis of primitives do not permit reduction of the number of premisses below the minimum of the input argument(s). Hence the system has no logical truths as theorems, nor can it demonstrate any one-premiss arguments, even when one proposition follows in Chrysippus’s sense from another. In particular, it cannot demonstrate conjunction elimination or simple disjunction elimination using arguments. It also has no repetition rule, even though on Chrysippus’ criterion every proposition follows from itself. 3.2 Acceptance of Argument Contraposition Chrysippus used Aristotle’s rule of argument contraposition as his first thema. In order to avoid generation of one-premiss arguments using this thema, he had to regard an argument with one premiss repeated twice as a two-premiss argument. Thus repetitions count; the premisses of a Stoic syllogism are not a set. Hence, in order to avoid generating unwanted repetitions in chaining two arguments together, Chrysippus needed three versions of the cut rule, to cover cases where
236 David Hitchcock
the premisses of the superordinate argument (other than the conclusion of the subordinate argument) came entirely from the subordinate argument, entirely from outside the subordinate argument, or partly from it and partly from outside it. 3.3 Connexionism Chrysippus was a connexionist about true conditionals, about true disjunctions, and about valid arguments: in a true conditional there had to be a connection between antecedent and consequent, in a true disjunction an incompatibility between each pair of disjuncts, and in a valid argument a connection between premisses and conclusion. The connexionist criterion for a true conditional rules out the usual deduction theorem (conditional proof), as well as proofs of a conditional from its consequent or from the contradictory of its antecedent. It also rules out hypothetical syllogism, a mood discovered by Theophrastus that is invalid on the connexionist account. Similarly, the connexionist criterion for a true disjunction rules out proofs of a disjunction from one disjunct and the contradictory of the other, as well as dilemma. The connexionist criterion for a valid argument rules out ex falso quodlibet and e quolibet verum. Apparently because of Chrysippus’ acceptance of argument contraposition, it rules out redundant premisses, except in cases where the argument has a mood without redundant premisses. Hence the system is non-monotonic and close to being countermonotonic. 3.4 Inadvertent Omissions Most of the apparent peculiarities of the system are thus necessary consequences of Chrysippus’ assumptions about the concepts of argument and of validity. The Chrysippean-valid, multi-premiss arguments that are not demonstrable in his system appear to have been overlooked as a consequence of the way Chrysippus selected his five types of primitives, each with one complex premiss and one simpler premiss and simpler conclusion. It is rather surprising, in fact, given Chrysippus’ numerous pre-systematic restrictions on the concept of a valid argument and the apparent absence of any effort to ensure completeness, that his system allows one to prove the validity of practically all the formally valid moods expressible in the system which we use in real reasoning and argument. From this point of view, the Chrysippean
The Peculiarities of Stoic Propositional Logic 237
restrictions on valid arguments may not be as unmotivated as we might suppose. notes Abbreviations A.L. = Sextus Empiricus, Against the Logicians (= Against the Professors 7–8) Alexander In An. pr. = Alexander of Aphrodisias, In Aristotelis Analyticorum Priorum Librum I Commentarium Alexander In Top. = Alexander of Aphrodisias, In Aristotelis Topicorum Libros Octo Commentaria Diocles = Diocles of Magnesia (ca. 40 bce). Excerpt from his Survey of the Philosophers (Epidromê tôn philosophôn) = D.L. 7.49–82. D.L. = Diogenes Laertius FDS = Hülser, Die Fragmente zur Dialektik der Stoiker I.L. = Galen, Institutio Logica (original Greek title Eisagôgê Dialektikê) P.H. = Sextus Empiricus, Outlines of Pyrrhonism (Pyrrhôneiôn Hupotupôseôn) 1 Topics I.1.100a25–27. Cf. Aristotle’s Sophistical Refutations 164b27–165a2 and Prior Analytics I.1.24b18–20. Here and elsewhere, translations are my own. For Aristotle’s works, I use the version of the Oxford Classical Texts, unless otherwise indicated. 2 John Woods, Aristotle’s Earlier Logic (Oxford: Hermes Science, 2001). 3 George Boole, The Mathematical Analysis of Logic (Cambridge: Macmillan, Barclay & Macmillan, 1847). 4 Anthony Speca, Hypothetical Syllogistic and Stoic Logic, Philosophia Antiqua Volume 87 (Leiden: Brill, 2001). 5 Benson Mates, Stoic Logic (Berkeley and Los Angeles: University of California Press, 1953). 6 Michael Frede, Die stoische Logik (Göttingen: Vandenhoeck & Ruprecht, 1974). 7 Karlheinz Hülser, Die Fragmente zur Dialektik der Stoiker, 4 vols. (StuttgartBad Cannstatt: Friedrich Frommann, 1987–8). Cited henceforth as ‘FDS’ followed by the fragment number. 8 Susanne Bobzien, ‘Stoic Syllogistic,’ Oxford Studies in Ancient Philosophy 14 (1996): 133–92. Susanne Bobzien, ‘The Stoics,’ in The Cambridge History of Hellenistic Philosophy, Kempe Algra, Jonathan Barnes, Jaap Mansfeld, and Malcolm Schofield, eds. (Cambridge: Cambridge University Press, 1999), 92–157.
238 David Hitchcock 9 Cf. Diocles of Magnesia, Survey of the Philosophers (Epidromê tôn philosophôn), excerpted in Diogenes Laertius, Vitae Philosophorum, vol. 2, H.S. Long, ed. (Oxford: Clarendon Press, 1964). 7.69. The excerpt from Diocles is cited hereafter as ‘Diocles,’ followed by the book and section number in Diogenes Laertius’ work. Other parts of Diogenes Laertius’ work are cited hereafter as ‘D.L.’ followed by the book and section number. See also Sextus Empiricus, Against the Logicians, trans. R.G. Bury (Cambridge: Harvard University Press, 1935), 2.88–90. This work of Sextus Empiricus is cited hereafter as ‘A.L.,’ followed by the book and section number or numbers. 10 Cf. Diocles, Survey of the Philosophers, 7.71, A.L. 2.109–111. See also Aulus Gellius, Noctes Atticae, P.K. Marshall, ed. (Oxford: Oxford University Press, 1968), 16.8.9 = FDS 953. 11 Cf. Diocles, Survey of the Philosophers, 7.72. 12 Diocles, Survey of the Philosophers, 7.65; Marcus Tullius Cicero De Fato, trans. H. Rackham (Cambridge: Harvard University Press, 1932), 38. 13 A.L. 2.103. 14 A.L. 2.125; Gellius, Noctes Atticae, 16.8.11 = FDS 967. 15 Chrysippus, Dialectical Definitions, apud Diocles 7.71. 16 Diocles 7.73; Sextus Empiricus, Outlines of Pyrrhonism, trans. R.G. Bury (Cambridge: Harvard University Press, 1933), 2.111. The latter work is cited hereafter as ‘P.H.,’ followed by the book and section number or numbers. 17 There are also truth-functional (Diocles, Survey of the Philosophers, 7.72) and fully connexionist (Gellius, Noctes Atticae, 16.8.13 = FDS 976) versions. 18 P.H. 2.191. 19 D.L. 7.45. 20 A.L. 2.443. 21 Diocles, Survey of the Philosophers, 7.77. 22 P.H. 2.137, A.L. 2.415–417; cf. A.L. 2.112. 23 Galen, Institutio Logica, Karl Kalbfleisch, ed. (Leipzig: Teubner, 1896), 4.2. Hereafter cited as ‘I.L.,’ followed by the chapter and section number. Apollonius Dyscolus, De Coniunctionibus, in Apollonii scripta minora, vol. 1. fasc. 1 of Apollonii Dyscoli Quae supersunt, Richard Schneider, ed. (Leipzig: Teubner, 1878), 218 = FDS 926. Gellius, Noctes Atticae, 16.8.14 = FDS 976. 24 Diocles, Survey of the Philosophers, 7.73. The claim is in fact false. North of the Arctic Circle, human beings distinguish day and night even in the middle of winter, when it is dark for weeks at a time. I thank Rani Lill Anjum for the counter-example. In what follows, I shall occasionally pretend that it does not exist. 25 P.H. 2.111. 26 Cf. ibid.
The Peculiarities of Stoic Propositional Logic 239 27 Apollonius Dyscolus, De Coniunctionibus, 218 = FDS 926. 28 In forthcoming work, Mauro Nasti de Vincentis has proposed to define conflict between two propositions p and q as follows: p and q are incompossible, and, whenever contrary to one another, they are also contingent, provided that a simple intuitive deduction, say of not-q from p (possibly by means of supplementary assumptions), is also available (e-mail communication, 21 February 2004). See also Mauro Nasti De Vincentis, ‘La validitá del condizionale crisippeo in Sesto Empirico e in Boezio (Parte I),’ Dianoia 3 (1998): 45–75; Logicha della connessivitá: Fra logica moderna e storia della logica antica (Bern-Stuttgart-Wien: Verlag Paul Haupt, 2002); and ‘From Aristotle’s Syllogistic to Stoic Conditionals: Holzwege or Detectable Paths?’ Topoi 23 (2004): 113–37. 29 The usual translation is indemonstrable. But the suffix -tos is ambiguous, corresponding either to the English suffix -ed or to the English suffix -able/ible. We are told that the Stoics’ arguments are anapodeiktoi because they do not need demonstration (Diocles, Survey of the Philosophers, 7.79; cf. A.L. 2.223), an explanation which makes far more sense if we translate anapodeiktoi as undemonstrated than if we translate it as indemonstrable. Further, the latter translation implies (falsely) that the validity of arguments of these types cannot be demonstrated, whereas the former translation merely indicates (correctly) that no demonstration of these arguments is given; such arguments, as Sextus Empiricus reports, ‘have no need of demonstration because of its being at once conspicuous in their case that they are valid’ (A.L. 2.223). 30 Cf. Diocles, Survey of the Philosophers, 7.80, A.L. 2.224, P.H. 2.157, I.L. 6.6. 31 Cf. Diocles, Survey of the Philosophers, 7.80, A.L. 2.225, P.H. 2.157, I.L. 6.6. 32 Cf. A.L. 2.226, Diocles, Survey of the Philosophers, 7.80, P.H. 2.158, I.L. 14.4. 33 Cf. Diocles, Survey of the Philosophers, 7.81; cf. P.H. 2.158, I.L. 6.6. 34 Cf. D.L. 7.81, P.H. 2.158, ps.-Galen Historia Philosopha, in Doxographi Graeci, Hermann Diels, ed. (Berlin: Georg Reimer, 1879), 595–648; 15, 608,1–2 = FDS 1129. 35 Marcus Tullius Cicero, Topica, in his Rhetorica, vol. 2, Augustus S. Wilkins, ed. (Oxford: Clarendon Press, 1902–3), 54 = FDS 1138; Ioannes Philoponus, in Aristotelis Analytica Priora Commentaria, Maximilian Wallies, ed. (Commentaria in Aristotelem Graeca vol. 13, part 2) (Berlin: Georg Reimer, 1905), 245,19–246,14 = FDS 1133. 36 Galen refers to the Stoics’ first, second, third, and fourth themata (On the Doctrines of Hippocrates and Plato, 3 vols., Phillip de Lacy, ed. and trans. [Berlin: Akademie, 1981–1984], 2.3.18–19 = FDS 1160) in a polemical context in which it suits his purposes to mention all their themata. Further, we have
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37 38
39 40 41 42
43 44 45
46 47
48
references elsewhere to the first (ps.-Apuleius, De interpretatione, in De philosophia libri, vol. 3 of Apulei Platonici Madaurensis Opera quae supersunt, Claudio Moreschini, ed. [Leipzig: Teubner, 1991], 191, 5–10 = FDS 1161), second (Alexander of Aphrodisias, In Aristotelis Analyticorum Priorum Librum I Commentarium, Maximilian Wallies, ed. [Commentaria in Aristotelem Graeca vol. 2, part 1] [Berlin: Georg Reimer, 1883] [hereafter cited as Alexander, In an. pr.], 164,30–1; 284,15), third (Alexander, In an. pr. 278,6–7,11– 14; 284,15; Simplicius, In Aristotelis De Caelo Commentaria, Johan L. Heiberg, ed. [Commentaria in Aristotelem Graeca vol. 7] [Berlin: Georg Reimer, 1893], 236,33–237,4), and fourth (Alexander, In an. pr. 284,15) themata, but to no other. 8.14.163a32–6. 33.182b37–183a2. Following the entire western logical tradition, Woods, Aristotle’s Earlier Logic, calls this rule argument conversion. The label is misleading, since conversion of a proposition involves changing two components (subject and predicate, antecedent and consequent) without changing their sign. The label argument contraposition signals the analogy to contraposition of propositions, which involves changing the sign of two components that are interchanged. Galen (I.L. 6.5) mentions this variant formulation for multi-premissed syllogisms. Ps.-Apuleius, De interpretatione, 191,5–11 = FDS 1161. Cf. Diocles, Survey of the Philosophers, 7.73, A.L. 2.89. The descriptions of the undemonstrated arguments, which were apparently more authoritative, allow for these variations. The first thema makes it unnecessary to complicate the moods to provide for them. In an. pr. 284,13–15 = FDS 1165. Alexander In an. pr. 274,21–24 = FDS 1166; 278,8–11 = FDS 1167; cf. 283,15– 17 = FDS 1165. Gerhard Gentzen, ‘Investigations into Logical Deduction,’ in The Collected Papers of Gerhard Gentzen, M.E. Szabo, ed. (Amsterdam and London: North Holland, 1969), 68–131. Bobzien, ‘Stoic Syllogistic.’ Bobzien’s choice is unusual; it rests on the fact that existing reconstructions using Alexander’s version are difficult, if not impossible, to apply in practice. In forthcoming work, I propose a reconstruction on the basis of Alexander’s version that is practically workable. The extension is justified by the parallel extension to the first thema and by the fact that Alexander’s version allows a second input argument (in his case, the subordinate argument) with more than two premisses.
The Peculiarities of Stoic Propositional Logic 241 49 Simplicius, In Aristotelis De Caelo Commentaria, 237,2–4 = FDS 1168. 50 The abbreviations are mnemonic for the simple propositions in an argument (Not both it is light and it is night; and if it is day it is light; but it is day; therefore not it is night) which Sextus Empiricus analyses (A.L. 2.234–8) using a so-called ‘dialectical theorem.’ 51 Bobzien, ‘Stoic Syllogistic,’ 151. 52 In this respect she follows Katerina Ierodiakonou, Analysis in Stoic Logic, (dissertation, University of London, 1990), 72–4. The present author made a similar proposal in ‘The Missing Rules of Stoic Logic,’ unpublished paper presented at the annual congress of the Canadian Philosophical Association, Ottawa, Canada, 7–10 June 1982. I have changed Bobzien’s wording to conform to my translation of Simplicius. 53 The capital letters are mnemonic for the constituents of the ‘argument through two moodals’ reported by Origen, Contra Celsum, Paul Koetschau, ed. (Leipzig: J.C. Hinrich, 1899), 7.15 = FDS 1181: If you know that you are dead, you are dead; if you know that you are dead, you are not dead; therefore, you do not know that you are dead. Galen reports (On the Doctrines of Hippocrates and Plato 2.3.18 = FDS 1160) that such syllogisms were analysed using the first and second themata. A ‘moodal’ (tropikon) is a type of non-simple proposition which can be the leading premiss of a mood (tropos) of undemonstrated argument, i.e., either a conditional or a negated conjunction or a disjunction. 54 Frederic B. Fitch, Symbolic Logic (New York: Ronald Press, 1952). 55 Bobzien, ‘Stoic Syllogistic,’ 151. 56 The argument that the second, third, and fourth themata jointly have the force of an unrestricted cut rule is labelled ‘supporting considerations’ because it does not constitute a rigorous proof of the ‘conjecture,’ which has been doubted. Having attempted without success to construct a counterexample, I suspect that the conjecture is correct. But proof is needed. 57 Diocles, Survey of the Philosophers, 7.78. 58 P.H. 2.137, A.L. 2.415–417; cf. A.L. 2.112. 59 The Stoics recognized some ‘non-differently concluding’ (adiaphorôs perainontes) syllogisms, e.g., Either it is day or it is light; but it is day; therefore it is day (Alexander of Aphrodisias, In Aristotelis Topicorum Libros Octo Commentaria, ed. Maximilian Wallies [Commentaria in Aristotelem Graeca vol. 2, part 2] [Berlin: Georg Reimer, 1891] [hereafter cited as Alexander In Top.], 10,11–12), which can be reduced to a fourth and a fifth undemonstrated argument using the second thema. 60 Peter Milne, ‘On the Completeness of Stoic Logic,’ History and Philosophy of Logic 16 (1995): 41.
242 David Hitchcock 61 A.L. 2.429–32. 62 The system permits demonstration of some arguments with a redundant premiss, for example, If there exists a sign, there exists a sign; if not there exists a sign, there exists a sign; but either not there exists a sign or there exists a sign; therefore, there exists a sign (A.L. 2.281), where the third-mentioned premiss is redundant. But this argument has the same form as the argument If it is day, it is light; if it is night by lamplight, it is light; either it is day or it is night by lamplight; therefore it is light, which is valid and has no redundant premiss. Similarly for all other demonstrable arguments with a redundant premiss. Hence, to preserve the soundness of the system, one needs to create an exception for arguments with the same form as a valid argument without a redundant premiss. Such an exception, unlike that proposed by Bobzien, (‘Stoic Syllogistic,’ 180), covers not only syllogisms but also other types of valid arguments. 63 Duplicated propositions are propositions in which the same proposition occurs twice (Diocles, Survey of the Philosophers, 7.69, A.L. 2.109). The Stoics recognized as valid ‘duplicated arguments’ in which such propositions occur as premisses (Alexander, In an. pr. 164, 28–31 = FDS 1169; In an. pr. 18,17–18; 20,10–12 = FDS 1171; In Top. 10,8–10 = FDS 1170). 64 Milne’s complaint (‘On The Completeness of Stoic Logic,’ 52–3) that this mood is indemonstrable is thus misplaced. 65 Cf. ibid., 39–64. 66 For (a), cf. ibid.
14 On the Substitutional Approach to Logical Consequence MATTHEW MCKEON
1. Introduction The characterization of logical consequence in terms of substitutions is generally thought to be inadequate and is not taken seriously in the literature as a rival to the model-theoretic approach. While I am a fan of model theory, it seems to me that the substitutional approach to logical consequence has been mistreated. My aim in this paper is to defend it. I shall consider a well-known criticism of Quine’s version of the substitutional approach that is based on the claim that it expands the class of logical truths in first-order logic with identity beyond what is sanctioned by the model-theoretic account. Briefly, I argue that at best the criticism is shallow and can be answered with slight alterations in Quine’s account. At worse the criticism is defective because, in part, it is based on a misrepresentation of Quine’s account. My defence of Quine’s substitutional account to logical consequence serves not only to clarify Quine’s position, but also to crystallize what is and what is not at issue in choosing the model-theoretic account of logical consequence over one in terms of substitutions. I begin by sketching Quine’s account, distinguishing it from the model-theoretic account. Then I present two objections and my replies. 2. Quine’s Substitutional Account of Logical Consequence For ease of exposition, we follow Quine and define logical consequence from logical truth in the standard way.1 The task then is to characterize Quine’s notion of logical truth. His substitutional account fixes the extension of logical truth only for interpreted languages. Accordingly,
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we first define the notion of an interpreted first-order language with identity. Then we give the familiar model-theoretic account of logical truth, followed by an exposition of Quine’s account. Let a first-order language L be the pair áLex, Gñ. The first element is a first-order lexicon, which includes variables, individual constants, and n-place predicates. The second element is a grammar that generates all the well-formed formulas (wffs) of L. Included in L’s grammar is a list of particles, that is, lexical items used in constructing wffs from other wffs. The particles in L’s grammar are the functional equivalent of the truth-functional connectives (‘~,’ ‘&,’ ‘,’ and ‘o’), the quantifiers (‘v,’ ‘v’), and the identity symbol (‘ = ’). An interpretation I of L is the pair áM, Vñ, whose first element is a model M = áD, Rñ, and whose second element V is a valuation function. D is an arbitrary set of individuals and R is the set of Rn (n for which there are n-adic predicates), where each Rn is the set of all subsets of Dn. A valuation function V assigns elements of D to the variables of L, and elements of Rn to the n-place predicates. We define truth relative to the model M under I in the standard way in terms of an extension V9 of V which maps interpreted wffs onto the set {true, false}. On the model-theoretic account, a logical truth is a sentence true in every model under every interpretation. Here we may ignore the intended interpretation of a language2 in fixing the extension of logical truth. This allows us to clearly display the partial dependence of truthvalue on domain. For example, in a language about the natural numbers, the sentence ‘xy(y is less than x)’ is false, but we can show that it is not logically false by ignoring the intended domain of discourse and appealing to the fact that the sentence ‘xy(y is less than x)’ is true when the variables range over the domain of all integers. Since the cardinality of the ranges of quantifiers is a non-logical element in a quantification, we want to portray the truth of a logically true quantification as invariant across a range of assumptions about the cardinality of the domain over which its variable(s) range. Now we turn to Quine’s substitutional account of logical consequence. My sketch of the account centres on Quine’s presentation of it in Philosophy of Logic (PoL).3 There, Quine gives two equivalent formulations of his substitutional notion of logical truth. For purposes of this paper we rely on the first characterization:4 A logical truth ... is definable as a sentence from which we get only truths when we substitute sentences for its simple sentences.5
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On Quine’s substitutional (S) account, a sentence from an interpreted first-order language L is a logical truth if and only if (iff) we get only truths when we substitute well-formed formulas (wffs) for its simple wffs. A simple wff is one in which no particle from L’s grammar appears. No logical terminology appears in a simple wff. The substitutions may involve complex (possibly open) wffs replacing simple wff, the same complex (possibly open) wff replacing the simple wff throughout the sentence. We represent the nuances of admissible substitutions by developing Quine’s definition as follows. Let E and S range over the well-formed formulas (wffs) of L. Let the n-tuple (S1 , ... , Sn) be the distinct simple wffs occurring in an L-sentence P. P is a logical truth iff each sentence P9 that results by replacing a subset of (S1 , ... , Sn) with admissible (E1, ... , Ek) (k d n) is true. As stated, it needn’t be the case that each Si is replaced by some Ei. It is to be understood that if an Ei does replace an Si in sentence P, then Ei is to be substituted uniformly for Si wherever it occurs in P. Two constraints on admissible substitutions are required in order to ensure that the definition is not too restrictive. First, no variable free in Ei , other than ones that are also free in Si, is bound in Ei when Ei replaces Si in P. If this is ignored, then ‘xy(x is a dog) o yx(x is a dog)’ is no longer a logical truth since substituting ‘x = y’ for ‘x is a dog’ results in a falsehood assuming that the domain of discourse is greater than one. The second constraint on admissible substitutions is that no variable that occurs bound in Ei appears in any occurrence of Si. Without this, the Quine account is unable to reflect that ‘x(yFy o Fx)’ is a logical truth. For in a language whose domain is greater than one we would be able to replace ‘Fy’ and ‘Fx’ in ‘x(yFy o Fx)’ with ‘x ~ (x = y)’ and ‘x ~ (x = x)’ to get the falsehood ‘x(yx ~ (x = y) o x ~ (x = x)).’ With these two constraints on admissible substitutions in place, we say that an L-sentence P is a logical truth iff there are no admissible replacements Ei (from L) for Si in P that result in a false sentence. We illustrate Quine’s account by applying it to a first-order language in which the predicates that occur below have their ordinary interpretations and whose domain of discourse is the collection of integers. In sentence (A) ‘x(x is a natural number) o xy(x is a natural number & y is a natural number & ~(x = y))’ the simple wff is (x, y: v is a natural number); ‘(x, y: v is a natural number)’
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represents the fact that ‘v is a natural number’ occurs with x in the variable position – denoted by v – and it occurs with y in the variable position. (A) is not logically true since (A9 ) ‘x(x is an even prime number) o xy(x is an even prime number & y is an even prime number & ~(x = y))’ is false and (A9 ) results when we replace (x, y: v is a natural number) in (A) with (x, y: v is an even prime number) – that is, we replace each occurrence of ‘x is a natural number’ and ‘y is a natural number’ with ‘x is an even prime number’ and ‘y is an even prime number,’ respectively. Similarly, we may show that (B) ‘(x (x is an even number) & x (x is greater than three)) o x (x is an even number & x is greater than three)’ is not logically true by replacing the simple sentence (x: v is greater than three) in (B) with (x: ~( v is an even number)), that is, ‘~(x is an even number).’ The resulting sentence, ‘(x (x is an even number) & x ~ (x is an even number)) o x (x is an even number & ~ (x is an even number)),’ is false. The sentence, (C) ‘~xy(x is a natural number & y is a natural number & ~(x = y))’ is not logically false – that is, false solely by virtue of its logical form – for we get the true sentence ‘~xy(z ~ (x = z) & z ~ (y = z) & ~ (x = y))’ by replacing (x,y: v is a natural number) in (C) with (x,y: z ~ (v = z)). On the model-theoretic account, a sentence P from a first-order language L is a logical truth because there is no model M and interpretation I such that P is false in M under I. While, according to Quine’s substitutional account, a sentence P from L is a logical truth because there are no admissible wffs Ei from L such that uniformly substituting the Ei in place of the component simple wffs Si in P (consistent with the above two constraints) results in a false sentence. Are these two accounts extensionally equivalent with respect to what is logically true in first-order logic with identity? There are two reasons for answering in the negative. First, there are first-order languages for which the fact that there are no admissible wffs Ei that generate a false sentence when they replace
On the Substitutional Approach to Logical Consequence 247
Si in P is due to the paucity of L’s lexical resources and not due to the pattern of the logical constants appearing in P. For example, consider the language L about my only son Matt Jr., with just the one-place predicate letter is Matt Sr.’s son. The interpretation of L is as follows: M = á{Matt Jr.}, {{Matt Jr.}, }ñ, and V(is Matt Sr.’s son) = {Matt Jr.}. Then, ‘x (x is Matt Sr.’s son) x ~ (x is Matt Sr.’s son)’ is an S-logical truth, that is, a logical truth according to Quine’s substitutional account. There is no admissible expression E from L such that replacing (x: v is Matt Sr.’s son) with E results in a false sentence.6 On the model-theoretic account we can, of course, ignore the intended interpretation of L and falsify ‘x (x is Matt Sr., son) x ~ (x is Matt Sr., son)’ in a larger domain by, say, adding my daughter Shannon to the domain and keeping the extension of V(is Matt Sr.’s son) as before. The second reason for thinking that the substitutional and modeltheoretic accounts are extensionally distinct is that if a first-order language L allows a sentence P with no simple wffs, then if P is true, P is logically true. For such a P, since there are no simple wffs Si in P, there are obviously no wffs Ei such that uniformly substituting them in place of the Si in P results in a false sentence. For example, since the above language L about my only son Matt Jr. is a first-order language with identity, ‘ = ’ is a logical particle. But then the substitutional account makes ‘~xy(x z y)’ a logical truth in L. Since the model-theoretic account allows changes in the domain of discourse, we can falsify ‘~xy(x z y)’ by expanding the domain. In contrast to the model-theoretic account, there is nothing in Quine’s S account that justifies varying the domain of the quantifiers in order to falsify ‘~xy(x z y)’ in L. More generally, there is nothing in Quine’s account that justifies restrictions and expansions in the intended domain of discourse. Critics have appealed to the apparent discrepancies between the two accounts of logical consequence to argue that Quine’s account is seriously problematic as a means of capturing logical truth in first-order logic with identity. The critics include Boolos, Field, Hanson, Hinman et al., Pap, and Read.7 I have identified two related objections to the Quine account, which I now simply list: 1. In order to capture first-order logic with identity, we must define logical truth in a first-order language L in which the identity symbol is a primitive logical constant (i.e., a particle in the logical grammar). But then this allows the formation of sentences consisting of only logical terminology. Intuitively not all such sentences are logi-
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cally true (e.g., intuitively, ‘x(x = x)’ is logically true; ‘xy(x z y)’ isn’t), but they all are S-logical truths in language with a denumerably infinite domain. Since it illegitimately inflates the class of logical truths, Quine’s account is extensionally inadequate in first-order logic with identity.8 2. Intuitively, whether a sentence is a logical truth should depend only on the interpretation and arrangement of its logical components and the pattern of the remaining expressions. Hence, logical truth in first-order logic with identity should be invariant from one firstorder language to another. However, this does not happen on Quine’s account. As we have seen, what is logically true here turns on the wealth of a language’s (extra-logical) lexicon, including the cardinality of the intended domain of discourse.9 I now respond to these criticisms. I shall argue that the first criticism relies on a misrepresentation of Quine’s treatment of ‘=.’ The second criticism can be answered by slightly altering Quine’s account so that a logical truth in a language L is a sentence that cannot be turned false even under supplementation of L’s lexical resources. 3. Response to Criticism 1 Criticism 1 turns on considerations pertaining to a class of first-order sentences. Letting n be any positive integer, I shall call any sentence of the form there exists n individuals an existential sentence. Reasonably enough, critics of Quine construe the logical form of existential sentences as consisting of only logical terminology. For example, the logical form of (D) ‘There are two things’ is taken to be (D9) ‘xy (x z y).’ On Quine’s account, no sentence with the same form as (D9) is false since every such sentence just is (D9). More generally, since no simple wff occurs in any existential sentence and since each such sentence is, in fact, true, existential sentences remain true on all substitutions for the simple sentences that occur in them. Therefore each existential sen-
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tence turns out to be an S-logical truth in a first-order language whose domain of discourse is denumerably infinite. The criticism here is that the S analysis is extensionally inadequate: each existential sentence is, intuitively, logically contingent; that is, for each n, if there are n things, it is not a matter of logic that there has to be n things.10 However, this misrepresents Quine’s view of logical truth, which prohibits treating any symbol denoting identity as a primitive logical constant. Quine does not construe the logical form of existential sentences in the way that the criticism requires. Nevertheless, Quine does include identity theory as a part of logic, and so he does treat logical consequence in first-order logic with identity. Let me spell this out in terms of Quine’s discussion of logical truth in PoL. According to Quine, logical truth requires semantic ascent. To say that a sentence from some first-order language L is true solely in virtue of its form (i.e., logically true) is to state a generality in L’s meta-language which quantifies over L-sentences (hence the ascent to a metalanguage in order to demarcate logical truths from other kinds of truths), and which is not reducible to a generality expressed in the object language L.11 A sentence P is true by virtue of its form alone iff there are no admissible replacements Ei from L for Si in P that result in a false sentence. Hence, a sentence P from a first-order language L is true in virtue of its form (i.e., it has a valid sentence form) just in case there is a true universal quantification P9 in L’s meta-language stating that all L sentences which result from P by the process of substitution (i.e., all L-sentences with the same logical form of P) are true. Important here for Quine is that the meta-linguistic generalization P9 is not reducible to an L-sentence, that is, not reducible to a generality expressed directly in the object language L. To illustrate, suppose that we are considering whether to let ‘men’ and ‘mortal’ be logical terms. If they are, then no non-logical terminology occurs in ‘All men are mortal.’ Consequently, to say, All sentences of the form ‘All men are mortal’ are true would be equivalent to saying All men are mortal since any sentence of the form ‘All men are mortal’ just is ‘All men are mortal.’ But then the meta-linguistic generalization by semantic ascent is reduced to the (logical) generality expressed directly in the object language: for all objects x, (it is logically true that) if x is a man then x is mortal. This would suffice for Quine to discount ‘men’ and ‘mortal’ as logical constants insofar as ‘All men are mortal’ is not true in virtue of form on Quine’s understanding of logical form. Quine must rule out any logical truth containing only logical termi-
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nology. The connection between logical truth and semantic ascent requires that the form of a logical truth must have more than one instance. If a sentence P is logically true, then P contains at least one occurrence of a simple wff for which we may make substitutions, and for each admissible substitution, the resulting sentence is true. Otherwise, Quine’s talk of semantic ascent is superfluous to his view of logical truth. If we make existential sentences logically true, then logical generalities become expressible by direct quantification in the object language, and we lose the idea of a logical truth being true in virtue of form. Quine’s view, therefore, prohibits the logical form that criticism 1 assigns to existential sentences. As an aside, this response to criticism 1 forces Quine to mitigate his holism: semantic ascent helps to define a fairly sharp dividing line between logical truth and other kinds of truth. Quine does count the laws of identity as logical truths in PoL,12 and a first-order logic with identity is a first-order logic that represents the laws of identity as logical truths.13 As Quine acknowledges in PoL, the problem he faces is that the inclusion of identity theory as part of logic seems to result in abandoning the essential connection between logical truth and semantic ascent by reintroducing a primitive identity constant. Quine avoids this by simulating identity so that all laws of identity become logical truths of a purely quantificational sort. His strategy is to treat identity as equivalence: sameness-with-respect-to-somefinite collection-of-predicates. For example, consider a first-order language L with just two oneplace predicates, ‘P,’ ‘Q,’ and one two-place predicate ‘R.’ Then, ‘xy (x = y)’ becomes, ‘xy ((Px l Py) & (Qx l Qy) & z((Rxz l Ryz) & (Rzx l Ryz)))’. ‘xy x = y’ tells us that x and y are indistinguishable by the three predicates; that they are indistinguishable from each other even in their relations to any other object z, insofar as these relations can be represented as simple sentences in the object language L.14 Since Quine restricts a language’s lexicon of predicates to a finite list, his simulation of identity is a recipe for turning the sentence ‘x = y’ from any firstorder language L into a quantification stating that x and y are indistinguishable with respect to the predicates of L’s lexicon.15 Using this recipe, Quine can assign logical forms to existential sentences that allow substitutions that yield falsehoods.16
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For example, consider a first-order language L with just two oneplace predicates, ‘P’ and ‘Q.’ The formal representation in L of the two existential sentences, (E) ‘There are two things’ and (F) ‘There are three things,’ is not (E9) ‘xy x z y’ and (F9) ‘xyz (x z y & y z z & x z z),’ but something like (E0) ‘xy{(Px l Py) (Qx l Qy)}’ and (F0) ‘xyz[{(Px l Py) (Qx l Qy)} & {(Py l Pz) (Qy l Qz)} & {(Px l Pz) (Qx l Qz)}]’. Suppose ‘Q’ is ‘is a male’ and ‘P’ is ‘is a republican.’ (E0) and (F0) are both true. We may establish the contingency of each as follows. Replace (x,y: Pv) and (x,y: Qv) in (E0) with (x,y: Pv & Pv), and (x,y: Qv & Qv). The resulting sentence, ‘xy{((Px & Px) l (Py & Py)) ((Qx & Qx) l (Qy & Qy))},’ is false. Hence, (E0 ) is not a logical truth. In (F0 ), we need only replace (x,y,z: Qv) with (x,y,z: Qv & Qv) (e.g., put ‘(Qx & Qx)’ in for each occurrence of ‘Qx,’ ‘(Qy & Qy)’ for each occurrence of ‘Qy,’ etc.) in order to generate a falsehood. The sentence that results from (F0) by such replacements is false because at least one of the disjunctive formulas is not satisfied. For at least one of ‘(Px l Py),’ ‘(Px l Pz),’ or ‘(Py l Pz)’ is false for all x, y, and z, and each disjunctive formula contains an open sentence ‘((Qv & Qv) l (Qv9 & Qv9))’ which is satisfied by no objects a, ß. In general, for each existential sentence P, we may replace the simple wff Si in P with Ei that are open sentences that either every object satisfies or none does. This will produce a false sentence that serves as a counter-example to the claim that P is a logical truth.17 Key here is that even though it serves as an analysis of first-order logic with identity, Quine’s S account does not treat ‘=’ as a primitive logical constant.18 One objection to Quine’s complex paraphrase of ‘x = y’ is, of course, that it does not endow ‘x = y’ with the sense of genuine identity. For example, let L’s domain be the real numbers, and suppose that the only predicate is ‘<.’ Then if a real number x and a real number y satisfy ‘z ((x < z l y < z) & (z < x l z < y))’ then x and y satisfy ‘x = y.’ However, ‘x = y’ does not come out with the genuine sense of identity in a language L9 produced by just replacing the lone predicate ‘<’ in L with ‘is a natural number.’ Within the confines of L9 , Quine’s manner of
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defining identity equates all natural numbers on the one hand and all non-natural numbers on the other. So, as he acknowledges, whether or not Quine’s plan endows ‘x = y’ with the sense of genuine identity depends on the discourse at hand. According to Quine, for the purpose of fixing the extension of the first-order consequence relation in L9, the treatment of ‘=’ as ‘indistinguishable with respect to the predicate of L’ is a serviceable facsimile of identity in that the laws of identity are converted via Quine’s translation recipe into L9 sentences that are logically true. The objection here is that Quine’s approach may get the logic of identity right for each first-order language, but it doesn’t preserve the truth of all identity sentences in every first-order language. The force of the objection turns on thinking that in fixing the extension of first-order logical truth the required characterization of the identity symbol is adequate for a given language L only if distinct elements from L’s domain of discourse do not get incorrectly identified. Quine rejects this. There are advantages outside of logic to acquiescing to the treatment of what is distinguishable as the same from the vantage point of a given lexicon.19 Furthermore, we may blame the fact that Quine’s recipe doesn’t define genuine identity in, say, L9 on the paucity of L9 ’s lexicon, and not on the recipe itself. If you want to give ‘x = y’ the sense of genuine identity, don’t look to L9. Surely there is more to say here. However, to the best of my knowledge, nobody in print who faults Quine’s account for making existential arguments valid offers a substantive challenge to Quine’s rationale for his treatment of identity.20 Why in securing the logic of identity is it inadequate to characterize ‘x = y’ so that the values of ‘x’ and ‘y’ are the same if these values are indistinguishable with respect to a particular lexicon? As part of a critique of Quine, an answer must be sensitive to the fact that Quine’s method does not entail any robust thesis about the nature of identity. In various places, Quine argues for his treatment of identity not on the basis of any robust metaphysical thesis but on pragmatic grounds,21 and so it cannot be easily dismissed by a quick rejection of the identity of indiscernibles. 4. Response to Criticism 2 Criticism 2 claims that defining logical truth in terms of substitutions makes what is logically true vary from one first-order language to another. Hence, whether a sentence P from a language L is logically true turns not only on the arrangement of logical constants that appear
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in P and the pattern of the remaining expressions, but also on how robust L’s lexical resources are. Quine doesn’t seem particularly bothered by this. His response is to blame the impoverished language and not his account of logical truth. He tells us that to secure first-order predicate logic with identity, we should look at the yield of the substitutional account for a reasonably rich language. Specifically, for a first-order language L that is adequate for number theory, and whose logical particles are no more than the Boolean connectives, first-order quantifiers and variables, Quine’s substitutional definition and the model-theoretical definition of logical truth do not diverge.22 Quine doesn’t seem to think that the invariance of logical truth from one first-order language to another is a feature of the intuitive notion of logical truth. However, not everyone agrees. For example, Hinman et al., complain that the sentence (H) ‘xy((x is an even number) & (y is an even number) & ~(x = y)) x (x is an even number) x ~ (x is an even number)’ is intuitively contingent but it comes out logically true on Quine’s account in an impoverished L with just one predicate, is an even number, and L interpreted in an abbreviated way as D = {1, 2, 3, 4} and V(is an even number) = {2, 4}.23 (H) should not be a logical truth in any firstorder language. However, this counter-example is easily dismissed since, as previously discussed, Quine does not take ‘=’ to be a primitive logical constant. It is required that (H) be transformed into ‘xy((x is an even number) & (y is an even number) & ~((x is an even number) l (y is an even number))) x (x is an even number) x ~(x is an even number),’ which is false, and so not logically true. However, we cannot make all counter-examples to Quine’s account disappear in this way. Furthermore, for those who don’t buy Quine’s paraphrase of ‘x = y,’ it would be nice to answer criticism 2 independently of the response to criticism 1. Towards this end, the response to criticism 2, suggested in part by Quine himself, is to drop the relativization to the lexicon of one language in Quine’s definition of logical truth. That is, we define logical truth as follows: An L-sentence P is a logical truth iff there are no admissible replacements of Ei from any first-order L for Si in P that result in a false sentence.
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What this amounts to is that we may appeal to the variables and predicates from another interpreted language in making substitutions. For example, we may show that (H) is not logically true by adding, say, is greater than three to L’s lexicon. Replacing (x,y: v is an even number) in (H) with (x,y: v is greater than three) results in a false sentence. It is simple enough to make the yield of Quine’s account for a given language insensitive to the availability of predicates in L’s lexicon. However, in order to ensure that the substitutional account does not ‘bloat the extension of logical truth in first-order logic,’24 its yield for a given language should be invariant with respect to assumptions as to the cardinality of the domain. For example, consider the language L about my only son Matt Jr. Just supplementing L’s lexicon with predicates is not enough to keep ‘x (x is Matt Sr.’s son) x ~ (x is Matt Sr.’s son)’ from being a logical truth for there is no admissible expression E of any L such that replacing (x: v is Matt Sr.’s son) with E results in a false sentence. We supplement L with variables from the interpreted language L9 to make room for exchanging interpreted variables for interpreted variables. Recall that variables of an interpreted language L range over L’s intended domain of discourse, that is, the chunk of the world that L is about. Within the framework of the substitutional approach, we capture the idea of changing the domain of discourse, by appealing to substitutions for variables occurring in a sentence. Recall the interpreted language L: M = á{Matt Jr.}, {{Matt Jr.},}ñ, and V(Matt Sr.’s son) = {Matt Jr.}. Let the interpretation of a language L9 identical to L be the same as the interpretation for L except that the domain of discourse for L9 includes my daughter Shannon (the intended model of L9 is: M = á{Matt Jr., Shannon}, {{Matt Jr.},, {Shannon}, {Matt Jr., Shannon}}ñ. The important difference here is that the variables of L range over just Matt Jr., while those from L9 range over the set consisting of Matt Jr. and my daughter Shannon. For ease of exposition, we distinguish distinctly interpreted variables using subscripts that pick out the cardinality of the domain over which they range; that is, the L variables are x1, y1, z1, etc., and the L9 variables are x2, y2, z2, and so on. The L-sentence ‘x1 (x1 is Matt Sr.’s son) x1 ~ (x1 is Matt Sr.’s son)’ is true, but it isn’t a logical truth, for replacing (x1: v (v is Matt Sr.’s son)) with (x2: v(v is Matt Sr.’s son)) results in a false sentence. For another quick example, consider a language L0 which is just like L except for the inclusion of is taller than in the lexicon. The intended model of L0 is just like that of L. The interpretation of L0 includes that of L along with V(is taller than) = . Consider sentence (I):
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(I) ‘(x1 ~ (x1 is taller than x1) & x1y1z1((x1 is taller than y1) & (y1 is taller than z1) o (x1 is taller than z1))) o x1y1 ~ (y1 is taller than x1)’. (I) is true in L0 . In order to establish that (I) is not logically true in L0 , we supplement the variables of L0 with those from a language about the natural numbers – that is, x0, y0, z0, etc. – and add ‘>,’ which has its ordinary extension over the domain of natural numbers. Then we show that (I) is not logically true by replacing ‘taller than’ with ‘>’ and ‘x1’, ‘y1’, and ‘z1’ with ‘x0’, ‘y0’, and ‘z0’, respectively. These substitutions yield the false sentence (x0 ~ (x0 > x0) & x0 y0 z0 ((x0 > y0) & (y0 > z0) o (x0 > z0))) o x0 y0 ~ (y0 > x0). In sum, by allowing the replacement expressions Ei to come from any first-order language, we may supplement a given L not only with predicates outside of L’s lexicon but also with interpreted variables from another language as well.25 One advantage of the amended definition over Quine’s original one is that it represents the fact that the truth of a logically true quantification is invariant across the range of assumptions of cardinality.26 We have departed from the details of Quine’s account. Nevertheless, I think that we have shown that criticism 2 is shallow: a substitutional definition can make logical truth in first-order logic with identity invariant from one first-order language to another given that we can supplement the lexicon of a language with predicates or variables from another interpreted language. Certainly the criticism in conjunction with the first is not grounds for claiming, as Field seems to think, that ‘definitions in terms of substitution are seriously problematic.’27 As far as I know, the alteration of Quine’s account does not contradict any view espoused by Quine in print on the nature of logical truth, and the revised Quinean account is invulnerable to objections that face the unrevised account when we consider its extension to logical notions kindred to the notions of logical consequence and logical truth for the ordinary first-order languages considered in this paper.28 Before concluding, I address the question of Quine’s rationale for choosing the substitutional account over the model-theoretic account of first-order logical consequence. The short answer is that Quine’s account offers savings on ontology. ‘Sentences suffice, sentences even
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of the object language, instead of a universe of sets specifiable and unspecifiable.’29 But if we rely on the lavish resources of set theory in mathematics, why begrudge logic from helping itself to full-blown set theory? Quine answers as follows: Parts of mathematics, however, require less lavish resources in set theory than other parts, and it is a good plan to keep track of the differences. In this way, when occasions arise for revising theories, we are in a position to favor theories whose demands are lighter. So we have progressed a step whenever we find a way of cutting the ontological costs of some particular development. This is true equally outside of mathematics, and it is true in particular of the definition of logical truth.30
Quine acknowledges that the substitutional account is not free from sets: to ensure the extensional equivalence of his account and the modeltheoretic one, Quine appeals to a language adequate for elementary number theory, and the latter relies on the theory of finite sets. His final assessment of the rationale for retreating from the model-theoretic account to the substitutional one is that ‘it renders the notions of validity and logical truth independent of all but a modest bit of set theory; independent of the higher flights.’31 5. Conclusion Criticisms 1 and 2 are not decisive in showing that Quine’s substitutional account fails to capture first-order logic with identity. It seems to me that the most promising criticism must be centred on the claim the substitutional approach cannot represent all the intuitive features of the pre-theoretic concept of logical consequence even if it is extensionally adequate. Is there a feature of the ordinary, pre-theoretic notion of logical truth that not only Quine’s account misses, but that cannot be represented with a definition of logical truth in terms of substitutions? There is the claim that, intuitively, logical truth is non-substantive; that is, what is logically true does not turn on substantive facts about the world.32 Recall that a sentence is logically true on the substitutional approach because each sentence that results from admissible substitutions is true – given the way the world, in fact, is. If the world were finite, then by the purely substitutional criterion the above sentence (I) would be logically true. Also, it is claimed that there is a robust modal element in the concept of logical truth;33 that is, sentence P is a logical
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truth only if there is no way the world could be according to which P is false. However, the substitutional approach makes any modal element in logical truth disappear. The substitutional theorist, of course, has replies. For example, Quine rejects that there is a robust modal element in logical truth, and, given his holistic approach to logic, the fact that logic is substantive is not problematic for him. If our best theory of the world required that there be a finite number of worldly first-order particulars, then logic would be revised accordingly. Note that the model-theoretic characterization fairs no better than Quine’s substitutional account here. Indeed, the model-theoretic account has been faulted for making logic substantial and it has been faulted for not adequately representing the modal element in logical consequence.34 What this paper points to is that there is no solid basis for rejecting the substitutional approach to logical consequence because of its extension in first-order logic with identity. Slightly amending Quine’s account, properly understood, yields a substitutional notion of logical truth that is not ill-behaved contra Hinman, Stich, and Kim. It is inaccurate to think that just because an account is substitutional it turns first-order sentences that are, intuitively, logically contingent into logical truths. More fundamental claims about the features of the pre-theoretic concept of the logical consequence relation need to be brought to bear against Quine in criticizing his account. But that requires more philosophy of logic than I have seen displayed by the chorus of critics of Quine’s account in particular and of the substitutional approach to logical truth in general. notes This paper is a shorter and slightly altered version of my ‘On the Substitutional Characterization of First-Order Logical Truth,’ History and Philosophy of Logic 25 (2004): 195–214. The website for Taylor and Francis, publisher of History and Philosophy of Logic, is http://www.tandf.co.uk/journals. 1 The conclusion C of an argument is a logical consequence of its premises P1, ... , Pn iff ‘(P1& ... &Pn) o C’ is a logical truth. If the set of an argument’s premisses is the null set, then the argument is valid iff its conclusion is a logical truth. For his own reasons, which I will not get into here, Quine prefers to derive the notion of logical consequence from the notion of logical truth, relying on the above equivalences. See W.V.O. Quine, Philosophy
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2
3
4
5 6
7
of Logic (Cambridge: Harvard University Press, 1986), 49 (hereinafter listed as PoL). For discussion see S. Shapiro, ‘The Status of Logic’ in New Essays on the A Priori, P. Boghossian and C. Peacocke, eds. (Oxford: Clarendon Press, 2000), 333–6. The intended model of L is the set of things over which L’s quantifiers are intended to range. R is as before: it’s the set of all extensional n-place relations over D. The intended interpretation I = áM, Vñ of L in its intended M results from assigning, via the valuation function V, the intended referents and intended extensions to the terms and predicates of L. To say that a sentence from an interpreted first-order language L is true is to say that it is true under the intended interpretation in its intended model. This is by far Quine’s most substantial presentation of his favoured account of logical truth. Furthermore, the substitutional account in PoL differs from Quine’s earlier account (e.g., the one presented in ‘Carnap and Logical Truth,’ in The Ways of Paradox and Other Essays [Cambridge: Harvard University Press, 1976], 110), in that in the former we talk of wff replacing simple wff, while in the latter we say that any non-logical words can be varied in a logical truth without engendering falsity (i.e., in a logical truth, it is only the logical words that occur essentially). The account in PoL doesn’t give rise to objections that can be lodged against the earlier account. For the same point see D. Berlinski and D. Gallin, ‘Quine’s Definition of Logical Truth,’ Nous 3 (1969): 111–28, 115. The second formulation is in terms of schemas. A logical schema is a dummy sentence; it is like a sentence except that it has schematic letters in place of predicates. In other words: it is built up by quantifiers and truth functions from simple sentence schemata such as ‘Fxy,’ ‘Gz,’ etc. A logical schema is valid if every sentence obtainable from it by substituting sentences for simple sentence schemata is true. A logical truth is a truth thus obtainable from a valid logical schema. PoL, 51. PoL, 50. Replacing the universal quantifications in ‘x (x is Matt Sr.’s son) x ~ (x is Matt Sr.’s son)’ with their expansions in D we get ‘(Matt Jr. is Matt Sr.’s son) ~(Matt Jr. is Matt Sr.’s son),’ and there is no substitution for (Matt Jr.: is Matt Sr.’s son) which will yield a false sentence. See G. Boolos, ‘On Second-Order Logic,’ Journal of Philosophy 72 (1975): 509– 27; H. Field, ‘Metalogic and Modality,’ Philosophical Studies 62 (1991): 1–22; W. Hanson, ‘The Concept of Logical Consequence,’ Philosophical Review 106 (1997): 365–409; P. Hinman, J. Kim, and S. Stich, ‘Logical Truth Revisited,’ Journal of Philosophy 65 (1968): 495–500; A. Pap, Semantics and Necessary Truth (New Haven: Yale University Press, 1958), 130–3; S. Read Thinking
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8 9 10 11
about Logic: An Introduction to the Philosophy of Logic (Oxford: Oxford University Press, 1995), 41. E.g., see Hanson, ‘The Concept of Logical Consequence,’ 371. Hinman, Kim, and Stich, ‘Logical Truth Revisited’; Boolos, ‘On SecondOrder Logic.’ E.g., see Hanson, ‘The Concept of Logical Consequence,’ 371. As Quine tells us, ‘A ... salient trait of the logical truths was seen in our tendency, in generalizing over them, to resort to semantic ascent. This again is explained by the invariance of logical truth under lexical substitutions. The only sort of generality that can be managed by quantifying within the object language, and thus without semantic ascent, is generality that keeps the predicates fixed and generalizes only over the values of the subject variables. If we are to vary the predicates too, as for logical theory we must, the avenue is semantic ascent’ (PoL, 102). To further elaborate, We can generalize on ‘Tom is mortal,’ ‘Dick is mortal,’ and so on, without talking of truth or of sentences; we can say ‘All men are mortal.’ We can generalize similarly on ‘Tom is Tom,’ ‘Dick is Dick,’ ‘0 is 0,’ and so on, saying ‘Everything is itself.’ When on the other hand we want to generalize on ‘Tom is mortal or Tom is not mortal,’ ‘Snow is white or snow is not white,’ and so on we [must] ascend to talk of truth and of sentences, saying ‘Every sentence of the form ‘p or not p’ is true.’ What prompts this ascent is ... the oblique way in which the instances over which we are generalizing are related to one another. PoL, 11.
12 Quine’s reasons for treating the laws of identity as logical truths are as follows. First-order predicate logic with identity is complete. Second, truths of identity are peculiarly basic (i.e., they have a universality suggestive of logical truths). That is, truths of identity are true regardless of the values of the variables used to state them. Finally, as soon as we have merely specified the truth-functional notations, the variables, and the open sentences of a language, then we know enough what to count as an adequate definition of identity. PoL, 62. 13 Important to what follows is that we should not assume that any first-order logic with identity is a logic that counts some symbol referring to identity as a logical constant. This is false. See E. Mendelson, Introduction to Mathematical Logic, 3rd ed. (Monterey: Wadsworth & Brooks/Cole, 1987), 74. A first-order logic with identity is a first-order logic that represents the laws
260 Matthew McKeon of identity as logical truths. What Quine’s considered view illustrates (sketched below) is that we do not need to regard a symbol for identity as a logical constant in order to represent the laws of identity as logically true. 14 With respect to language L, the laws of identity, x(x = x) and xy (x = y o (Px l Py)) become x((Px l Px) & (Qx l Qx) & z((Rxz l Rxz) & (Rzx l Rzx))) and xy (((Px l Py) & (Qx l Qy) & z((Rxz l Ryz) & (Rzx l Ryz))) o (Px l Py)). The latter two sentences are first-order logical truths. 15 Of course this relativizes the logical form of an identity sentence to the lexical resources of a given language (e.g., the logical form of ‘there are two things’ varies from the two languages which are described in ‘Response to Criticism 1,’ above), and so it wouldn’t make sense to speak of the form of an identity sentence. 16 In Quine’s words, A reconciliation is afforded, curiously enough, by the very considerations that counted most strongly for reckoning identity theory to logic; namely, the definability of identity illustrated [in ‘Response to Criticism 1,’ above]. If instead of reckoning ‘=’ to the lexicon of our object language as a simple predicate, we understand all equations as mere abbreviations of complex sentences along the lines of [as sketched in ‘Response to Criticism 1,’ above], then all laws of identity become mere abbreviations of logical truths of the purely quantificational sort ... the structural view of logical truth is sustained. (PoL, 64) 17 A reply to an objection. First the objection. Consider a language L with just ‘P’ and ‘Q’ for predicates, and every object in L’s domain of discourse possesses the properties they name. Consider the conditional (G), xy{(Px l Py) (Qx l Qy)} o xyz[{(Px l Py) & (Py l Pz) & (Qx l Qz)} {(Px l Py) & (Px l Pz) & (Qy l Qz)} {(Px l Pz) & (Pz l Py) & (Qx l Qy)}] This tells us that if there are at least two things, then there are at least three. Is this an S-logical truth in L? The strategy for generating substitutions that I appealed to in the body of the text will not work in generating a falsehood. That is, replacing the simple wff Si in P with Ei that are open sentences every object satisfies or none do will produce a conditional with a false antecedent which doesn’t show that (G) is logically contingent. The problem is that Quine’s account of logical truth restricts the availability of substitutions to the lexical resources of a given language. The objection fails, however, because it ignores a nuance in Quine’s characterization of logical truth that we appeal to in answering criticism #2: we say that a sen-
On the Substitutional Approach to Logical Consequence 261 tence is a logical truth because there are no substitutions, even under supplementation of L’s lexical resources, for its component simple wff, which yield a false sentence. (See PoL 59.) We may show that (G) is not logically true in L by adding a predicate F to L’s lexicon which is true of some elements of the domain and false of others. This allows us to produce replacements that generate a falsehood from (G). Replace (x,y,z: Pv) with (x,y,z: Fv) and (x,y,z: Qv) with (x,y: Qv & Qv). For cases in which the domain of discourse is exactly one thing and there is no such F, see below. 18 As Hanson, ‘The Concept of Logical Consequence,’ points out, in Quine’s Methods of Logic there is a glaring discrepancy, not acknowledged by Quine, between his account of the pre-theoretic notion of logical consequence and the model-theoretic one according to which all existential arguments are invalid. In the introduction to Methods of Logic, Quine explicitly adopts the substitutional characterization of the intuitive, pre-theoretic notion of logical truth. The trouble is that in the later development of the model-theoretic account, Quine treats identity as a primitive logical constant even though it is clear that this cannot be so on the substitutional approach. Surely we should expect a more seamless transition in Methods of Logic from the prefatory pre-theoretic explanation of logical consequence to the model-theoretic account that follows. 19 See, for example, W.V. Quine, ‘Identity, Ostension, and Hypostasis’ in From a Logical Point of View (Cambridge: Harvard University Press, 1980), 65–79, 71. 20 For example, Read, Thinking about logic, remarks that, It is not a matter of logic that there are at least two things. None the less ... the purely substitutional criterion characterizes [the proposition, ‘There are at least two things’] as a logical truth, given the usual acceptance of the quantifier ‘some’ or ‘there are,’ negation and identity as logical expressions ... this is clearly absurd. (41) Read uses this observation to motivate the model-theoretic criterion of logical truth. But this, of course, doesn’t work unless the usual acceptance of identity as a primitive logical expression is defended. Read offers no defence. 21 E.g., see, W.V. Quine, ‘Logic and the Reification of Universals’ in From a Logical Point of View (Cambridge: Harvard University Press, 1980), 102–29. 22 To establish the equivalence of the substitutional and model-theoretic definitions of logical truth, what must be proven is the following equivalence: A sentence S remains true under all substitutions for its simple sentences iff S is true on each interpretation of its non-logical terminology in every non-empty domain.
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23 24 25
26 27 28
Quine’s argument for this uses a generalization of Löwenheim’s theorem that is due to Hilbert and Bernays: any satisfiable first-order sentence is satisfied by a model whose domain is the set of natural numbers and whose predicates are assigned relations on natural numbers that can be defined in arithmetic. The following is a sketch of the argument for the above biconditional. The only if part: The contrapositive of the Hilbert and Bernays generalization is: if no true sentence results from the admissible substitution of sentences of elementary number theory for S’s simple wff, then S is not satisfiable. Letting ~S represent the negation of S, the contrapositive of the Hilbert and Bernays generalization amounts to saying that if each sentence that results from substituting sentences of elementary number theory for the simple wff in ~S is true, then ~S is satisfied by all models. We have derived the only if part. The if part: If a sentence is true in all its interpretations in every non-empty domain, then, by the completeness theorem, it is provable in some standard system. Such a system is sound: its theorems are only those sentences that don’t yield a falsehood on some substitution. Obviously, Quine can’t point to model-theoretic proofs of soundness as evidence that the theorems are substitutional logical truths. This would presuppose the equivalence that he is trying to establish here. He says that the proof methods are ‘visibly such as to generate only schemata that come out true under all substitutions.’ This proves the if part. Hinman, et al., ‘Logical Truth Revisited.’ Ibid. Elaboration of the effects of such supplementation is beyond the scope of this paper. For further details see Berlinski and Gallin, ‘Quine’s Definition of Logical Truth.’ I offer a sample of what is involved. If the variables of a language L range over a domain D with cardinality n, then supplementing L with variables from another language L9 which range over a D9 with cardinality k (n z k) turns L into a many-sorted language L0 whose domain of discourse D0 = D D9 . The lexicon of L0 has three sorts of variables: those that range over just D (e.g., xn, yn, zn, etc.), those ranging over just D9 (e.g., xk, yk, zk), and variables ranging over just D0 (e.g., xn + k, yn + k, zn + k, etc.) This responds to a criticism of Quine’s account by Boolos, ‘On SecondOrder Logic,’ 519–20. Field, ‘Metalogic and Modality,’ 2. For example, Boolos’s ingenious proof (‘On Second-Order Logic,’ 526–7) of the existence of a set S of sentence from a first-order language L rich enough to express number theory which is satisfiable but which cannot be turned into a set of truths by admissible substitutions from L for the simple
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29 30 31 32
33 34
wff appearing in S. To suppose otherwise is to suppose that number-theoretic truths are recursively enumerable. On the amended Quinean definition proposed in this paper, we drop the restriction that admissible substitutions must be built from L’s lexicon. For another potential objection, consider the usual language L for natural number theory, containing a numeral for every number. Following Berlinski and Gallin, ‘Quine’s Definition of Logical Truth,’ 113, suppose we say, a sentence S(m,n...) is a general truth of number theory only if it is true and remains true regardless of what numbers the numerals represent. Call this the model-theoretic account of general truth. Let’s say that on the substitutional account, S(m,n...) is a general truth only if it is true and remains true regardless of which other numerals are substituted for the numerals m, n, ... Since L contains a numeral for every number, the two requirements turn out to be the same. However, if the same language is used to discuss the algebra of real numbers, the two requirements diverge and lead to two different classes of general truths of algebra in L. We make the substitutional and model-theoretic accounts concur by noting that each representation of the numerals in the domain of the reals determines a language L9 , and by allowing the appeal to the (interpreted) numerals of L9 to get replacements in fixing the extension of general truth for L. PoL, 55. Ibid. PoL, 56. See J. Etchemendy, The Concept of Logical Consequence (Stanford, CA: CSLI Publications, 1999); H. Field, ‘Is Mathematical Knowledge Just Logical Knowledge?’ in Realism, Mathematics and Modality (Oxford: Basil Blackwell: 1989), 79–124. E.g., see Etchemendy, The Concept of Logical Consequence and Hanson, ‘The Concept of Logical Consequence.’ E.g., see Etchemendy, The Concept of Logical Consequence and Field, ‘Is Mathematical Knowledge Just Logical Knowledge?’
15 The Fallacy of Transitivity for Necessary Counterfactuals: On Behalf of (Certain) Non-Transitive Entailment Relations JONATHAN STRAND In recent years various relations have been proposed as useful standards by which to judge the validity of deductive inferences. It has generally been assumed, however, that only transitive relations need apply.1 This essay challenges that assumption. There is at least one important use to which we put deductive inference which calls for a non-transitive standard. When we deduce one thing from another our purpose is very often to demonstrate that, of necessity, if the latter were the case the former would be also; we use deductions as conditional proofs to establish the necessary truth of a ‘subjunctive’ or ‘counterfactual’ conditional. But there is good reason to believe that the propositional relation expressed by such ‘necessitated’ conditionals is not in general transitive; ‘necessary counterfactual implication,’ like ‘counterfactual implication’ simpliciter, is non-transitive. As a result, the relations which provide the most sensible standards of validity for inferences in ‘necessary counterfactual proofs’ are non-transitive. 1. The Various Uses of Deductive Inference Call for Various Standards of Validity We typically distinguish good from bad inferences on the basis of whether their premisses are true and their conclusions bear the right sort of logical relationship to their premisses. We say that the best relationship, yielding the strongest arguments, is the ‘deductive’ relationship of ‘entailment,’ ‘validity,’ or ‘logical consequence.’ What exactly this relationship amounts to, however, has been the subject of some controversy. The dominant view has been that ‘strict implication’ provides the correct standard by which deductive inferences are to be
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judged; an inference is ‘valid’ exactly when it is logically impossible for its premisses to be true while its conclusion is false. In common notation, an inference of q from p1 – pn is valid exactly when u ((p1 & ... & pn) q), or u (~( p1 & ... & pn)q), or ~e ((p1 & ... & pn) & ~q).2 (Italicized letters and formulas will be used as names of, or variables for, propositions or other abstract objects. Non-italicized formulas are being used; i.e., asserted.) Many rival theories of logical consequence have arisen, however, often motivated by ‘The paradoxes of strict implication’: necessarily false or contradictory premises strictly imply every proposition, and every proposition strictly implies every necessary truth. Since premisses can then strictly imply conclusions that seem irrelevant or even contradictory to them,3 many have thought that strict implication is too weak to qualify as the sanctioned deductive relationship. Though logicians have often sanctioned single standards, by which, apparently, every deductive inference, in every context, should be judged, there is good reason to think this is a mistake. Inferences are used for various purposes, and it would be foolish to reject an inference that accomplishes what it is being used for. This suggests a variety of standards of adequacy. When we are trying to establish that certain things are certainly or probably true, for example, by deduction from other things we find certain or highly probable, strict implication seems a perfect standard. It’s simple, transitive, and the ‘uncertainty’ of a conclusion strictly implied by a set of premisses cannot exceed the sum of the uncertainties of those premises.4 When we are trying to decide what else to believe, however, on the basis of contradictory information, strict implication is useless. Contradictions strictly imply everything, but one cannot safely infer just anything from contradictory information. A more discriminating standard, one restricting adjunction at least, is called for.5 Perhaps there is one ultimate relation which can accurately be used as the standard by which good deductive inferences can be distinguished from bad, in every context. If so, however, there will almost certainly be contexts in which there is another relation which can more easily, but still accurately, make the needed distinction. Strict implication is a non-starter for dealing with contradictory information. But its simplicity and its indisputable success in separating the sheep from the goats in many contexts make it perfectly suited for use in those contexts. What relation provides the best standard of valid inference, then, probably varies with the inference and its use.
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2. Necessary Counterfactual Proof One use to which we very often put deductive inference is to deduce one thing from another for the purpose of demonstrating that, necessarily, if the latter were true the former would be also; we use deductions as conditional proofs to establish the necessary truth of ‘subjunctive’ or ‘counterfactual’ conditionals.6 Using ‘u ’ as the necessity operator and ‘>’ as the counterfactual conditional operator, we often use a deduction of q from p (alone) to establish that u (p > q). More generally, we often use arguments of the following form: ‘p1. p2. ... pn. ... ?q. ?u ((p1 & ... & pn) > q).’ But what sort of relation should we use as our standard of valid inference when deducing q from p for the purpose of proving ~(p > q)? Obviously, if the whole point of a deduction is to establish what else would, of necessity, be the case if p were, it would be silly to endorse an inference of something from p which wouldn’t, necessarily, be the case if p were. So we can establish, as a condition of adequacy, that no standard of inference adequate for this purpose will endorse as valid any series of inferences by which q can be deduced from p, unless u (p > q). Now so long as it is possible for the initial, assumed premises in a ‘necessary counterfactual proof’ to all be true, Strict Implication meets this condition with flying colours. It’s simple, and, uncontroversially, so long as p1 – pn could have all been true, that ~((p1 & ... & pn) > q) is true exactly when p1 – pn jointly strictly imply q.7 Since necessary falsehoods strictly imply everything, however, Strict Implication is the unexceptionable standard by which to judge all inferences in all necessary counterfactual proofs if and only if every counterfactual with a necessarily false antecedent is necessarily true. Unfortunately and notoriously, it is far from clear that this is so. There is, in fact, very good reason to believe otherwise. 3. Why Strict Implication Is Inadequate On the most popular, possible-worlds-type, semantics for modal claims and counterfactual conditionals, ~(p > q) is true exactly when, from the point of view of every possible world, the ‘most similar’ (or selected) p-worlds (worlds in which p is true) are also q-worlds.8 But since there are no possible worlds in which a necessary falsehood is true, possible-worlds semantics must deal with counterfactuals with such antecedents in an ad hoc manner. They usually count every such ‘counterpossible’ as true, for simplicity’s sake, or because necessary
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falsehoods strictly imply everything and it is assumed that whenever p strictly implies q, u (p > q). But this is the point at issue, of course. Consider two contingently false propositions, P and Q, which are ‘counterfactually independent’ (i.e., both P and ~P counterfactually imply neither Q nor ~Q, and vice versa). (We’re capitalizing ‘P’ and ‘Q’ here because we’re using them as names of specific propositions rather than as variables.) Suppose P and Q are about a (normal) penny and quarter, each of which were flipped once and landed ‘tails up.’ Let P be The penny landed heads up and Q, The quarter landed heads up. In this perfectly possible situation, would (P & ~ P) > (Q & ~ Q), or even (P & ~ P) > Q, be true at all, to say nothing of necessarily true? As reflected in typical possible-worlds semantics for counterfactuals, p > q is true when q is satisfied in certain states of affairs in which p is satisfied but which otherwise, in some relevant sense, differ minimally from the actual state of affairs. States of affairs in which p is satisfied but which contain other, superfluous differences from actuality are irrelevant to the truth-value of the conditional. If p and q are both actually but independently false, for example, the existence of states of affairs in which both are satisfied in no way renders p > q true. The satisfaction of q in those states of affairs is a superfluous difference between them and actuality, rendering them irrelevant to the truthvalue of p > q. Now on the face of it, impossible states of affairs which somehow satisfy P & ~ P, but do not satisfy Q, are the P & ~ P-states of affairs which differ minimally from actuality. Those states of affairs in which Q & ~ Q or Q is satisfied differ from actuality in a superfluous way, rendering them irrelevant to the truth-values of (P & ~ P) > (Q & ~ Q) and (P & ~ P) > Q. Since neither Q & ~ Q nor Q is satisfied in the ‘P & ~ P-’ states of affairs which differ minimally (non-superfluously) from actuality, (P & ~ P) > (Q & ~ Q) and (P & ~ P) > Q look to be neither necessarily, nor even actually, true. To simply extend possible-worlds semantics to handle such conditionals, the P & ~ P-worlds which differ minimally from the actual world, D, would be impossible worlds which satisfy P & ~ P but not Q or Q & ~ Q. Since neither Q nor Q & ~ Q is satisfied in these worlds, (P & ~ P) > (Q & ~ Q) and (P & ~ P) > Q are neither actually nor necessarily true. Every world in which both P & ~ P and Q, or Q & ~ Q, are satisfied contains at least one superfluous difference from the actual world (since Q, or Q & ~ Q, is satisfied in that world, but not D). A counterfactual is true only when every antecedent-world which does not differ superfluously from the actual world is also a consequent-world.
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Since in this case every antecedent-world which is also a consequentworld does differ superfluously from the actual world, the conditionals in question are false. In fact, if ~(p > q) were true, for every q and impossible p, then the P& ~ P-world which would differ minimally from every possible world would have to be O, the world satisfying every proposition.9 On the face of it, however, such a world hardly differs minimally from D; it differs from D in many superfluous ways. Many propositions are satisfied in O but not D, whose satisfaction (arguably) neither contributes to nor is required by that of P & ~ P; though the satisfaction of P & ~ P requires the satisfaction of one contradiction (P & ~ P), for example, it in no way requires the satisfaction of every contradiction. So it seems not to be the case that, for every q and necessarily false p, u (p > q). Since some propositions, then (e.g., some contradictions), strictly imply propositions they do not necessarily counterfactually imply, strict implication is inadequate as a general standard of valid inference for necessary counterfactual proofs. 4. Two Objections Many have cast doubt on this sort of reasoning by casting doubt on the notion of impossible worlds.10 The Fall 1997 issue of the Notre Dame Journal of Formal Logic, however – a ‘Special Issue on Impossible Worlds’ – more than adequately answers this objection.11 It provides several perfectly coherent notions of impossible worlds. It demonstrates that, taken as abstract objects, impossible worlds are no more objectionable than possible ones. If one accepts the use of possibleworlds semantics for counterfactuals, there is no good reason to reject their extension into impossible worlds. For the purposes of logic, possible and impossible worlds are at least very useful fictions. Another objection to the argument above, however, is likely to be the following. By what standard do the P & ~ P-worlds in which either Q or Q & ~ Q is satisfied, and by what standard does O, differ superfluously from the actual world? After all, there is at least one clear standard according to which O would be the P & ~ P-world differing least from D: O is the only P & ~ P-world in which the satisfaction of propositions is closed under strict implication. 5. A Concession Those who study counterfactuals tell us that which similarity relation among worlds determines the truth-value of a counterfactual is a con-
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text-dependent matter. There are occasions when a counterfactual conditional sentence will express a truth because the antecedent-worlds most similar to D in the respects relevant in that context are consequentworlds as well. But often, in other contexts, that very same sentence will express a falsehood; the antecedent-worlds most similar to D in the respects relevant in that context include worlds which are not consequent-worlds. The same sentence can be thought of as expressing significantly different propositions in different contexts, because different similarity relations are invoked. Now there may very well be contexts in which logicians (for example) use counterfactuals, invoking similarity relations which entail closure under strict implication. In such contexts, every sentence of the form ‘(p & ~ p) > q’ will express a necessary truth. There are almost certainly other contexts, however, in which other similarity relations are invoked on which many such sentences are false. 6. The Maximal-Absurdity of l Let us conceive of an ‘evaluation’ of a class of propositions å as a state of affairs ‘composed of’ states of affairs of the sorts its being the case that p and its not being the case that p. Say that a state of affairs is ‘included in’ another when the former is among the states of affairs of which the latter is composed, or it just is the latter. And say that a proposition p is ‘(dis)satisfied in’ or ‘by’ a state of affairs s, or s ‘(dis)satisfies’ p, exactly when s includes its being the case that p (its not being the case that p). (Proposition p is actually (dis)satisfied, then, of course, when it is (dis)satisfied by a state of affairs that actually obtains.) Conceive of an evaluation of å, then, as such a state of affairs which either satisfies or dissatisfies (but not both) p, for every p å. In keeping with the standard conception of possible worlds as total ‘ways things could have been,’12 then, let us conceive of ‘worlds’ as such evaluations of the class of all propositions. The possible worlds will be the worlds which could have obtained, the impossible worlds the rest. We can represent any such evaluation E of a class of propositions å as a partition of å into a pair of classes áå *, å **ñ such that å * å ** = å and å * å ** = I; the members of å * are satisfied, and the members of å ** are dissatisfied, by E. If we say that one evaluation áå *, å **ñ ‘contains’ another áT*, T**ñ exactly when T* å * and T** å **, we can also represent every ‘absurdity’ contained in any impossible world as an evaluation áå *, å **ñ such that it is not possible for all the members of å * to be actually satisfied while the members of å ** are not.
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Now if we think of ‘absurdity’ as entailing, and being entailed by, impossibility, but admitting of degrees, some absurd states of affairs seem even ‘more absurd’ than others. (E.g., a square’s being both round and triangular seems even more absurd than a square’s being round.) But by at least one natural standard, no world can be more absurd than O. For any absurdity áå *, å **ñ, áå * å **~ å **, Iñ is at least as absurd (where å **~ is the class of the negations of the members of å **). But for any world, áå *, å **ñ, áå * å **~ G**, Iñ just is O. Moreover, in fact, every absurdity áå *, å **ñ contained in any impossible world is matched by an absurdity at least as great (áå * å **~ å **, Iñ) in O. Hence O is at least as absurd as any world; its absurdity is unsurpassable. But since no possible world is absurd at all, absurdity is a kind of dissimilarity from any possible world. Hence, ceteris paribus, how absurd a world is is a measure of how dissimilar it is to any possible world. Hence, very plausibly, any world which is no more absurd than another is no less similar than that other to some possible world. But then every P & ~ P-world which does not satisfy Q & ~ Q, or Q, is no less similar than O to some possible world. But (p & ~ p) > q is a necessary truth, for every p and q, only if no p & ~ p-world is as similar to any possible world as O. Hence, by this standard, it is not true that (p & ~ p) > q is a necessary truth, for every p and q. To maintain that every counterpossible conditional always expresses a necessary truth is to implausibly maintain that by every standard of similarity we could properly invoke to evaluate counterfactuals, and for every proposition p, O is the most similar p & ~ p-world to every possible world. 7. Another Way For all propositions p and q, let ~p be the proposition that it is not the case that p, p & q the proposition that it is the case that p and the case that q, and p q the proposition that it is either the case that p or the case that q, or both. In every possible world, then, and for all p and q, ~p is satisfied exactly when p isn’t (so is dissatisfied), p & q is satisfied when p and q are, and p q is satisfied when either p or q (or both) are. Call these ‘the satisfaction conditions’ for these negations, conjunctions, and disjunctions. One cannot suppose any proposition p and its negation ~p satisfied, without ‘violating the meaning of’ ~p – or ‘violating’ ~p itself, assuming propositions just are meanings. To suppose that both p and ~p are satisfied is to suppose that (it is the case that) it is not the case that p,
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while supposing, nonetheless, that it is the case that p. Similarly, any world which satisfies p & ~ p violates (the meaning of) either p & ~ p or ~p; it satisfies p & ~ p without satisfying p, or without satisfying ~p, or satisfying ~p while satisfying p as well. Impossible evaluations frequently violate propositions in such ways; they (1) satisfy a proposition without meeting – in fact while ruling out the meeting of – its ‘satisfaction conditions,’13 or they (2) meet a proposition’s satisfaction conditions without satisfying – in fact while dissatisfying – the proposition. For any p and q, any evaluation áå *, å **ñ violates ~p whenever either (1) ~p å * and p å * (so p å **), or (2) p å ** ( å *) and ~p å **. It violates p & q whenever either (1) p & q å * but either p å ** or q å **, or (2) p å * and q å * but p & q å **. And it violates p q whenever either (1) p q å * but p å ** and q å **, or (2) either p å * or q å * but p q å **. But consider any such evaluation of any set of propositions å which is closed under negation, conjunction, and disjunction, and whose atomic members are logically independent.14 How many violations of propositions such an evaluation contains provides an easy way to measure how absurd it is. Among such evaluations, if E and E* are evaluations of the same set of propositions, E is more (less) absurd than E* whenever E contains more (fewer) such violations than E*, and E is no more (less) absurd than E* whenever E contains no more (less) violations than E* (and its violations are not a strict super(sub)set of E*’s). Since the atomic propositions in these sets are logically independent, any absurdities contained in any such evaluations will be fully accounted for in terms of their propositional violations. Such evaluations of such sets can never satisfy a necessary falsehood, dissatisfy a necessary truth, or absurdly satisfy certain propositions while dissatisfying others, without violating one of those propositions or their ‘components.’ In fact, given those violations, these evaluations’ other absurdities are no absurdities at all; given that ~p is satisfied, for example, though p is as well (violating ~p), there is nothing further absurd at all in p & ~ p being satisfied. On the contrary, it would be a further absurdity for p & ~ p not to be satisfied then. So the degree of absurdity of such an evaluation can be fully accounted for in terms of its violations of propositions – its satisfying a proposition while dissatisfying its satisfaction conditions, or satisfying its satisfaction conditions while dissatisfying the proposition. Now consider the closure å under negation, conjunction, and dis-
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junction, of {P, Q}. And consider the evaluation E of å which violates only ~P (by satisfying P and ~P) and dissatisfies Q. E satisfies P & ~ P, it contains only one violation of one proposition, there is no evaluation of å satisfying P & ~ P but containing fewer such violations, and E satisfies neither Q nor Q & ~ Q. So E is no more absurd than any other evaluation of å which satisfies P & ~ P. In fact, every evaluation of å which also satisfies Q & ~ Q is more absurd than E. Now very plausibly, for any two such evaluations E and E* of such a set of propositions å, if E is no more absurd than E*, there is a world containing E which is no more absurd than any world containing E*. But then there is a world satisfying P & ~ P but not Q or Q & ~ Q, which is no more absurd than any world satisfying P & ~ P and Q or Q & ~ Q. But then neither ~[(P & ~ P) > (Q & ~ Q)] nor ~[(P & ~ P) > Q] is true. The P & ~ P-worlds which satisfy Q, or satisfy Q & ~ Q, all differ superfluously from some possible world – from the actual world, in fact. Hence, it is not the case that, in every context, ‘u (p > q)’ expresses a truth, for every q and impossible p. And, hence, there are contexts in which some propositions (e.g., simple contradictions) strictly imply things they do not necessarily (or even actually) counterfactually imply. And from this it follows that strict implication is not a generally adequate standard of validity for necessary counterfactual proofs. 8. Another Objection The dominant view seems to have been quite the opposite, however. As C.I. Lewis demonstrated,15 any proposition can be easily and straightforwardly deduced from any explicit contradiction, using only seemingly unimpeachable rules of inference. And surely, it is thought, if ‘q’ can be thus deduced from ‘p & ~ p,’ ‘u ((p & ~ p) > q)’ must be true – in every context. Hence, in every context, 8 must be the most similar p & ~ p-world to any possible world after all (for any p & ~ p), and the satisfaction of no proposition is superfluous to that of any contradiction. Hence strict implication is in fact generally equivalent to ‘necessary counterfactual implication.’ So strict implication is the correct standard of validity by which to judge the inferences in necessary counterfactual proofs. Lewis’ deduction can be represented as follows: Suppose,
1. p & ~ p
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Therefore, by simplification from 1, Therefore, by addition from 2, Therefore, by simplification from 1, Therefore, by disjunctive syllogism from 3 and 4,
2. p 3. p q 4. ~p 5. q
To conclude on the basis of this deduction, however, that u ((p & ~ p) > q), for any p and q, is to use a necessary counterfactual proof, begging the question at issue here. To know whether this is a valid proof of ~((p & ~ p) > q), we must first know when inferences in such deductions should be considered valid, for this purpose. 9. Toward a Better Standard The criterion of adequacy we used above entails that any standard adequate for judging the inferences in such proofs must be at least as strong as necessary counterfactual implication itself (which, if our reasoning above was correct, is strictly stronger than strict implication; for all p and q, ~(p > q) entails ~(p q), but not vice versa). How about necessary counterfactual implication itself, then? Call an inference of s from r valid (in a deduction of q from p, for the purposes of demonstrating ~(p > q)) when u (r > s). In fact, by our absurdity and similarity standards above, and given two additional charitable assumptions,16 every inference in Lewis’ deduction passes the test; each is a necessary counterfactual implication.17 And one might reasonably think that if q can be deduced from p & ~ p by a series of such implications, p & ~ p so implies q. That is, one might reasonably think that necessary counterfactual implication is transitive. In fact, those who use Lewis’ deduction to prove ~((p & ~ p) > q), for all p and q, seem to be assuming just this. But given our plausible similarity standards, and even those additional charitable assumptions, every p & ~ p does not so imply every q. For there will be worlds satisfying P & ~ P, P, P Q, and ~P, but not Q, which are no more absurd than any world satisfying P& ~ P, P, P Q, ~P, and Q. Hence, like counterfactual implication simpliciter,18 but unlike strict implication, necessary counterfactual implication is not a transitive relation. Taking deductions like Lewis’ as proof that u ((p & ~ p) > q) suffers from a fallacy of transitivity for necessary counterfactuals. This also shows, however, that not even necessary counterfactual implication itself passes the test as an adequate standard of valid inference for necessary counterfactual proofs.
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10. Adequate Standards Thankfully, necessary counterfactual implication, like counterfactual implication simpliciter, does enjoy a limited sort of transitivity; there is a simple condition under which it is transitive. A chain of counterfactual implications is transitive whenever those implications would still hold if its first term were satisfied; p > r is true whenever p > q and q > r are, so long as p > (p > q) (which is always the case when p > q is) and p > (q > r).19 Likewise, and hence, a chain of necessary counterfactual implications is transitive whenever, necessarily, those (simple) implications would still hold if the first term in the chain were satisfied; u (p > r) whenever u (p > q) and u (q > r), so long as u (p > (p > q)) (which is always the case when u (p > q) is) and u (p > (q > r)). Lewis’ deduction fails to satisfy this condition. Given our charitable assumptions, the first three inferences pass the test, but the last does not. From any possible world, and for any counterfactually independent p and q, the most similar p& ~ p-worlds, by our standards, will be worlds which satisfy p, ~p, and p q, and from those worlds they themselves will be the most similar p & ~ p-, and ~p-, worlds. Hence, u [(p & ~ p) > ((p & ~ p) > p)], u [(p & ~ p) > (p > (pq))], and u [(p & ~ p) > ((p & ~ p) > ~ p)]. But it is not the case that u ([p & ~ p] > [([p q] & ~ p) > q]). For the minimally absurd p & ~ p-worlds not violating p & ~ p will all satisfy (p q) & ~ p, but some will satisfy q and some will not. And the former will be no less absurd than the latter. Hence, though the final inference in Lewis’ deduction is actually a necessary counterfactual implication, it wouldn’t necessarily have been a counterfactual implication at all if p & ~ p were satisfied. The following, then, would be a generally adequate standard of valid inference for deductions in necessary counterfactual proofs. Where p is the assumed premise – the initial term in the deduction and antecedent of the conclusion – an inference of r from q is valid exactly when u (p > (q > r)). Equivalently, and easier to apply, such an inference is valid when u ((p & q) > r).20 More generally, then, in a necessary counterfactual proof of ~((p1 & ... & pn) > s), an inference of r from q1 – qn is valid exactly when u ((p1 & ... & pn & q1 & ... & qn) > r).21 Of course these schemata do not give us single relations, but rather whole classes of relations by which to judge such inferences – a different relation for every proposition p, or set of propositions {p1, ..., pn}. And these relations are not generally transitive.22 They do enjoy a limited sort of transitivity, however, which saves them as useful standards
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of validity for necessary counterfactual proofs. A series of such implications is transitive so long as p is (or p1 – pn are) the first term(s) in the series.23 Of course this entails that there are generally transitive relations that can do the job that needs doing here, but those relations are more complex and less easily applied as standards.24 And the fact that the relations specified above are not in general transitive does not in the least inhibit their usefulness as standards of validity for inferences in necessary counterfactual proofs; the relations of p-entailment (let’s call them) are transitive when restricted to chains of implications beginning with p – which is the context in which we’re proposing to use them. 11. In Conclusion If our purpose in using a series of deductive inferences is to establish what else would, of necessity, be the case if some initial assumption(s) were the case, the relations which provide the best standards of valid inference for that task are relations which are not generally transitive. notes 1 See, e.g., R. Routley, V. Plumwood, R.K. Meyer, and R.T. Brady, Relevant Logics and Their Rivals, part I (Atascadero, CA: Ridgeview Publishing, 1982), 74–6. 2 Where ‘u ’ and ‘e ’ are the necessity and possibility, and ‘~,’ ‘&,’ ‘,’ and ‘’ the truth-functional negation, conjunction, inclusive disjunction, and material conditional operators. 3 E.g., in arguments of forms like ‘p & ~ p, ? q,’ ‘p & ~ p, ~ q, ?q,’ ‘p, ? q ~ q),’ or ‘~(p ~ p), ?p ~ p.’ 4 See Alan Hájek, ‘Probability, Logic, and Probability Logic,’ in The Blackwell Guide to Philosophical Logic, Lou Goble, ed. (London: Blackwell, 2001), 377– 8, and Ernest Adams, A Primer of Probability Logic (Stanford, CA: CSLI, Stanford University, 1998), ‘Four Probability-Preserving Properties of Inferences,’ Journal of Philosophical Logic 25 (1996): 1–24, and The Logic of Conditionals (Dordrecht: Reidel, 1975). 5 See, e.g., P. Schotch, and R. Jennings, ‘Inference and Necessity,’ Journal of Philosophical Logic 9 (1980): 327–40. 6 I take and use these as equivalent terms, picking out a single class of conditionals. 7 At least if we take ‘p > q’ as expressing the proposition that q would (or would still) be the case if p were.
276 Jonathan Strand 8 See, e.g., Alvin Plantinga, The Nature of Necessity (Oxford: Oxford University Press, 1974), 55; Robert Stalnaker, ‘A Theory of Conditionals,’ in Studies in Logical Theory, Nicholas Rescher, ed. (Oxford: Blackwell, 1968), 98–112; reprinted in Ifs: Conditionals, Beliefs, Chance and Time, W.L. Harper, R. Stalnaker, and G. Pearce, eds. (Dordrecht: Reidel, 1981), 41–55; and David Lewis, Counterfactuals (Cambridge: Harvard University Press, 1973), 8–16. 9 See Stalnaker ‘A Theory of Conditionals,’ 45–6. 10 See, e.g., David Lewis, ‘Postscripts to “Anselm and Actuality,”’ in Philosophical Papers, vol. 1 (New York: Oxford University Press, 1983), 21; and David Lewis, On the Plurality of Worlds (Oxford: Basil Blackwell, 1986), 7n. 11 See, in particular, Edwin Mares 1997, ‘Who’s Afraid of Impossible Worlds?’ Notre Dame Journal of Formal Logic, 38, no. 4 (1997): 516–26; Greg Restall, ‘Ways Things Can’t Be,’ Notre Dame Journal of Formal Logic 38, no. 4 (1997): 583–96; David Vander Laan, ‘The Ontology of Impossible Worlds,’ Notre Dame Journal of Formal Logic 38, no. 4 (1997): 597–620; and Edward Zalta, ‘A Classically-Based Theory of Impossible Worlds,’ Notre Dame Journal of Formal Logic 38, no. 4 (1997): 640–60. 12 See Plantinga, The Nature of Necessity, 44–5, and Lewis, Counterfactuals, 84. 13 No world containing that evaluation meets that proposition’s satisfaction conditions; the evaluation dissatisfies what it must satisfy, and/or satisfies what it must dissatisfy, in order to meet that proposition’s satisfaction conditions. 14 Whenever p å and q å, then ~p å, p&q å, and p q å. And å’s members, which are neither negations, conjunctions, nor disjunctions, are such that any, all, or none of them could have been actually satisfied while the rest were not. 15 See C.I. Lewis and C.H. Langford, Symbolic Logic, 2nd ed. (New York: Dover, 1959; 1st ed. 1932), 250. 16 The charitable assumptions are that p and q are logically independent, and that one is invoking a similarity relation ‘ruling out’ worlds violating p& ~ p itself. Otherwise, where p = q, e.g., many p q-and- ~ p-worlds which don’t satisfy q are no more absurd than any that do. And many p& ~ pworlds which don’t satisfy p are no more absurd than any that do. 17 There is an even quicker deduction of q from p& ~ p (for logically independent p and q), whose inferences are all necessary counterfactual implications even without that second charitable assumption; u ([p & ~ p] > [(p & ~ p) q]), and u ([(p & ~ p) q] > q), though ~u ([p & ~ p] > q). 18 See Lewis, Counterfactuals, 32–5. 19 See Jonathan Strand, ‘Arguing for English Conditionals: A Simple Evalua-
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20
21
22
23
24
tion Procedure for Natural Language Arguments with Conditional Conclusions,’ in New Studies in Exact Philosophy: Logic, Mathematics and Science, B. Brown and J. Woods, eds. (Oxford: Hermes Science Publishing, 2001), 113– 38, esp. 116, 119, 125. It is not that ‘u (p > (q > r))’ is equivalent to ‘u ((p & q) > r)’ or that the relations between q and r expressed by these formulas are equivalent. Rather, any series of inferences satisfying either standard will also satisfy the other, so long as p is the first term in the series. Though these two standards will not always sanction the same individual inferences, they will sanction the same necessary counterfactual proofs. To define this relation in model-theoretic terms, for the purposes of formal logic, we could say that q1 – qn
R s)), for every possible world w in M, the most similar p1 – pn and q1 – qn-satisfying worlds to w (according to R) also satisfy r. Under otherwise standard assumptions, this is when ~((p1 & ... & pn & q1 & ... & qn) >R r) will be true in every possible world in every such model structure. q can imply r, and r can imply s, by these standards, without q thus implying s. ~(p > (q > r)) and ~(p > (r > s)) can be true without ~(p > (q > s)) being true. (Let p = P, q = P & ~ P, r = (P & ~ P)Q, and s = Q.) Likewise for ~((p & q) > r), ~((p & r) > s), and ~((p & q) > s) (let p = P, q = ~P, r = (P & ~ P)Q, and s = Q). This is because ~(p > (q > r)) and ~(p > (r > s)) cannot be true without ~(p > (q > s)) being true, so long as p = q. Likewise for ~((p & q) > r), ~((p & r) > s), and ~((p & q) > s). E.g., let q1 – qn < p1–pn r whenever (1) u ((p1 & ... & pn & q1 & ... & qn) > r) and (2) there is a series of inferences from p1 – pn to q1 – qn which all meet the same standard. This defines a generally transitive relation which can accurately distinguish valid from invalid deductions in necessary counterfactual proofs. (I am indebted to Donald Nute for this point.) But to use this relation as a standard of valid inference for such proofs unnecessarily prevents one from judging an inference in such a deduction valid unless one has already judged each previous inference in the deduction (or some other deduction) valid. Moreover, such a standard would reject as invalid individual inferences which seem best taken as valid. If a series of inferences contains an invalid inference (by this standard) then every subsequent inference in the series would be judged invalid, no matter how valid it may seem (so long as there isn’t another series of inferences which leads validly from the initial pre-
278 Jonathan Strand mises to that conclusion). Even subsequent instances of ‘repetition’ (inferring q from q) would be judged invalid. It seems most informative and least misleading to say that only the invalid inference of q from p (for example) is invalid, not also a subsequent inference of q from q (or p & q from p and q). This transitive standard would not misidentify any whole deductions as invalid, however.
16 Vagueness and Intuitionistic Logic: On the Wright Track DAVID DEVIDI
1. Preamble The basic structure of the sorites paradox is familiar to most philosophers and, what with the ever-expanding literature on vagueness, is by now painfully familiar to some. The discussion in this paper centres around a closely related paradox – the ‘no sharp boundaries’ paradox – and what Crispin Wright has to say about it. I therefore reluctantly begin by sketching the sorites once again, so that the motivation for Wright’s introduction of this paradox can be made clear and so that the paradox can be contrasted to the standard sorites paradoxes. At the centre of any sorites paradox is an argument of the following form: 1. M(0) 2. n(M(n) o M(n+1)) Therefore, 3. M(10,000) The validity of this argument, presented in this form, would seem to be beyond question. Indeed, as Michael Dummett emphasized in his seminal discussion of vagueness,1 if an argument of this form might be invalid, the consequences are dire. For we don’t really need the universally quantified second premiss. We can replace it by 10,000 conditional statements, each an instantiation of (2), and the conclusion would be derivable from those premisses by 10,000 applications of modus ponens.
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So if we want to claim invalidity, it would appear that we will wind up rejecting either modus ponens itself or the transitivity of deducibility. The plausibility of the premisses, the implausibility of the conclusion, and hence the appearance of paradox obviously depend chiefly on how we interpret the predicate M. A favourite choice for interpreting it is the predicate ‘a person with x hairs on his or her head is bald’ – each premiss seems obviously true, the conclusion obviously false, and hairs are nice, discrete objects that give us an obvious way of applying natural numbers to the phenomena in question. The sentences ‘x grains of sand do not make a heap’ and ‘x is a small number’ share most of these virtues, though it is a bit less obvious how large a number we need in the conclusion to make them obviously false. Some candidates, like ‘shade number x is red,’ require somewhat more work so that we can make numbers apply – dividing a range of shades running from obviously red at one end to obviously orange at the other through a series of 10,000 shades, each perceptually indiscriminable from the next so that the natural numbers apply, for instance. But it is often suggested that any vague predicate, with a bit of this sort of massage, can be squeezed into this basic form. Taking stock, we seem to have a sound argument with a false conclusion. Peter Unger2 takes this to be an accurate description of the situation and notices that with suitable introduction of negation signs we can run the argument in the other direction and so show that each of the first premiss and the conclusion are both true and false. Short of this counsel of despair, and presuming that we do not want to find ourselves rejecting either modus ponens or the transitivity of proof, we seem to have only two options: we can refuse to grant the truth of the first premiss or of the second premiss. Since the rejection of ‘a person with 0 hairs is bald’ seems a rather tough position to defend, the rejection of the second premiss seems our best bet. That would be that, but for the following considerations: If (2) is not true, then it must be false, and so 4. x(M(x) o M(x+1)) must be true. But that statement is equivalent to 5. x(M(x) M(x+1)). A bit of reflection on what (5) amounts to when M has any of the
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sorites-susceptible interpretations mentioned above is enough to convince most, though interestingly not all, that it is unacceptable – the idea that there is a specific number of hairs such that a person moves from not-bald to bald by the loss of a single hair, or that n could be a small number but n+1 not-small, strikes most people as outlandish. This reasoning depends on principles we might call ‘classical’ in at least two ways. First, the move from the untruth of (2) to its falsity – hence the truth of (4), its negation – depends on the assumption that a statement of this sort must be either true or false hence, it would seem, on the principle of bivalence. Second, the move from statement (4) to (5) is legitimate in classical logic but not in some other logics. These dependencies on classical principles have had the consequence that to a great extent the literature on vagueness consists of people trotting out every available non-classical logic as a ‘solution’ to the sorites paradox. For our purposes, what matters is that neither of these classical assumptions is valid in intuitionistic logic. It is Hilary Putnam who is best known for having advocated intuitionistic logic as a key to dissolving the paradox, emphasizing that in intuitionistic logic one cannot make the inference from (4) to (5).3 As paradoxes are wont to do in the face of rather simple solutions, this one reappears only a step or two away. Shortly after Putnam’s paper appeared, Crispin Wright and Stephen Read pointed out that one doesn’t need (2) to generate a paradox.4 For that, the negation of statement (5) suffices. In intuitionistic logic, from (1) and 6. x(M(x) M(x+1)) we do not get M(1), but we do get M(1). Now if we suppose M(2), we get from (6), even in intuitionistic logic, that M(1), and thus a reductio of our supposition. Thus we have M(2). And so on. We thus have M(10,000), which, while not intuitionistically equivalent to (3), does stand in contradiction to the evident fact that M(10,000), and so we have a paradox. In his more recent writings on vagueness, Wright dubs this cousin of the sorites paradox the ‘no sharp boundaries’ paradox.5 Notice that the way the counter-intuitiveness of rejecting (2) was made evident was by showing that it had (5) as a consequence. This consequence seems clearly unacceptable for any genuinely vague predicate M. So the nosharp-boundaries paradox has at least a couple of advantages over the usual sorites paradox: first, it has as its major premiss a claim which it
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is plausible to contend is more obviously something anyone would have to accept for any vague M than is principle (2); second, it is immune to the most obvious attempts, like Putnam’s, to dissolve the paradox by adopting intuitionistic logic. Of course, this does not mean that there is no further room for manoeuvre for someone who is intent on advocating intuitionistic logic as a way of solving puzzles about vagueness. The obvious first step is to regard the no-sharp-boundaries argument as a reductio of (6), hence as a proof of x(M(x) M(x+1)). Since this is not intuitionistically equivalent to (5), one is not committed thereby to the assertion that M has a sharp boundary. However, while this observation is quite correct, by itself this response to the no sharp-boundariesparadox is insufficient, according to Wright: [T]o treat the paradox as a reductio is to deny a premiss which seems to say merely that [M] is vague. Any genuine solution to the paradox has therefore to explain how that appearance is illusory – how the major premiss fails as a schematic presentation of vagueness. There is no way around the obligation and no reason to think, once it is met, that any further purpose will be served by imposing intuitionistic restrictions in this context.6
Wright’s own response to the no-sharp-boundaries paradox is that ‘when dealing with vague expressions, it is essential to have the expressive resources afforded by an operator expressing definiteness or determinacy.’7 For, using Def to symbolize such an operator, the proper expression of the vagueness of a predicate M is not (6), but instead is the principle: 7. x[Def(M(x)) Def(M(x+1))]. And this, unlike (6), does not yield a sorites argument. So, the second respect in which (6) is crucial is that it is a statement which seems to capture something essential about vagueness, but only by virtue of being easily confused with another, similar principle. As Dorothy Edgington has pointed out,8 Wright doesn’t tell us where this leaves principle (6) – we are only told that it does not capture the intuition which lies behind vagueness. But if (6) is true, we nonetheless get the no-sharp-boundaries paradox, because we presumably still want to hold that M(10,000) is true. Perhaps the fact that
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Wright does not say anything about the fate of (6) is evidence that he takes his replacing it by (7) as an explanation of why it is that (5) ‘fails as a schematic expression of vagueness,’ and why it is an illusion to think that it does express it, and so thinks he has met the challenge he puts to the advocate of intuitionistic logic. On the other hand, it is not clear that Wright can hold that (6) is false if he also accepts the classical inference from its falsity to the truth of (5). For he says of the operators expressing definiteness which occur in (7) that it is ‘undeniable’ that ‘there is no apparent way whereby a statement could be true without being definitely so.’ Now it would seem that the truth of (5) would entail that, for some n, both M(n) and M(n+1) are true, so Def(M(n)) and Def(M(n+1)) are true, whence x[Def(M(x)) Def(M(x+1)] is true. So it looks rather as though Wright is going to need to reject bivalence when accounting for the status of (6). What I propose to do in this paper is to help out both Putnam and Wright. Since Wright needs to reject bivalence, and so needs to accept some non-classicality in his treatment of vagueness if he is to answer questions about the status of (6), I shall recommend to him a suitable non-classical alternative. I shall, in fact, recommend that he join Putnam in advocating intuitionistic logic for this purpose. In doing so, I shall further recommend that he accept the double negation of (5), but not (5) itself, and shall suggest ways in which one might meet the challenge to advocates of intuitionistic logic posed by Wright. In part, this justification will overlap with Wright’s claims about (5), which depends on the introduction of a ‘definiteness’ operation. What I shall suggest is that the u operator familiar from discussions of modal logics, when transferred to an intuitionistic setting, admirably performs the role of ‘definiteness’ operation. 2. Intuitionistic Logic and Vagueness Wright’s challenge to those who advocate one or another alternative logic as a solution to the sorites paradox is, at least in part, a challenge to show that their account is not an ad hoc patch of the problem by providing some independent motivation for it. In the case of intuitionistic logic, one way of doing so is by reference to the standard algebraic interpretation of intuitionistic logic in terms of the open sets of topological spaces. Under this interpretation, the semantic value of a formula is simply one of the open sets of the topological space, which, depending on one’s tastes and purposes, can be thought of as the truth-
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value or the extent of the formula under the interpretation: the former being perhaps more natural for sentences, though typically there will be many truth-values and the values will not be linearly ordered, while the latter is most natural for formulas with one free variable. For present purposes, it is useful to think of intuitionistic logic as providing a logic of concept extensions. The interpretation of and is the familiar one in terms of intersections and unions. The most obvious way to arrive at the thought that intuitionistic logic might have something to do with vagueness is to appeal to the common intuition that what the existence of borderline cases shows is that there are sentences P for which the statement ‘P or not-P’ is not (entirely) true. As is well known, the law of excluded middle is not valid in intuitionistic logic; hence that logic allows for the failure of sentences of this form to be (entirely) true. If one is thinking of the semantic value of P under a particular interpretation as the extension of the concept P, then the explanation of the failure of excluded middle links with vagueness in a most satisfactory way: P P is not (entirely) true because the extensions of the concepts P and P do not, between them, include everything – that is, one might tendentiously say, there are borderline cases for the concept P. I certainly would not want to denigrate this line of argument, having ridden it hard elsewhere.9 However, the force of this argument is muted by the failure of many other authors to share the intuition that borderline cases show excluded middle to fail: supervaluationists, after all, go to some lengths to preserve excluded middle even while giving up bivalence, while others feel that in a case where P asserts some object to have some property while that object is a borderline case for the property, one ought to say that both P and P are true.10 A slightly different way to see the virtue of intuitionistic logic as a logic of concept extensions requires briefly considering the interpretation of o and in algebraic semantics. A o B is the join – that is, the least upper bound – of the set of all those open sets whose intersection with (the interpretation of) A is a subset of (the interpretation of) B. Thinking of as ‘implying in this interpretation,’ and recalling the connection between intersection and conjunction, this amounts to the requirement that for any P, P A implies B if and only if P implies A o B. This condition can be naturally construed in either of two ways that make its naturalness evident – as the conjunction of the deduction theorem and the validity of modus ponens, or as encoding the standard introduction and elimination rules for o in systems of natural deduc-
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tion. Negation is interpreted by equating A with A o A, where A is an absurd proposition which is uniformly interpreted by (which is open in every topological space). So A is interpreted to be the join, that is, as the possibly infinite disjunction, of all the extensions for concepts (which interpret those) P such that (the interpretation of) P A implies (, i.e., the interpretation of) A. In other words, the negation of A is the smallest concept extension which includes the extensions of all those concepts which contradict A or, again, as the weakest concept such that if anything falls under it, that thing contradicts A as well. That this is a very natural way to think of negation when one is considering the phenomena of vagueness is nicely suggested by this passage from a paper by R.M. Sainsbury: what a concept excludes is graspable in a positive way, mediated by other concepts. A grasp of red attains grasp of what is not red at a derivative level, via a grasp of yellow, green, blue and so on. A system of such concepts is grasped as a whole, as can be seen in the way paradigms are used in learning. There are paradigms of red, but nothing is non-derivatively classifiable as a paradigm of not-red. Any paradigm of another colour will serve as a paradigm of how not to be red, but only in virtue of its positive classification as another colour.11
This suggests a very plausible account of the negation of colour terms, not least because it makes rather clear why one ought not to expect all the red things plus all the non-red things to include all coloured things: the fact that something is not a clear case of a red thing doesn’t mean that it will be a clear case of a thing of some other colour. Not everybody will be happy with Sainsbury’s move from here to the conclusion that the concept red has no boundary, for the topological interpretation might lead us to say that the interesting concepts – that is, the ones for which excluded middle fails – are precisely the ones with non-empty boundaries. But we shall not pursue this matter here, partly due to the suspicion that it would amount to nothing more than a terminological turf dispute. This discussion could be greatly extended. In particular, I think the case that intuitionistic logic is a useful tool for thinking about the formation of complex concepts in the presence of vagueness can be strengthened by considering the role of relations and quantifiers, and not merely restricting attention to the operators of propositional logic. However, moving on to consider quantifiers increases the complexity
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of the discussion considerably, so for present purposes I hope that the couple of considerations provided are enough to show that an advocate of intuitionistic logic has a case to make in the face of the charge that such advocacy is merely an ad hoc patch designed to circumvent paradox. One task remains: earlier I said I would suggest that it was possible for an advocate of avoiding the sorites paradox by accepting only intuitionistic logic rather than classical to also skirt the no-sharp-boundaries paradox. The nub of the issue, recall, is that while intuitionistic logic allows one to avoid the derivation in general of the principle 5. x(M(x) M(x+1)), it doesn’t allow one to avoid the derivation of its double negation 8. x(M(x) M(x+1)). And the problem with accepting this is that it is the denial of 6. x(M(x) M(x+1)), which, as Wright notes, ‘seems merely to state that [M] is vague.’ While there are quantifiers in these formulas, it seems to me that there is no need to go beyond considering the propositional logical operators to see what needs to be said in reply. In intuitionistic logic, however formulated, P is provably equivalent to P o A. So, whatever the semantics we happen to be using to interpret the intuitionistic operators, P must mean that P doesn’t merely fail to be true, but in some sense of ‘necessarily’ it means that P is necessarily false. That is, intuitionistic negation has, and must have, a modal aspect to its meaning – though, of course, the exact nature of the relevant modality will vary depending on the semantics in use. This, by itself, is enough to suggest a reason why the intuitionistic negation of (5) seems to, but does not, merely state that M is vague. For it says not just that x(M(x) M(x+1)) is false, but that it is, in some sense, impossible. And that would be enough to show that what is in question is a stronger claim than merely that M is vague. But how much help is this? For what is required is not just to show why (6) is not merely the claim that M is vague, but to show why the negation of that claim – that is, (8) – should be accepted even when (5)
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is not. For the lesson of the previous paragraph, reapplied, shows that x(M(x) M(x+1)) says that, in some appropriate sense of ‘impossible,’ it is impossible that it is impossible that x(M(x) M(x+1)). And this, in a rough and ready way, we can read as saying that it must be the case that the formula in question could be true. And why, one might wonder, should we believe that? I don’t pretend to have a compelling argument that this is something to be welcomed by the advocate of intuitionistic logic in this context. However, there are a few things to be said for it. First, and rather weakly, the double-negated claim is clearly quite different from the claim with those negation signs removed. While even the unnegated claim (5) has its influential defenders in the vagueness literature, it is surely easier to believe that there could be a sharp boundary between the M and the non-M, or even that it must be the case that there could be such a boundary, than that one is already in place. Indeed, such a supposition is required if the supervaluationist accounts of vagueness, with their appeals to acceptable sharpenings – which are exactly choices of such a boundary for every such predicate – are going to get off the ground. So the advocate of intuitionistic logic, it seems, is not alone in needing to defend this supposition. But what are the prospects for a more direct defence of the claim that (8), the double negation of (5), must be valid for any vague M? Here’s one suggestion for the shape such an argument could take: If (8) fails, then the reason that (5) does not hold cannot be merely that M is vague, but must be some other (more serious?) defect. The reason for thinking this is very much in the spirit of supervaluationism – if M is vague, then the meaning of M could be modified in such a way that the boundary of the concept becomes precise. Or, at least, in cases where there might be a continuum of possible values which we have blocked into chunks so that the sorites argument can be made to apply, the meaning can be made as precise as it needs to be to make (5) come out true for any choice of units. For if not, the failure of (5) is not due merely to there being the possibility of borderline cases between M and M; rather, the nature of what is going on in the border region must be very different from what is going on elsewhere. For sorites and no-sharp-boundary arguments only get off the ground if we are dealing with the sort of property which can be ordered in suitable ways. If this ordering continues through the border region, then an arbitrary line could be drawn separating the M from the non-M. So if there is no possibility of drawing such a line then, whatever properties are named by M and M, they are
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behaving very differently in the region of the borderline cases than elsewhere – for the cases in both the M and non-M regions need to be orderable while the border region is not. So the difficulty here is not merely that no sharp boundary exists, but rather that M is some property which is regular enough to be ordered within the M and the non-M but which behaves very differently in between. I merely suggest this as a way in which an advocate of intuitionistic logic as a cure for vagueness paradoxes might defend the claim that (8) is acceptable while (5) is not. I do not mean to endorse it, nor am I sure whether there is any better argument for the conclusion. I do think that the considerations advanced here show that this approach to responding to the paradoxes is worth further investigation. 3. The Logic of ‘Definitely’ To begin with, let us suppose that ‘definitely’ is a statement-forming operator – that is, if ‘P’ is a statement then so too is ‘Definitely, P,’ for which we shall write ‘Def(P).’ What would the logic of such an operator need to be like? To start, we note that Def(P) clearly implies P, and so we should expect Def(P) o P to be a valid scheme. On the other hand, as noted, Wright is willing to grant that there is no apparent way for P to be true when Def(P) is not, but he is not willing to grant the equivalence of the two statements. Such a relationship is naturally, and familiarly, captured for such an operator by allowing that the inference rule P, Therefore, Def(P) is valid but the scheme P o Def(P) is not. Finally, supposing that it is definitely true that P o Q and that P is definitely true, it seems reasonable to conclude that Q is definitely true as well. And this amounts to saying that, barring some reason for thinking otherwise, we should accept that the scheme Def(P o Q) o (Def(P) o Def(Q)) is valid. To readers with some familiarity with classical modal logic it will be clear that ‘definitely’ is akin to a u operator in a normal modal logic which is at least as strong as KT. An obvious further question is which of the other familiar modal principles holds when u is interpreted as ‘definitely.’ We shall return to this question, in particular to the question of whether Def(P) o Def(Def(P)) is valid, that is, whether Def is an S4 operator.
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This terminology – ‘normal modal logic,’ ‘KT,’ ‘S4,’ and so on – is at home in discussions of classical modal logics. There are various complications involved in transferring these notions to the case where the background logic is intuitionistic rather than classical. Here we mention only a few. First, in classical modal logic one usually works with a pair of mutually dual operators, u and e . These are interdefinable according to the familiar equivalences: (1) uP l eP (2) eP l uP In intuitionistic logic things do not work out so simply. For instance, if we take u as a primitive operator and introduce e by definition using equivalence (2), equivalence (1) will not be true in general, and vice versa. On the other hand, if we insist on u and e being a pair which satisfies both these equations, we can easily prove u P o u P and e P o e P. This was noted long ago by R.A. Bull, who considers these results ‘intuitionistically implausible.’12 The result is that he, and many others who have considered intuitionistic modal logic, have concluded that it is preferable if at least one of (1) and (2) fails. Relatedly, the familiar parallel between the interaction of u and e in classical logic and the interaction of the interdefinable classical quantifiers breaks down in intuitionistic cases. Given the standard Kripkesemantic understanding of these operators, where u P means ‘P is true in all accessible worlds’ and e P means ‘P is true at some accessible world,’ it is natural to regard the u as a sort of universal quantifier and e as a sort of existential quantifier. In the intuitionistic case, we are left with a choice. One strand in the literature tries to maintain the parallel between u and e and and , except, of course, that now the quantifiers must be intuitionistic rather than classical. So, for instance, u P o e P would be expected to be invalid, since x Px o x Px is not intuitionistically valid. However, the Kripke semantics which makes this the case needs to give up the idea that u P means ‘P is true in all accessible worlds’ in favour of a more complicated condition.13 On the other hand, if one prefers to preserve the interpretation under which u P is true at some point if and only if P is true at all points accessible to it, the result is that u P o e P comes out valid, but e P o u P does not. Thus we have competing candidates for the title ‘normal modal intuitionistic logic.’
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Second, there are many distinct but equivalent ways of characterizing normal modal logics in the classical case – some stated in terms of u, some in terms of e, and some with reference to the appropriate Kripke models, for instance. However, if one attempts to characterize the normal intuitionistic modal logics by insisting that the same conditions be met but that the underlying logic be intuitionistic rather than classical, one gets quite different systems depending on which of the various characterizations of normality one chooses from the classical case. That is, in the absence of the Law of Excluded Middle, they are no longer equivalent, so some choice of what is to count as ‘normal’ needs to be made.14 The upshot of this for us is that one will need to be a bit guarded in the conclusions to be drawn from the arguments below. The notion of ‘normal intuitionistic modal logic’ is not unambiguous, so it is not quite right to simply advance the claim that the u operator of normal intuitionistic modal logics behaves like a ‘definitely’ operator, at least if one only considers a particular candidate for normal intuitionistic modal logic in arguing for the claim. So the reader is advised to take any such claims with an appropriate dose of salt – we shall restrict attention to one sort of normal intuitionistic modal logic in what follows without pausing to state the appropriate provisos. Most discussions of modal operators in intuitionistic logic which begin, as we did above, with algebraic semantics work with operators for logics stronger than KT: that is, the u will not merely satisfy necessitation, K, and the T scheme u A o A. In particular, it is usual to consider operators which also satisfy the scheme 4, u A o u u A – that is, to consider intuitionistic S4 operators. It is not useful for present purposes to restrict attention in this way. For given T and 4, we have the validity of the scheme u A l u u A, which, reading the u as ‘definitely,’ seems to rule out higher-order vagueness, at least according to one presentation of higher-order vagueness. The identity of ‘Definitely red’ and ‘Definitely definitely red’ would seem to rule out the vagueness of ‘Definitely red.’ Investigations of intuitionistic modal logics weaker than S4, and in particular of K and KT, have usually employed a different sort of semantics.15 It involves welding together the usual Kripke semantics for intuitionistic logic with Kripke semantics for modal logics. In frame semantics for intuitionistic logic one uses an accessibility relation to interpret the sentential operators which must be reflexive and transitive, while in modal logic one uses an accessibility relation to interpret
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the u and e operators. Obviously, in the case of K or KT, one cannot use the same relation in anything like the standard way to interpret the sentential operators and the modal operators, since in K one cannot require the relation to be either reflexive or transitive. This is just one example of the key difficulty in developing this semantics: one needs to employ two accessibility relations, one for each purpose, and to specify how these relations need to be related to one another to preserve desirable features of intuitionistic logic or of modal logic. For instance, one presumably wants to keep the persistence property – that any statement which is true at some point remains true at all ‘later’ points – which is characteristic of frame semantics for intuitionistic logic. Something rather interesting happens in the course of these investigations. If we write d for the accessibility relation which is used to interpret the intuitionistic connectives, and read x d y, as is commonly done, as ‘x is a state of information which is extended by the state of information y,’ and write R for the accessibility relation which we shall use to interpret u and call it the relation of ‘modal accessibility,’ the most straightforward way to get such properties to work out might be stated thus: acquiring information doesn’t affect modal accessibility. More precisely, if x has modal access to y, it also has modal access to all states of information extending y, and if some state of information extending x has access to y, then x also has access to y.16 If one begins with e rather than u, different conditions are required. What is particularly interesting for current purposes, though, is what happens if one puts both sets of conditions in place. For if one adds the hard-to-state condition that if any z which is extended by x has access to y, then x has access to something which extends y,17 one makes valid this scheme: u P u P. If one strengthens this to the requirement that if any z earlier than x has access to y, then x also has access to y,18 then one also makes valid the scheme u P u P. Now, all of these results are from investigations carried out on (a particular sort of) frame semantics for intuitionistic modal logic, while we used the algebraic semantics of intuitionistic logic to motivate it as useful for discussions of vagueness. Nothing essential rides on this, though. For these results, showing as they do the axiomatizability of the differing sorts of intuitionistic modal logic, are sufficient to show that there are operators on Heyting algebras sharing the properties of each sort of modal operator. We shall not present any translation into algebraic terms here, as this paper is already rather technical for the
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forum for which it is intended. I intend to pursue the question of whether these operators have any independent algebraic interest elsewhere. Instead, we shall simply help ourselves to operators on the algebras which validate the same schemes. The behaviour of these operators has led some authors, including me, to suggest that u in such systems is somehow related to vagueness. In particular, it looks to be some sort of precisifier, that is, an operator which takes vague concepts as input (i.e., concepts which are uncomplemented) and returns precise (i.e., complemented) concepts as output. It is intriguing to notice that we get two different sorts of operator here: under the first condition if we begin with uncomplemented P, so P P is less than completely true, then u P need not be complemented, though u P will be. Under the stronger condition, both u P and u P will be complemented, of course. But notice that neither of these conditions validates u P o u u P: we can introduce a precisifier which obeys the condition T (so that everything which is definitely P is P) without ruling out the possibility of higher-order vagueness. The compatibility of the stronger sort of precisifier with higher-order vagueness suggests the need to mark a distinction: it is possible that not everything which is definitely P is also definitely definitely P even though ‘definitely P’ picks out a precise (i.e., complemented) concept. It seems to me to be a worthwhile question whether this would count as higher-order vagueness. If it does, then we must be careful not to confuse the higher-order vagueness of M with the vagueness of Def(M). For this to be of any help to Wright, though, it must minimally offer some account which makes it reasonable to refuse to accept (5),
x(M(x) M(x+1)), yet to accept its double negation (8),
x(M(x) M(x+1)), and at the same time to accept (7), the negation of (5) to which we have judiciously inserted ‘definitely’ operators:
x(u M(x) u M(x+1)). The end of the previous section was devoted to suggesting a direction one might take in the attempt to show the first two of these to be com-
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patible. Extending that discussion to the third formula, we see that the challenge is to explain why, for a vague M, it is not merely false but could not be true that there is some x which is definitely M and yet its successor is definitely not M. The likeliest shape for an explanation, using the sort of semantics at which I have waved my hands in this paper, is as follows. Even in those cases where u generates a precise concept from M, u M will always pick out a concept whose extension is a subset of M. Similarly, of course, for u M. So, speaking pictorially, if M and M have a fuzzy boundary which need not be unpopulated, u M and u M will be concepts (perhaps with sharp boundaries, depending on the fine-tuning of the semantics) which have moved back from that boundary. If the original boundary was genuinely fuzzy – in the sense that the steps involved in generating the sorites sequence are sufficiently small that somewhere along the line it becomes indeterminate whether one is on one side of the gap or one is in the fuzzy region, and similarly it is eventually indeterminate when one begins to come out the other side – then it cannot be the case that one can proceed in one step from one side of the divide to the other. 4. Conclusion The main lesson I would like to urge from all of this is that regarding the definitely operators as akin to a u operator in intuitionistic modal logic gives a leg up to accounts of vagueness which follow Wright’s line. For by requiring the validity of necessitation, but not the triviality of the u operator which would follow from granting the validity of P o u P, we explain in familiar terms how to admit, as Wright wants to, that ‘there is no apparent way a statement can be true without being definitely so’ while the introduction of such an operator still has a point. The fact that the background logic is intuitionistic rather than classical leaves Wright room to answer the question, ‘Even given that the correct replacement for (6) in the sorites argument involves a ‘definitely’ operator, what is the truth-value of (6)?’ For, as we have noted, (6) can’t be either asserted or denied without causing serious problems, whether the language has a definiteness operator or not. Accepting intuitionistic rather than classical logic will give us the means to avoid having either to assert or to deny (6). This, coupled with the independent grounds for thinking intuitionistic logic is a useful way to look at the formation of concepts – one which avoids commitment to
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the unlikely claim that whenever one has a grasp on the concept P one has a grasp on a concept which is the complement of P – leaves us with good grounds for thinking that u operators in intuitionistic logic are worth serious investigation by those interested in vagueness. notes The financial support of the Social Sciences and Humanities Research Council of Canada during the work for this paper is gratefully acknowledged. I would also like to thank Windsor Viney for his diligent and expert research help on the project which gave rise to this paper and for some helpful discussions of vagueness. 1 ‘Wang’s Paradox,’ reprinted in Vagueness: A Reader, Rosanna Keefe and Peter Smith, eds. (Cambridge, MA: Bradford/MIT Press, 1997), 99–118. 2 ‘There Are No Ordinary Things,’ Synthese 41 (1979): 117–54. 3 Hilary Putnam, ‘Vagueness and Alternative Logic,’ Erkentniss 19 (1983): 297–314. Other discussions of Putnam’s proposal include Stephen Schwartz, ‘Intuitionism and Sorites,’ Analysis 47 (1987): 179–183, and ‘Intuitionism versus Degrees of Truth,’ Analysis 50 (1990): 43–7; Schwartz and William Throop, ‘Intuitionism and Vagueness,’ Erkentniss 34 (1991): 247–56; and Timothy Williamson, ‘Putnam on the Sorites Paradox,’ Philosophical Papers 25 (1996): 47–56. 4 ‘Harrier Than Putnam Thought,’ Analysis (1985): 56–8. In this note they say ‘Putnam admits that, at the time of writing, he had not thought this idea through. What will already be apparent to the alert reader is that, in order to see serious difficulties for the proposal, Putnam would not have had to think very far’ (56–7). 5 ‘Further Reflections on the Sorites Paradox,’ Philosophical Topics 15 (1987): 227–90. Reprinted with omissions in Vagueness: A Reader. See also ‘Is Higher Order Vagueness Coherent?’ Analysis 52 (1992): 129–39. 6 ‘Is Higher Order Vagueness Coherent?’ 129, n.3. 7 Ibid., 130. 8 ‘Wright and Sainsbury on Higher-Order Vagueness,’ Analysis 53 (1993): 193–200. The cited remark is in footnote 2 on page 193. 9 David DeVidi and Graham Solomon, ‘u in Intuitionistic Modal Logic,’ Australasian Journal of Philosophy 75 (1997): 201–13. 10 For instance, Dominic Hyde in ‘From Heaps and Gaps to Heaps of Gluts,’ Mind 106 (1997): 641–60. 11 ‘Concepts without Boundaries,’ in Vagueness: A Reader, 258.
Vagueness and Intuitionistic Logic 295 12 ‘Some Modal Calculi Based on IC,’ in Formal Systems and Recursive Functions, J.M. Crossley and Michael Dummett, eds. (Amsterdam: North-Holland, 1965), 3–7. To show u P o u P, begin with u P o u P, substitute equivalents on the right to get u P o e P, recall that in intuitionistic logic triple negations are equivalent to single negations, so u P o e P, i.e., u P o u P. The proof for the diamond case is similar. (We shall see below that for our special purposes the condition u P o u P need not be regarded as alarming.) 13 Though advocates of this approach would point out that the extra complication of the condition is essentially the same as the complication required when interpreting the universal quantifier in Kripke semantics for intuitionistic predicate logic, one is simply treating the u as a universal quantifier of the appropriate (i.e., intuitionistic) sort. 14 For instance, if one characterizes normality in the most usual way in terms of u – that is, by requiring the validity of the rule of necessitation ‘P, therefore u P,’ and of the scheme K, u (P o Q) o u P o u Q – one counts different logics as normal than if one characterizes normality in terms of e by requiring the validity of the rule ‘P o Q, therefore e P o e Q,’ of the formula e A, and of the scheme e (P Q) o e P e Q. Indeed, one captures different systems if one uses ‘P o Q, therefore u P o u Q,’ u A, and the distribution of u over conjunction in place of necessitation and K. Further concerns arise as one moves to stronger logics. For instance, as is shown by H. Ono in ‘On Some Intuitionistic Modal Logics,’ Publications of the Research Institute for Mathematical Sciences (Kyoto University) 13 (1977): 687–722, within the first framework there is a canonical choice for intuitionistic S4, but there are infinitely many inequivalent choices for intuitionistic S5 – that is, there are infinitely many sets of formulas which result in distinct systems if added to intuitionistic logic, though each results in the usual S5 if added as axioms to classical logic. 15 This discussion employs the results from M. BoÌiñ and K. Došen, ‘Models for Normal Intuitionistic Logics,’ Studia Logica 43 (1984): 217–43, and K. Došen, ‘Models for Stronger Normal Modal Intuitionistic Logics,’ Studia Logica 44 (1985): 39–70. 16 The condition in BoÌiñ and Došen is, d + R = R + d = R. That this is equivalent to the statements in the text follows easily from the reflexivity of d. 17 That is, d –1 + R R + d –1. 18 That is, d–1 + R = R.
17 The Semantic Illusion R.E. JENNINGS
1. Introduction On any plausible use of the word phenomenon, language must be regarded as a physical or more specifically a biological phenomenon. We emit sequences of sounds or we commit inscriptions, and those sounds and those inscriptions have biological and other physical effects. I say, in the course of a lecture, ‘Would someone please be good enough to open a window?’ and with sufficient luck and patient repetition, someone will rise to a semi-upright position, move to the transparent side of the room, whichever side that is, fumble with a catch and push out a casement, or throw up a sash, or change the position of a slider, all depending upon physical circumstances. The production of the sound stream was in effect like tripping a low-energy relay that partially tripped a lot of other relays, and thereby in one case triggered a relatively high-energy response. A sociopathic lecturer might describe himself as having opened the window, as it were, by remote control. The semanticist will reply that independently of meaning there is nothing that physically fits that stream of sound to the opening of a window. Depending upon the linguistic mix of students, different, physically quite unrelated, streams of sounds might have had similar or better success. What range of sound streams work is, as it were, entirely incidental: it is the meaning that counts. To put the same point differently, the connection between the sound streams that work and the work that they do is what is called conventional. Now what is meant by this characterization is unclear. There has been no convention in the ordinary understanding of the word, no coming together of the linguistic family or some elected set of its representatives to settle the
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terms of agreement between, say, the particular combinations of sounds available in English speech and the details of their regulation and coordination of other physical activities of English speakers. What can be the force of the claim that the connection between speech and other activities, states, and so on is conventional? The physical theorist will answer that the character of language that prompts us to say that its connection with the rest of nature is conventional is this: that the causal relationships that constitute the connection of language with the world themselves have a causal history. This is only what we ought to have noticed all along: we produce the sound and inscriptional complexes that we produce with the effects that they have as the combined result of two factors: (a) because our linguistic ancestors (that is, everyone whose linguistic productions have affected our own) produced the sound and inscriptional complexes that they produced with the effects to which they gave rise, and (b) because of the facts of engendering. The effects of present speech are engendered by the effects of earlier speech. An explanatory physical theory of language must therefore be one that tells us how the effects of earlier speech and inscription have engendered the effects of present speech and inscription. A central topic of research in such a theoretical framework will be the nature of the engendering relation. An important body of data for such a theorist will be the ancestry of vocabulary and the ancestry of syntax; for in the larger story, the causal role of items of vocabulary is, through engendering, the product of the causal roles of their ancestral items, and those roles were and are played out within larger vocalized and inscribed entities, namely sentences. To sum up, an explanatory theory of language will interest itself in the causal significance of items of spoken and written language, but its explanatory force will derive from its success in telling us how that causal significance was engendered by the character of ancestral causal connections. It can of course adopt the language of meaning, but in such a theory a meaning will be a physical type, or more precisely a type of physical effect, and in the early stages of the theory’s development, the exact character of the effect-types that constitute particular meanings will remain unsettled. A suitable language for characterizing them must await a suitable functional language for explaining the workings of the human brain, not just Wernicke’s and Broca’s areas, but all of the regions implicated in the productions and apprehensions of the wider range of physical activities within which linguistic activities are integrated and find their biological fit. The most that can be
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said of these effect-types at the outset is that the effects in question include neural effects that are available to the motor control structures that regulate speech production. As we shall see in the sequel, these effects, however they may eventually be subcategorized in whatever functional meta-language, must be regarded as giving us the core effect-types of an explanatory theory of language. Fortunately, we can make substantial progress in the absence of any such detailed functional characterizations. We need only assume that however such neural effects are eventually typed, the following is true: in general an English (or L) sentence addressed to group of English (or L) speakers will produce in each of them about the same complex of neural effects, as such effects are categorized by the functional neural account. That is, we must assume that whatever functional neural account is given of the relevant areas of the brain, it will enable us to speak non-trivially of a population of effects, of a type licensed by that functional account. 2. The Biological Model of Meaning Just from what we have said so far, it follows that within the theoretical framework proposed, a meaning is a species, where by species is meant: the union, (P) of a set P of populations, P being temporally ordered by an engendering relation. What we have not yet said is that a species figuring as subject matter of such an explanatory theory bears three important similarities to biological species as they are ordinarily understood. First, with few exceptions, such a species is a non-classical set, a set whose characteristic function is not 2-valued; that is, there must be items whose status as members of the set cannot non-arbitrarily be assigned a 1 or a 0. Second, a related fact, with few exceptions, every member of any such species of neural effects has ancestors that are not members of that species of neural effects. The third similarity is that members of the same species have similar morphological profiles. In organic species this amounts to their having corresponding morphologies at corresponding ages of development, and the comparisons are to be made modulo sexual differences. For effects that are elements of meanings, they are the effects of vocal sound streams or inscriptions within which there are many variations, all of which are comprehended by the idea of a morphological profile. Modulo those distinctions, elements of the same meaning are the effects of items of speech having the same morphological profile. Let me stress that our use of the language of species is not metaphori-
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cal. The term species is to be understood according to the definition given. There is some point to the warning, for having come this far, it is tempting to look to biology for concrete and specific as well as formal correspondences; however, when we do, although we do in fact find individually striking parallels, they are biologically heterogeneous. The few observations offered here are neither complete nor biologically consistent. Of course once the fancy has our lapels in its grip, our attention is forcibly drawn to numerous parallels even at the most fundamental level: first, (a) the prolific serial composition of words from a limited vocabulary of phonemes, and (b) the more prolific serial composition of sentences from that vocabulary of words are suggestive of (a9) the serial composition of codons from triples of amino acids, and (b9) the composition of chains of codons in a reading frame. (Here the molecular biologists’ lapels were first gripped by the parallels.) Second, it is tempting to think of parts of speech as requiring, or at least supporting, a distinction akin to the distinction between genes as DNA molecules on the one hand, and their expression in protein synthesis on the other, for we must distinguish the basic phonemic ingredients of speech from their neural effects in the cerebral cortex of their recipient. Third, it is striking that, as the expression of genes is regulated at numerous levels (their transcription, the manner of their processing, their transport to site, and so on), so in speech the recipient’s neural expression of a speaker’s phonemic string is regulated at numerous levels. Of these, I mention only two: First, the effect of the utterance of a sentence is altered by the circumstances of its utterance and by surrounding matter of speech. The point is obvious when the roles of pronouns and anaphora in general are considered, but equally, very general contextual and background matter plays a role in regulating the effects of speech. Consider the sentence 2.1 I’ll have breakfast only if there is insufficient food. Second, the effects of the phonemic matter of speech are regulated by variations within the three prosodic dimensions of its presentation: stress, pitch contour, and lengthenings of component sounds. To illustrate, contrast the pair 2.2 What is this thing called love? and
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2.3 What is this thing called, Love? BBC talk (with 2.2) There is, too, a rare element of indeterminacy in which different effects in distinct recipients are not the results of discerned differences. Consider the two readings of 2.4 No trees have fallen over here. Mary Shaw William Calvin1 has hypothesized that the production of speech involves a Darwinian process in which alternative sequencings compete for a limited neocortical workspace. There is no reason to suppose that, in general, speech reception is any less a Darwinian process than speech production. If it is, then we may look upon the cues that serve to promote one set of effects and to forestall others as something like environmental pressures that select for and so preserve one lineage while selecting against and so extinguishing others. One other point of contact between speech and organic biology will assume importance later in this story. This is that both speech and organisms are similarly modular. As nature does not produce larger creatures by constructing them of larger cells rather than more, so novels are not distinguished from short stories in virtue of their having longer rather than more sentences. In neither case is this an accidental feature. The synthesis of proteins by ribosomes and polyribosomes depends upon bonds and weak forces that are effective only at the small molecular distances within the endoplasmic reticulum and other cell substructures. The apprehension of syntactic structure requires that sentences not exceed certain variable limits of length, that is, that its parts not be distant from one another beyond the capacity of the prosodic and contextual cueing of structure at various levels. In both cases, however, there is a non-zero error rate. In both cases, as we shall see, such errors have more than momentary consequences. Now as I have already hinted, we are rightly suspicious of such pairings of features. From such observations in themselves, nothing of an explanatory nature can be inferred, since we have not claimed, let alone shown, that there is any physical connection between the features of the organic structures and what we have selected as corresponding characteristics of speech. And even if there is, it may be only that familiar recurrence of structural patterns that one finds at different levels of complexity in organic nature: even if there is some deep understanding
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to be had of it, the understanding may still yield little of immediate practical explanatory value. If there is theoretical utility in the observations, it must derive from the linguistic features themselves, independently of any supposed parallels with organic biochemical phenomena. Above all, there must exist linguistic data that give us the correct means of understanding. 3. Some Observable Facts: Polysemy and Logicalization 3.1 Some English Connectives My own main application of the biological approach I have been describing has been in the study of English connectives. To give a thorough account of that study will certainly take us far from the main purpose of this essay, but for various reasons that will become apparent, it will be worthwhile to pause over that topic long enough to make some specific remarks about the apparent place of semantics in linguistic understanding, and thereby to see the need for an explanatory theory of the sort envisaged here. English-speaking philosophers and logicians, at least the authors of just about all of the introductory logic texts in that mons copiosus that I have been able to examine, have, as it seems, settled and confident opinions on the subject of the so-called ‘logical’ vocabulary of English. No doubt in most cases they have inherited the opinions of their own teachers, but they share the attitude that since they can accurately use connective vocabulary in speech, their professional opinions have the warrant of whatever semantic understanding enables them to do so. So it is not surprising that so many philosophers, and such a high proportion of textbook authors, content themselves with illustrating the correctness of their assumptions rather than actually studying the English language. My intention is by no means to disoblige the authors in question. After all, natural-language semanticists since the early Wittgenstein have looked to the successful logical treatment of the connectives as a source of inspiration. In the quest for a semantic theory of natural language, the connective part is supposed, on that reckoning, to have been the easy beginning. And if anyone can be supposed to have such a theory, formalists can. It follows that if even logic text authors almost universally get even these matters wrong, there is some deep-seated error in some very common philosophical assumptions. These two topics – the errors on the one hand, their significance on the other – provide the subject matter for the remainder of this essay.
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Now it is no argument against the textbook views that they conflict with the views of Paul Grice, but the conflict is perhaps worth pointing out (though not at length, since it is discussed so fully elsewhere).2 I content myself with one instance, that of the English word or, that being Grice’s central example and a recurrent textbook theme. According to too many of the textbooks for it to be a coincidence, English has two or’s. The point is variously put: in English or has two meanings, two senses, two uses, and so on. One corresponds to the function, the other to the ¢ function, that is, to xor. In their 1971 paper3 Barrett and Stenner set out the requirements for establishing the existence of a ¢-like use of or. Since the table of ¢ differs from that of only in the first row, the demonstration of the existence of such an or requires that there be an or-sentence which is false but both of the clauses of which are true. No one has produced such an example. Reichenbach,4 twenty-four years earlier, had pointed out that iterated xor behaves like a quantifier: it outputs 1 if any odd number of 1’s is input, and outputs 0 else. So a sentence composed with xor of five simple sentences will be true iff exactly one or exactly three or all five of the sentences are true. (Notice that iterated material biconditional (l) also behaves like a quantifier, outputting 1 if any even number of 0’s is input, outputting 0 else. So, for example, the five-argument case with ¢ is equivalent to the corresponding five-argument case with l.) Linguists agree that there is no such naturally occurring connective vocabulary in any known natural language and a fortiori no such sense or meaning or use of or in English. (One might add that for corresponding reasons, there is no use of if and only if in English corresponding to l.5 That we don’t in general know such facts as these, even after we have studied or taught propositional logic, would tell us something about the general nature of human linguistic competence, if we would but listen.) Again, briefly, since it is also fully considered elsewhere, consider the or of such constructions as 3.1 You may have pie or you may have cake, variations on which appear in many introductory texts as examples of xor uses of English or.6 Since from such a remark the addressee correctly infers that he may have pie, this must be a sort of conjunction, and so neither an inclusive nor an exclusive disjunction. Why is this not noticed by textbook authors? The reason is partly that the language of disjunction and conjunction is not a part of the language of folk-
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semantics; nor is that distinction the salient distinction between and and or for English speakers. But why then, when we consciously set out to codify the semantics of natural language or, do we get the matter so wildly wrong even on so simple a case? I know of no empirical study that purports to answer that question. Most philosophical logicians have not noticed the fact. It is fair to say that the philosophical logicians who write the texts make quite contrary assumptions, assumptions for which no case has ever yet been made. One is that truth-conditions do play a central role in folk-semantics, as the abstract counterpart of valuations and compositions of functions do in formal semantics. Another is that a folk-semantical theory plays a role in the compositions of speech. That the same philosophical logicians, who are also speakers of natural language, get the matter so wrong when they try to articulate fragments of the folksemantics is evidence that if such a semantical theory exists, it is not accessible to trained experts. On the other hand, if we consult the illusive common man, say by reading what he says in press interviews, we tend to find such references as to ‘either-or situations,’ which unreasonably exclude our having our cake and eating it. The evidence is anecdotal, but in my experience exclusivity seems to form a natural component of spontaneous explanations of or; truth-conditions do not. Moreover exclusivity is taken to apply, even by the textbook authors, to cases in which or is in fact truth-conditionally conjunctive. A separate point urges an early mention here: truth is itself a folktheoretic notion of which we have little useful semantic understanding. In formal settings, it gives a convenient reading of formalist artefacts, but not a mathematically informative interpretation of them. We know as much as we need to know about 1 and 0 if (a) we know that they are distinct, and (b) we know which of them is designated. Outside of such settings truth remains a subject of philosophical puzzlement. In introductory philosophy courses it is invariably a topic, but the discussion dwindles after mention of one or two inadequate and incomprehensible theories. I return to the subject under a more general heading later. 3.2 The Gricean Urn: Did the Hand of the Potter Shake? The acceptance of an evolutionary rather than a creationist view (however scientific), in language as elsewhere in biology, would condition us to certain expectations. In the first place we would expect a great
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and messy prolificacy of meaning arising from comparatively simple origins, by a few simple principles acting in diverse environments that the earlier meanings have themselves partly produced. We would expect the theoretical constituents of its most fundamental explanations to be the same constituents as those of other sciences. Biology is fundamentally chemistry (or, if you like, fundamentally physics), though the nature of the chemicals is such that they sponsor a large number of specifically biological types. Polynucleotide-types are cheap, and no one invokes Occam’s razor, neat or modified, to compel biologists to limit the number of them. Biologists, by contrast, would themselves readily invoke the name of Occam in defence of limiting fundamental theoretical types. Occam, though he was himself no doubt a devout creationist, would be an eligible patron for evolutionary biologists. Theoretical foundations are expensive; life forms are cheap, provided that their existence is explained by the theory. Much the same Occamist sentiment would shape a biological approach to language. One wants to keep the number of new fundamental theoretical constructs at a minimum but accept whatever diversity of meaning they dictate. Theoretical foundations are expensive, meanings cheap if their existence is explained by the theory. I do not know how to understand the word sense as various authors apply it to words. So I do not know whether I am in conflict with Paul Grice over what he calls ‘Modified Occam’s Razor’: Senses are not to be multiplied beyond necessity.7 The two points of difficulty with the principle are, of course, at the occurrence of the word sense and at the occurrence of the word necessity. And again, as he confesses, there is some difficulty in sorting out what would constitute an overriding necessity. The exegetical task is further knotted by Grice’s inexplicable preoccupation with his putative strong sense of or, the sense in which it implies non-truth-functional grounds for its use. Here at least the difficulty is easily cut through, for even if he is successful in demonstrating that or has no such sense, this would not prove that it has only one. As the Yiddish saying has it, a for-instance is not an argument. In fact it is very easily demonstrated that all natural-language connectives have more than one truth-conditionally distinguishable use. Even but, which for Grice is a model case of a word that carries a conventional implicature, has both conjunctive and disjunctive uses. Contrast, for example, 3.2 It never rains, but we all own umbrellas anyway with
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3.3 It never rains but it pours. In fact it has other non-connectival uses as well. Consider 3.4 My, but it’s a beauty and 3.5 No one but his mother calls him Hulon. Why, one wants to ask, should words have only one connectival sense but be allowed many non-connectival ones? Or is another case in point. Simply contrast 3.6 He is thirty-nine or he is forty with 3.7 He may be thirty-nine or he may be forty or 3.8 You may have tea or you may have coffee. There is no reason to suppose that the correctness of the inference from this to I may have tea is founded in anything other than a distinct, truthconditionally expressible use of or: the one in which it means alternatively. And then consider 3.9 He must have left early or I would have seen him, in which the or could be replaced by otherwise, but not alternatively. And contrast 3.10 Lou is taller than Mary or Nancy, from which we properly infer that Lou is taller than Mary, with 3.11 Lou is the sister of Mary or Nancy, from which we do not properly infer that Lou is Mary’s sister.
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Two further points about or must be given their place in the assessment. The first is that, statistically, the disjunctive uses of or constitute a very small minority of its uses. The second is that, historically, it seems likely that a species of conjunctive uses is the ancestor of the species of disjunctive ones. It is a commonplace of philosophical logic that there are, or appear to be, divergences in meaning between, on the one hand, at least some of what I shall call the formal devices – , , , , (x), (x), (Lx) (when these are given the standard two-valued interpretation) – and, on the other, what are taken to be their analogues or counterparts in natural language – such expressions as not, and, or, if, all, some (or at least one), the. Some logicians may at some time have wanted to claim that there were in fact no such divergences; but such claims, if made at all, have been somewhat rashly made, and those suspected of making them have been subjected to some pretty rough handling.8
No such debate could have taken place before the advent of the truthtable. Equally, Grice’s own contribution to it, that the common assumptions of the contestants that the divergences do in fact exist is (broadly speaking) a common mistake, and that the mistake arises from inadequate attention to the nature and importance of the conditions governing conversation,9
as beguiling as it is, as influential as it has been, and fraught though it undoubtedly is with useful ideas, represents an oversimplification that could not have occurred to anyone before truth-tables were introduced. It is certainly a singular feature of Grice’s theory that the involvement of truth-tables should so precisely demarcate the authority of its dictates. The theory applies only to the vocabulary that gives us English readings of connectives. It does not apply to any of the other vocabulary of logic, the illative adverbs, for example. It is not insisted that the since of 3.12 I have been longing to meet you since I first read your book must be dependent for its meaning upon the logical role of the since of 3.13 Since I’ve read your book, you must read mine
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or that the so of 3.14 I knew his name and address, so I consulted the phone book must apply for its meaning to the so of 3.15 I knew his name and address, so I knew his name. Nor are any claims issued on behalf of the English word necessarily and the u of E.10 When a colleague says Necessarily D, and then Necessarily E, do I infer Necessarily D and E because of E or M11 plus a convention, or because of a principle of K?12 The answer is, of course, not forthcoming. Only the English words that happen to be given as convenient readings of certain of the truth-functional connectives are subject to these strictures. That unless might have been given as a reading of seems not to enroll it in this club. That had Polish notation gained the ascendancy the word either might have been given as the most convenient reading of A or given for C does not impose this requirement upon either or given, though presumably it might have. Had as well as been adopted for , had reverse Polish notation become the vogue ... there seems hardly any point in jogging further on this spot. There is but one other point to make: in his development of the notion of implicature, Grice has undoubtedly altered the professional lives, certainly the professional vocabulary, of many linguists. But where it matters, they place no great weight upon its intentional components, taking the implicatures of acts of speech to be effects, perhaps (conversationally speaking) even intended ones, but not with any special theoretical reliance upon the notion of intention. The difference between a Gricean account and a biological one is the former’s theoretical reliance upon the notions of truth-condition and intention, not its theoretical reliance upon implicature understood as linguists understand it. The implicatures are precisely what we learn when we learn language. In this Grice could not have got the matter righter. Moreover, the idea that acts of speech have effects that may incidentally invite the application of intentional and cognitive language is common to both approaches. The point of the explanatory theory is to explain conversation, not to put a stop to it. A word about that vocabulary. For a biological linguist, the uses of the words intend and true and their cognates are just more observational
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data. Uses of that vocabulary have effects; that they have those effects requires explanation. The theory presumes that they have the effects that they have because ancestral items of vocabulary had the effects that they had, and because of the particular character of the engendering of the later effects by the earlier. They have no special theoretical status. There is no guarantee or even presumption that they can be given a truth-conditional semantics. In fact all the comprehension (or indeed all the comprehensibility) that is guaranteed is what is required for the transmission of that part of the language from one generation to the next. Anyone who insists that there must be some method for finding out what intentions are or what truth is must overcome this challenge: to demonstrate that the transmission of the use of the vocabulary of intention and truth from one generation to another requires that there be such a method; for all we can be certain of is that we use the vocabulary and that our linguistic ancestors used vocabulary somewhat like it in somewhat the same way. 3.3 Where ‘Logical’ Connectives Come From All the functional vocabulary of any natural language has descended from ancestral lexical vocabulary. Sometimes functionalized items retain their old lexical morphology, sometimes not: contrast, for example, the auxiliary uses of have and go in the formation of perfect and progressive future tenses in English with the vestigial have and go endings of Latin past and simple future verbs. All natural-language logical connectives are functionalized (logicalized) versions of lexical vocabulary, usually vocabulary of physical relationship. So, for example, if is descendent from gyfan (give), or from other (second as in every other day), and but from butan (outside). They are as often as not morphologically reduced by the exigencies of frequent use, which explains why so many of the logical connectives of so many languages are such short words. The explanation of their uses in modern English is the explanation of their descent from that lexical vocabulary. The supposition that, for example, or or if has, truth-conditionally speaking, exactly one meaning must entail that only one descendent of other has survived, that it only ever had one line of descendants or that all the others must have been extinguished before the present morphology was achieved. Any weaker hypothesis must accept more than one meaning in the past, and any such concession must render arbitrary any a priori insistence such as Grice’s on one meaning now (the truth-tabular one) for each connective word.
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3.4 But Consider this list of sentences 3.16 I don’t fall asleep but I dream of Jeannie. 3.17 I fall asleep, but I wake up almost immediately. 3.18 My but Jeannie is beautiful. I make no claim that that list exhausts the distinguishable uses of but. It does not include such uses as that of 3.19 It is but Jeannie or 3.20 It is not Jeannie, but Sarah. However, it will do for the purposes of illustration. Observe first that the but of examples 3.3 and 3.16 is (on a superficial examination, anyway) disjunctive (Either it doesn’t rain or it pours; Either I don’t fall asleep or I dream of Jeannie.) When it is recalled that the connective but descends from butan (outside as in current northern ‘but the house’), it will be almost immediately apparent that, from a purely historical point of view, as between this but and the allegedly implicature-fired conjunctive but of 3.2, this disjunctive use of but is primary. The descent of the disjunctive but is not difficult to trace. The main stages of the development see the extension of spatial but first to abstract categories of items (no reward but glory) then to circumstantial classes: 3.21 No course is forgivable but that he should relent. The ellipsis of that gives us connectival but: 3.22 I will not be exalted but you shall have share in my glory. Now, notice, if historical order counts for anything, it is the conjunctive uses that ought to seem anomalous. If Grice had placed his faith in history rather than the truth-table, the Gricean line we would be debating would be whether we are justified in regarding even the truth-conditionally conjunctive aspect of conjunctive but as no genuinely distinct
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meaning, but rather as itself a mere implicature, and its implicaturial aspect as an implicature inside an implicature. 3.5 Mutations There seems to persist among some philosophers a vision of a largescale semantical theory of language. It rests upon an assumption that the semantic theory of the logical portion of language is now nearly complete and will form the foundation upon which some larger semantical edifice will be built. But, having taken no account of linguistic change, they have ordered their building materials the wrong way round. Connectives are not at all suitable for a lasting foundation, for historically, the connective vocabulary has shown itself to be extremely fragile, while under similar conversational pressures, lexical vocabulary has proved itself comparatively resilient. And although compositionality is widely claimed as a key condition both of our capacity for novel speech production and of our capacity for speech comprehension, this represents only a synchronic point of view. Diachronically, we see matters differently. First, for all our compositional potential, we actually compose very few sentences each day either in production or apprehension of speech, and this actual composition of speech is the main, brakeless vehicle (including the engine) of language change. But the compositional forces that can bring about compositionally significant changes of meaning in functionalized vocabulary such as connectives have less dramatic effects upon lexical items. The point, curiously enough, finds an illustration in Jane Austen: ‘Have you had any letter from Bath?’ [Henry Tilney to Catherine] ‘No, and I am very much surprized. Isabella promised so faithfully to write directly.’ ‘Promised so faithfully! – A faithful promise! – That puzzles me. – I have heard of a faithful performance. But a faithful promise – the fidelity of promising! It is a power little worth knowing however, since it can pain and deceive you.’13
What has Austen noticed? The construction, not an especially intelligent one, arises from a misconstrual of scope: in the word sequence promise faithfully to I the adverb faithfully, in the ancestral uses, modifies the infinitive. The promise, on the ancestral construal, is the promise faithfully to write. The other construal (reserved by Austen for the
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likes of the naive Catherine and the feckless Lydia), which takes faithfully to modify the main verb, permits these intransitive constructions and must therefore give to the element faithfully either a new meaning or no meaning at all. Of course the language finds an idiomatic use for the construction as a whole, a use which is suggestive of earnest, handon-heart asseverations and undertakings, but it is not one that relies upon composition of autonomous meanings. The word faithfully has never migrated with any such meaning to other environments. Negation-raising verbs such as believe, think, and so on present a similar phenomenon. The common construal of I don’t believe that D as I believe that not-D has not spawned a new meaning of the verb believe; in biological terms, idiomatic uses do not generally propagate except artificially. A non-English speaker who says I don’t hope you slip on the ice,14 by analogy with the negation-raising idiom, is more likely to be quoted than to be imitated. I mention in passing that the verb doubt may be an exception to this general claim. There is good reason to suppose that the present use of the verb is a mutation of an earlier, weaker use, one that is exemplified frequently, for example, in Pepys’ Diary: 3.23 There I found as I doubted Mr Pembleton with my wife15 and probably represents the correct understanding of every occurrence of the word in the King James Bible. It may well be this earlier, weaker use that persists in such constructions as 3.24 I do not doubt but that the Viet Cong will be defeated. Richard Nixon Functional vocabulary is less impervious to the effects of such scope misconstruals. Of the many instances I now know about I will mention only two here. The first involves the word unless, which is a reduced form of a longer construction on [a condition] less (than that). In its earliest inter-clausal uses, such a construction would be conjunctive in character. D on a condition less than that E would be representable roughly as D E. But now suppose (as seems to have been the case) that the construction is never used outside the scope of some prefixed negating item. Schematically we can represent this as Not D unless E (the underlining representing the original and ... not reading). On that reading of unless the sentence as a whole will have its present-day reading if the scope of the Not is taken to be as in Not (D unless E). But
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suppose that an emerging portion of the linguistic population agrees with its complementary portion in the construal of such a sentence as a whole, that is, agrees on the occasions of use of all such sentences, but takes the scope arrangements as (Not D) unless E. Then that emerging portion must give to the unless element of such sentences a new construal as or for purposes of new compositions, the new (succubinal) meaning that the overline is intended to express. Of course, since the two portions of the population never see (the ancestor of) unless unaccompanied by a preceding negation, the difference in their syntactic construals will never become apparent, and therefore the new construal remains uncorrected. But under such conditions a natural bias in favour of short-scope construals of negatives or simpler syntax more generally will eventually tip the balance of construals in favour of the innovation, and, if unless migrates to other, un-negated environments demanding an or construal, a sufficient portion of the population of language users will have already accepted the or construal of unless to ensure that it survives. Since the or reading will do for all instances, both the original (on the new syntactic construal) and the new, unnegated one, the and ... not construal is eventually extinguished. I have expressed this in the language of construals, but the phenomenon clearly has a neural substrate involving some form of imperfect replication of structure. One element of the apprehension of speech (or written text) involves the neural rehearsal of the motor sequencing involved in its production. I assume that the rehearsal of a spoken sequence under the auspices of one syntactic scheme is different from the rehearsal of the same spoken sequence under the auspices of another, that, for instance, the rehearsal of the Shaw example (2.4) is neurally distinct accordingly as over goes with fallen or here if one or the other is salient. In the case of functionalized vocabulary, the principle governing such changes seems to be (identical occasions of use) + (novel syntax) « (novel meaning) For lexical vocabulary such novel construals produce idiomatic constructions but seldom new independent meanings if, as is usual, we take it as a requirement of an independent meaning that it be stable through some range of distinct environments. We may of course speak of a meaning restricted to a single environment type if we wish, but within the theoretical framework I am presenting, this would just be a
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redescription of what we ordinarily call an idiomatic use. Even for much functionalized vocabulary its use is triggered only within a narrow range of environment types.16 (Think of universal any and its partiality for negative and modal environments.) 3.6 The Descent of But It seems that (a) a development paralleling the one just mentioned explains the slow emergence of disjunctive but, and that (b) then sometime over the sixteenth and seventeenth centuries, various conjunctive but’s emerged through developments of the same general character but from an ancestral base broadened by the earlier development. The first stage of this fragmentation more or less exactly parallels the emergence of disjunctive unless. Syntax schematically of the structure Not (D but E) is misconstrued as having the structure (Not D) but E, forcing an or or if not reading of but, which is initially concealed by coincidence of occasions of use, but which emerges when the new but migrates to environments in which no governing negation masks the new construal. As an example, consider 3.25 Damme but she’s a beauty. Again, the if not reading unifies such cases with those in which the preceding negation is present.17 The emergence of conjunctive but seems to have overlapped the emergence of the disjunctive one and to have involved once again a novel apprehension of scope, now sometimes involving negations, but also sometimes involving other non-negated environments licensed by the emerging if not reading, and abetted by the general trend toward ellipsis of that. As an example, consider 3.26 I would have gone but [that] I was afraid (that is, ... if [that] I had not been afraid). Certainly we find transitional uses in Pepys’s diaries and elsewhere. By the early eighteenth century, conjunctive but was established, with the result that one can, at this remove, be less confident, in many cases, as to which reading should be given. Consider the two but’s of Dafoe’s Journal of the Plague Year: 3.27 the man at the window said it had lain almost an hour, but that
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they had not meddled with it, because they did not know but the person who dropped it might come back to look for it.18 Compare, as an answer to ‘Where’s Bryson?’ 3.28 I don’t know [,] but he may be in Melbourne. With but as with other items of functionalized vocabulary that have been subject to parallel or similar developments, the production of new meanings by mutation is dependent upon there persisting sufficient similarity of effect of whole constructions that in early stages the two meanings can coexist undiscovered. The satisfaction-conditions need not be identical, or even capable of articulation for the two readings; they need only be at most negligibly different. Mainly it is required that the occasions of use of the construction under one construal be approximately the same as the occasions of use of the construction under the other. If the explanation is correct, then doubts may be registered as to the importance of semantics, truth-conditionally understood, to linguistic practice. If we look for parallels only to the uses of formal languages, we might take the significance of these disclosures to be that, as natural deductive rules are simply rules for extending proofs and not rules of inference, so whatever rules govern our use of language, they are really rules for extending conversation, not for the expression of the content of thought. In fact, though there is not space to develop the point fully here, the constraints on language use seem to be hybrid in character. To be sure, speech produces sensory and distinctively linguistic neural effects, and speech often serves as an intervention that modifies the effects of other speech. Accordingly, speech is constrained by previous or concurrent speech, as lines of proofs are constrained by previous lines. But many features of our surroundings, and not just the linguistic features, produce sensory effects, effects that include distinctively linguistic neural ones, since they can independently prompt us to speak, and do in fact constrain what we say. So linguistic acts should be understood as interventions within an interconnected field of stimuli, some of which are linguistic, some of which are not. That said, it can still be denied that the non-linguistic portion of the constraints on speech must be understood as semantic in character. All of these constraints, both linguistic and non-linguistic, are physical ones. But they are physical constraints that have been engendered by earlier
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sets of constraints upon ancestral neural structures. There is still room for understanding that goes beyond the systematic understanding of the neurophysiology of speech production and apprehension but that is nevertheless non-semantic. In this general connection it is a point insufficiently appreciated in philosophy that language is transmitted mainly through its being learned by children. In the early stages of language development, there need be no articulable supporting semantic or syntactic theory for connectives, no account of satisfaction-conditions for the sentences in which they occur. And when a habit of speech has been learned, there occurs no semantical audit through which the inexplicable habits of speech are rejected and the explicable ones retained. There is only the comfortable familiarity of adult speech. The transmission of language from one generation to the next does not depend upon understanding, only upon a growth of comprehension and the cultivation of a capacity for the production of comprehensible speech. I do not immediately see how, but would patiently hear it argued that a shared, detailed, and articulable semantic theory might help if we were all born into linguistic adulthood or acquired a first language in some such fashion as we do a second. But there is no evidence that any such theory is required for the acquisition of language by a child and therefore none that one is required for the continued possession of linguistic ability once it is acquired. There is certainly no evidence that such a theory is consulted in speech; in fact the automaticity of speech is all against the notion. Now all this might be taken to excuse us from the necessity of being able to give an account of but in its conjunctive uses. The development of conjunctive but does not depend upon any child’s understanding either the ancestral, disjunctive use or the new conjunctive one. As in the case of idiomatic faithfully, we are not required not to be puzzled, though neither need we assume that no account can be given for the role that conjunctive but plays. I know of no satisfactory such account in the literature. Grice seems to say only that the word carries a conventional implicature but does not venture a guess as to what in general that implicature is. He does give but not E as a formula for the cancellation of an implicature that E. So if his doctrines of implicature are intended as explanatory or even descriptive of the practices of speech, then it might be concluded that the cancellation of implicatures is a standard function of but in speech and written language. On such an account, but is used to implicate that there was an implicature to be cancelled or at least that the speaker or writer supposes that there was.19 But even if this is cor-
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rect, there remains the question as to why but plays this role, and how it does it. The following attempts an account. To begin, it should be said that as to the ancestral but, a crude schematic account of its construal is easy enough to make out, for it corresponds roughly to relative complementation. The semantic representation of It does not rain but it pours would be
__D __ – __ E __ = where __ D __ is the interpretation of It rains and __ E __ is the interpretation of It pours. The trail from the earlier use as outside at that historical juncture is still warm. So, as with faithfully in the earlier example, we can expect the descendent use to be regulated in some dimension or other and in some degree by that feature of the ancestral use. Now any account of conjunctive but must of course cover such cases as 3.29 I loved her, but she didn’t love me back that is, it must cover the cases fitting the roughly Gricean mould. And we can also admit on Grice’s behalf that a word (such as but or or) whose use we do not fully understand must be susceptible to many loosenings of semantic or pragmatic control. Nevertheless, any general account of conjunctive but must also take into account those cases in which one of the two clauses entails the other, and it must account for the conversational non-commutativity of such constructions, even of these cases. We can certainly form either of 3.30 He got here, but he got here late and 3.31 He got here late, but he got here. The two are clearly distinct. And in particular, there is nothing queer about the second case, even though the first clause does not implicate the negation of the second. (Nor is there aught queer about that observation, in spite of its content.) However, it must be admitted that uses of but can fail of queerness even where the two clauses are equivalent, as for instance in a sentence of the general shape
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3.32 If D, then E, but if not-E then not-D, even though 3.33 D but D without prosodic distinction is decidedly queer. Queerness in these matters is likely triggered by the failure of some neural anticipation, by some automatically initiated connection or arrangement requiring disconnection or dissolution. And these anticipations are probably schematic and syntactic rather than semantic in character.20 The presence of negation or of the element E or the combination of the two may satisfy the anticipation. I accept that in this case the judgement is a delicate one. As between trusting our automatic responses and trusting our settled judgements in such matters we must simply decide. In any case the following remarks may not account for all such examples. How then do we account for the conjunctive use of but in these cases, and how do we distinguish the one case from its commutation, He got here, but he got here late from He got here late, but he got here? On this account the element of relative complementation applies to conversational effects of the main clause, and the but is subtractive. Schematically the proposal presents but (as it would any connective) as having associated with it an instruction set to be acted upon by the recipient. As the instructions of the set apply to the D and E of an utterance of D but E, they require the following: Subtract from the effects of D, those effects that it shares with the negation of or salient alternative to E. So, as the rule applies to 3.34 I went home, but I went home, it subtracts from the effects of the main disclosure some effects that that disclosure shares with the alternative to going home. Without a fuller account of the imagined situation, we cannot say what the alternatives are; however, the story can be filled in in such a way as to tell us: being dragged or carried there, or staying where trouble was brewing. In another case, say
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3.35 I went home, but I went home, the salient alternative might be having gone to another bar or having wandered about in the wet. But again, having said this, we cannot give a priori an account of either minuend or subtrahend effects. 4. In the End Is the Beginning Whatever one may think residually about the place of semantics in the understanding of language, I trust that I have given sufficient reason to suppose that language can be studied from a physical, more particularly a biological, point of view without ignoring its essential features. I hope that I have illustrated that this point of view also reveals features of language that, to its cost, semantic theory has neglected. I admit that in the course of this piecemeal and inadequate discussion I have made free heuristic use of conversational semantic vocabulary. I don’t see how to do otherwise, nor do I see the necessity of avoiding the use of useful language for the latest moments of linguistic evolution. I do insist that the more neutral language of biological linguistics is essential for framing hypotheses about the earliest pre-linguistic developments that can be regarded usefully as ancestors of linguistic ones, and I claim for the idiom that we need not abandon it in framing hypotheses about later primitive stages. I would claim also that the study of logicalization can usefully recolour our aesthetic view of language. The earliest precursors of language represented relatively crude novel exploitations of natural effects by brains capable of such exploitations. The facts of logicalization suggest that what has centrally changed in the course of language–brain co-evolution is the diminished scale of the crudity, matched by an increased resolving power in the brain’s capacity to exploit effects in novel ways. The scale changes, but the crudity remains. A universal daily intoning of this central fact of linguistic crudity would have a salutary, not to mention steadying, effect upon philosophical outpourings. For more specialized philosophical interest groups, miniaturists in the style of Grice, there is also a useful new perspective, for it can be seen that this increased resolving power has a cost: linguistic transactions become liable to the kind of replicative error that we have sampled here, errors akin to the replicative errors of molecular biology. Inevitably, items long removed from their ancestral lexical uses and morphologically reduced beyond recognition have little resistance to mutations that
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result from errors of syntactic replication. We cannot cling to the simple Gricean view. If I am right, then that view is, in the words of one of my earlier respondents,21 no more than an imaginative fantasy. notes 1 William C. Calvin, The Cerebral Code: Thinking a Thought in the Mosaics of the Mind (Boston: MIT Press 1996). 2 R.E. Jennings, The Genealogy of Disjunction (New York: Oxford University Press, 1994). 3 Robert B. Barrett, and Alfred J. Stenner, ‘The Myth of the Exclusive “or,”’ Mind 80, no. 317 (1971): 116–21. 4 Apparently the first textbook author to do so. 5 There is a more fundamental reason: in English, neither or nor if and only if is binary. So neither the sentence Abe ambles or Mary marches or Ted tittups nor the sentence Abe ambles if and only if Mary marches if and only if Ted tittups is syntactically ambiguous. In a formal language, except by an abbreviative convention that also subsumes the properties of associativity, neither p q r nor p ¢ q ¢ r nor p l q l r is well-formed. 6 For their ubiquity and uniformity of content I have labelled it the argument from confection. 7 Paul Grice, Studies in the Way of Words (Cambridge: Harvard University Press, 1989). 8 Ibid., 22. 9 Ibid., 24. 10 The smallest classical modal logic, adding to the PL closure conditions, the rule [RE] £ D l E Þ £ u D l u E. 11 The smallest monotonic modal logic, adding to the PL closure conditions, the rule [RM] £ D o E Þ £ u D o u E. 12 The smallest normal modal logic, adjoining to the axiomatization of M, the axiom [K] £ u p u q o u (p q). 13 Jane Austen, Northanger Abbey (Ware, Hertfordshire: Wordsworth Editions, 1993; first published 1818), 209–10. 14 I owe the example to Charles Travis. 15 Henry B. Wheatley, The Diary of Samuel Pepys, 2 vols. (New York: Heritage Press, 1942), 1663-05-06. 16 The American use of internecine as meaning within a family or group is a useful foil. Notice, however, that it is (so far) restricted to conflicts, evidence of its origins in misconstrual, though in this case a misconstrual of extension not of scope.
320 R.E. Jennings 17 As a matter of interest, this explains a puzzle which has long (insufficiently) vexed linguists. The use of but here persists, when the curse is generalized to My god and the offending sacred reference expurgated, leaving only My. 18 Daniel Dafoe, A Journal of the Plague Year (London: J.M. Dent Everyman’s Library edition, 1908; first published 1722), 118. 19 I remark in passing that the idea would play mildly merry Ned with his account of the putative strong sense of or, since the construction ‘D or E, but I know which’ is decidedly queer whereas ‘D or E, but I don’t know which’ is not queer in the least. 20 Some such account may explain the effects of surprise endings as, in the right context and with the right prosodic presentation, that of It smells awful, but it may taste terrible. 21 Ronald De Sousa, at the Canadian Philosophical Association Conference, Ottawa, 1998.
Part Three: Respondeo JOHN WOODS
When examining an ancient theory, the investigator of intellectual history is faced with a choice between two main interpretations. One is that the theory is of merely antiquarian interest. The other is that it is still sufficiently alive to resonate with present-day concerns. When this is so, it is possible to see the theory as engaging those concerns and doing so in ways from which modern theories might learn something. We may then speak of the ancient theory as a contending theory. One of the attractions of the chapters by Darcy Cutler and David Hitchcock is their respective suggestions of different answers to these questions in the case of Aristotle and Chrysippus. Aristotle and Chrysippus shared a fundamental conviction, namely, that logic was to be a purpose-built theory for service as the central core of a comprehensive account of argument. For each, the shape that would be given to logic would of necessity be influenced by how arguments actually are. In the event, however, Aristotle’s logic is in several respects a contending theory, and Chrysippus’ logic is not. In Aristotle’s Earlier Logic1 I advanced the claim that, even in the early works of the Organon, the theory of syllogisms was a relevant, non-monotonic, paraconsistent, intuitionistic logic. Each of these traits flows from conditions that Aristotle imposes under the guiding principle that the ensuing logic produce a genuine elucidation of the give and take of argumentation. Aristotle wanted logic to help the theorist of argument attain a disciplined command of the distinction between good and good-looking arguments. Aristotle is the first logician to call attention to a class of errors that he called ‘fallacies.’ It must have been a wrenching discovery, for fallacies reproduce within logic itself that same untamed distinction between what looks good and what is good. A fallacy is an
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argument that appears to be a syllogism but is not in fact. This matters. If the property of being a syllogism is not a recognizable property, how can the theory of syllogisms control the distinction between good arguments and bad arguments that look good? The great meta-theoretical achievement of the Prior Analytics was the almost wholly successful proof of the recognizability of syllogisms. This being so, it is entirely natural to ask, with Darcy Cutler and others, whether in pressing his perfectibility thesis, Aristotle was anticipating the completeness results of modern logic. Part of the received answer is that, in significant respects, the modern quest cannot have been Aristotle’s, since modern completeness turns on a distinction that Aristotle did not have – that between syntax and semantics.2 Cutler is right to emphasize that completeness is not decidability. Knowing that if an argument is valid there is a proof of it is far from knowing what that proof is. Cutler is also right in suggesting that Aristotle’s perfectibility project cannot have been all of the modern project of effective recognizability. This is because Aristotle cannot have had the modern problem of infinity. It is also because Aristotle tries to make ample independent provision for what might be called ‘the minimization of finitude.’ This is essayed by two reductions, neither of which is adequately achieved and each of which appears to have been mistaken. Still, Aristotle thought that it was true, first, that everything stateable in Greek was stateable without relevant loss in the language of categorial propositions, of which there are only four types; and, second, that all logically correct reasoning could be captured without relevant loss in the logic of syllogisms. These are enormously simplifying assumptions. They considerably facilitate the further tasks with which Aristotle had total or near-total success. These are the proofs that a low finite and stateable number of arguments exhaust the class of syllogisms, and the proof that every argument that is a syllogism is either recognizably so as it stands or is provably so using premisses and rules that are recognizably appropriate and/or recognizably correct. David Hitchcock does an impressive job not only in detailing the respects in which Chrysippus’ logic is an odd thing by modern standards, but also in showing how these particularities flow from what Chrysippus took the structure of arguments to be. Also bearing on this matter is what Chrysippus took validity to be. This is interesting. Aristotle had nothing to say about validity (or what he called ‘necessitation’). Yet, as I argue in Aristotle’s Earlier Logic, there are reasons to suppose that Aristotle’s validity closely resembles the validity of mod-
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ern classical logic. Chrysippus, on the other hand, does have something to say about validity, in which it is obvious that he thinks it has a considerably non-classical character. In each case, we have a pair of facts or purported facts. In Aristotle’s case, validity (though not syllogisity) is classical, and there is a rich and relevant sense in which syllogistic logic is complete. In Chrysippus’ case, validity is non-classical and his logic is in various respects incomplete. In Chrysippus’ case we know the two facts to be connected. Might this not also be true of Aristotle’s? If we agree that modern logic was born in 1879, we may safely say that one of its principal tasks was to accommodate transfinite arithmetic, hence quantification over exceedingly large domains. Dale Jacquette reminds us in Part One of the extraordinary success of model theory in furnishing interpretations of such logics. Aristotle had no such task and, hence, no occasion to develop or anticipate the theory of models. This is true, but not very interesting. It is also true, but also very interesting, that some 2500 years later, logicians such as Quine would retain the requirement that logic accommodate transfinite arithmetic, and yet would also argue that the way of models is avoidably expensive. Consequence, Quine’s way, is substitutional, defined from the notion of a logical truth which is also substitutional. There is sometimes great value in Quine’s attempt to minimize logic’s dependency on model theory. It illustrates the prudence of resisting the Can Do Principle, discussed in comments on the previous section, in favour of a more conservative replacement, which we might call ‘Don’t Do Unless Unavoidable.’ Quine’s substitutional approach has attracted voluminous disapproval, which Matthew McKeon’s sure-footed chapter does much to answer. Apart from its employment in The Logic of Fiction, I am one of those who have long thought that there is more to the substitution thesis than meets the eye. Thanks to McKeon, we may now rest more easily in that view. Taken substitutionally or not, Quine’s consequence is transitive. So (I say) is Aristotle’s. By this I mean that Aristotle’s necessitation is transitive, even though his syllogistic consequence is not. Jonathan Strand rightly wonders whether the classical validity standard is appropriate for arguments from inconsistent premiss sets or databases. Quine’s advice is, in effect, to forbid inferences from inconsistencies. This is also Aristotle’s view rather more expressly; for nothing follows syllogistically from an inconsistent set of premisses. There is a point on which Aristotle and Strand are at one. They think that classical logic is unable to capture argumentation in a satisfactory way. This engenders
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a further point of agreement. Both think that the appropriate rules of argument (and of its corresponding inferences) can be got by constraining the classical consequence relation. Each way, argument-rules are restrictions of the truth-conditions on classical consequence. In both approaches, transitivity fails. A difference between Aristotle’s and Strand’s restrictions is that Aristotle’s are universally applied, whereas Strand’s is keyed to a particular class of arguments. In some ways, Strand’s is the smaller deviation from classical consequence; but in other respects, it is the more costly. Aristotle’s deviations are slight impediments to a smoothly complete theory, with comparatively little complexity. Yet as Strand’s own chapter clearly attests it takes quite some doing to get a whole theory right whose consequence relation is non-transitive. Here is a closer parallel that makes much the same point. Both syllogistic and relevant logic satisfy a relevance constraint (though not the same one). Neither syllogistic consequence nor relevant consequence is transitive. Aristotle was nevertheless able to marshall his syllogisms in ways that syllogisity is recognizable in a finite number of mechanical moves. We have in this at least a close approximation to decidability in the modern sense. But, as we have come to appreciate, even basic systems of relevant consequence are, in contrast to their classical counterparts, undecidable. True, there are instances in which decidability is achievable, but is so at mammoth computational cost. For some logicians, there is sometimes a way out of this problem. One is to sharpen the divide between consequence and inference and to abandon the idea that inference rules can be generated by tampering with the consequence relation. Another is to push the contrast between consequence and conditionality, in particular by abandoning (or anyhow qualifying) the presumption that an invariable truth-condition on conditional utterance is that its consequent be deducible from its antecedent. In like manner, some people propose that modal analyses of counterfactuality, like modal logic itself, be hived off from logic proper and treated as a part of linguistics. Whatever we might think of these strategic proposals, the fact remains that the conditional idiom is sometimes non-transitive. My point is only that this is a fact of some consequence, which requires adjustment to other consequential facts. The transitivity of consequence is an issue that plays centrally in the logic of vagueness, as David DeVidi reminds us in his valuable chapter. DeVidi’s mission is to disarm arguments that appear to show the
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susceptibility of all vague predicates to the sorites paradox. He does so in the face of Peter Unger’s insistence that such arguments shouldn’t be disarmed, not even when they are extended in ways that produce overt contradictions. This was much my line in ‘The Unassailability of Heraclitean Logic,’ and, whether in Unger’s way or my own, gives us a lovely (and extreme) example of Philosophy’s Most Difficult Problem. This is the problem of finding a non-question-begging adjudication of an argument which some regard as a reductio of its implying premisses and others see as a sound demonstration of a proposition hitherto considered surprising or even absurd. In ‘The Dialectical Unassailability of Heraclitean Logic’3 I make the dialectical point that the Heraclitean’s argument cannot be overturned short of begging the question. DeVidi is attracted to less dramatic remedies, and who can blame him? He gives up on the classical underpinnings of sorites reasoning and, following Putnam, seeks the safer harbour of intuitionistic logic in which the standard sorites proofs do indeed fail. He does this in ways designed to accommodate Crispin Wright’s well-known criticism of Putnam. One can only admire the adroitness of DeVidi’s exploitation of intuitionistic logic. But it all turns on how to handle quantificational premisses in sorites proofs. Since quantificational premisses aren’t needed to give the Heraclitean his head, DeVidi is left with harsher options. He must give up the transitivity of consequence or the validity of modus ponens or modus tollens. As we saw, the issues joined by Jacquette, Dive, Rodych, and McKeon touch on the appropriateness of model-theoretic approaches to semantics. In a striking turn-about, Ray Jennings goes them one better. He challenges the entrenched practice of trying to understand linguistic phenomena by constructing semantic theories for them, rather than by producing explanatory theories that emphasize the physical and causal significance of language. He thus forces the question, not whether model theory suffices for semantics, but rather whether semantics itself suffices for the elucidation of linguistic phenomena. Given that his own answer is in the negative, Jennings puts pressure on a particular variant of the residue problem. The residue problem is the problem of how to interpret those aspects of an idea that don’t make their way into a formalization of it. In such a case, there are two main options. One is to fault the formalization for its incompleteness. The other is to fault the residue for not being ‘important’ enough to find a place in the formal model. This, we said earlier, was a near-instance of Philosophy’s Most
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Difficult Problem. In the present case, the point that Jennings presses against the standard semantic treatment of (in particular) the connectives is a case that calls into question the adequacy of the standard formalization protocols of the usual run of logics, both classical and otherwise more or less mainstream. Jennings is saying, in effect, that the solution to the residue problem is to find fault with these formalizations. In The Death of Argument,4 I tried to show how seriously compromised such formalizations are. But neither there nor anywhere else that I know of is the case made with such depth and such ingenuity as by Jennings. This is not to say that there is no answer to this criticism. We noted earlier that sometimes a formalism that significantly distorts its analysandum can be defended on grounds that what the formalism manages to capture of it is sufficient for the formalizer’s theoretical purposes. Such, certainly, was Quine’s position in support of first-order distortions of actual English practice. It is not a wholly easy point to overturn, especially since Quine is happy to accommodate informally any subpar idiom should it prove useful to do so, even in the thinking up of the very theories in which they are allowed no place. Lingering nearby is the Heuristic Fallacy, which Quine is shrewd enough not to commit. The Heuristic Fallacy is, just so, the mistake of thinking that beliefs that it was necessary to have in thinking a theory up must be formally sanctioned by the theory itself. A case in point are the heavily intensional meta-languages required to properly set up strictly extensional theories. What this means is that Jennings could be (and I think is) right in saying that there are correct and important natural-language arguments that are wrecked by the truth-conditional construal of their embedded connectives. Such arguments would be left in the residue of standard formalizations. Such arguments would show the inadequacy of the formalization in question. For one thing, there would be good arguments that the formal model couldn’t recognize or express. But it was always thus with formal models. Formalizations distort what they are formalizations of; they are always empirically or descriptively deficient in some respect or other. But the question remains: Isn’t the regimented idiom adequate for its intended purposes? That, as is said, is another question, and an important one. notes 1 John Woods, Aristotle’s Earlier Logic (Oxford: Hermes Science Publications, 2001).
Respondeo 327 2 George Boger contests the received view in his chapter ‘Aristotle’s Later Logic,’ in Handbook of the History of Logic, vol. 1: Greek, Indian, and Arabic Logic, John Woods and Dov M. Gabbay, eds. (Amsterdam: North-Holland, 2003). 3 In Logic Journal of IGPL, forthcoming. 4 John Woods, The Death of Argument: Fallacies in Agent-Based Reasoning (Dordrecht: Kluwer, 2004).
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Part IV Reasoning
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18 Arguing from Authority LESLIE BURKHOLDER
There is wide (but not universal) agreement across many disciplines about how correct or good reasoning or arguing from medical and other diagnostic test results (such as lie detectors) should proceed. According to this view, when reasoning or arguing from a medical or other diagnostic test result is good or correct, it is so because it is deductively sound reasoning using probabilities. In particular, it is deductively sound reasoning using Bayes’ theorem to calculate a posterior probability based on the new information provided by the test result together with independent, old information available prior to the test. Here is a typical example and its analysis. Both the example and the analysis are borrowed (with some additions) from a popular textbook by John Woods and his colleagues.1 Pregnancy Example 1 Typical pregnancy tests, whether used at home or administered in a doctor’s office, work by detecting elevated levels of a hormone. The hormone is produced by the placenta during pregnancy. Women can test positive for various reasons but not really be pregnant. Sometimes the test comes out negative when a woman really is pregnant. Renée and Tom have been using a contraceptive that is 90 per cent effective. Suppose they have recently had sex during Renée’s fertile period and Renée has obtained a positive test result with a home pregnancy test. The test is said to detect 85 per cent of all pregnancies and have a false positive rate of 5 per cent. What should Tom and Renée think about the possibility that Renée is pregnant? If Tom and Renée are to correctly use the information about the preg-
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nancy test result to get an answer to this question then they need to combine it with the information about the effectiveness of the contraceptive according to Bayes’ theorem. In detail, their reasoning can be represented or analysed as follows: Pregnancy Example 1 Analysis Premiss 1: The prior probability that Renée is pregnant based on the contraceptive success rate with the contraceptive used is prob(pregnant/contraceptive) = 0.10. Premiss 2: The true positive rate for the home pregnancy test is prob(test_‘pregnant+’/pregnant) = 0.85 and the false positive rate for the test is prob(test_‘pregnant+’/~pregnant) = 0.05. Premiss 3: Renée tests positive on the home pregnancy test. Premiss 4: What Tom and Renée want is the posterior probability that Renee is pregnant based on the test result and the use of the contraceptive. This is the value of prob(pregnant/[test_‘pregnant+’ & contraceptive]). Premiss 5: By Bayes’s theorem, prob(pregnant/[test_‘pregnant+’ & contraceptive]) = [prob(pregnant/contraceptive)*prob(test_‘pregnant+’/pregnant)]/[[prob(pregnant/contraceptive)*prob(test_‘pregnant+’/pregnant)]+[prob(~pregnant/ contraceptive)*prob(test_‘pregnant+’/~pregnant)]] = [0.10*0.85]/ [[0.10*0.85]+[0.90*0.05]] | 0.65 Conclusion: The possibility (probability) that Renée is pregnant is about 65 per cent. Of course, some examples of reasoning or arguing from a medical or other test result are more difficult or complicated than this one. It may be difficult, for instance, to find or specify the prior probabilities. Or it may be that there are multiple relevant test results, either agreeing with one another or disagreeing. Consider, as an example of the latter, the following medical case: HIV Example 1 Tests for an HIV infection typically look for antibodies to the virus. That is, they look for signs that the body of the person who might be infected is trying to fight off the virus. Two tests that work this way are the ELISA test and the Western blot test. The ELISA test is much faster and less expensive than the Western blot test. On the other hand, the Western blot test is more reliable or accurate; it has fewer
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false positives and fewer false negatives. The result is that doctors usually administer the ELISA test first, and then only if the ELISA test is positive for HIV will they then also administer the Western blot test. Following this procedure, a patient can turn out to test positive for the virus on the ELISA test and then negative on the more reliable or accurate Western blot test. In principle the opposite can happen as well. Pierce has not been practising safe sex. This puts him among a group in the population where the rate of HIV infection is 10 per cent. He has an ELISA test and the result is unfortunately positive. His doctor now administers a Western blot test, and fortunately he is negative on this test. Suppose that the true positive rate for the ELISA test is prob(ELISA_‘HIV+’/HIV_infection) = 0.85 and the false positive rate for the test is prob(ELISA_‘HIV+’/~HIV_infection) = 0.05 and that the true positive rate for the Western blot test is prob (Western_‘HIV+’/HIV_infection) = 0.997 and the false positive rate for the test is prob(Western_‘HIV+’/~HIV_infection) = 0.015. What should Pierce and his doctor think are the chances of his being infected given these test results? According to the standard view, if Pierce and his doctor are to reason correctly then they should certainly use the results of the two medical tests to answer the question at the end of the example. But they should not ignore the background information about Pierce’s unsafe sexual behaviour. They should combine all of this according to Bayes’ theorem, calculating the posterior probability prob(HIV_infection/[Western_ ‘HIV-’ & ELISA_‘HIV+’ & unsafe_sex]). So long as the results of the Western blot test are independent of the results of the ELISA test, given an HIV infection, this is not a difficult calculation. Pierce and his doctor need to apply Bayes’ theorem only twice. On the first application they will get the posterior probability of his being infected based on the ELISA test result and the prior probability that he is infected. On the second application the probability of his being infected based on the ELISA test result and his unsafe practices will now be treated as the prior and the posterior probability of his being infected based on the Western blot test result and this new prior will be calculated. If the results of the Western blot test are not conditionally independent of the results of the ELISA test then things are a little trickier, and more or different numbers are needed than supplied in the example. In this essay I won’t question this way of understanding how correct
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or good reasoning from a medical or diagnostic other test result should work. I’ll take it for granted that what Woods and his co-authors and many others say on this point is right. What I want to do is extend this analysis to another kind of reasoning, reasoning from authority or expertise. The basic idea is simple. Reasoning from a diagnostic test result is just a special case of reasoning from what an authority or expert says. The authority or expert is not a person or human being, of course, but a test. But that shouldn’t matter. Further, if correct or good reasoning from what a medical or other diagnostic test says is deductively sound reasoning using probabilities and, in particular Bayes’ theorem for revising prior or old probabilities on the basis of new information or relevant evidence, so is correct or good reasoning from what any other kind of authority or expert says. 1 To start things off we should certainly have an example or two of an argument or reasoning from authority or expertise. Here is a case including some reasoning from what an authority or expert says about an issue: Chair Example You are trying to decide whether to buy a chair you have seen at a used furniture store. When you first saw the chair you thought it was a valuable antique. But it is pretty clear that the store owner doesn’t think so. He has priced it at $5. Since you know that the store owner is an expert about antique furniture, you wonder whether your first thought can possibly be correct. After all, if he thought it was an antique he would hardly have priced the chair at $5. Suppose when you first saw the chair you were 80 per cent sure that it was an antique. As for the store owner’s reliability or accuracy, if it is really a valuable antique then there is only 1 chance in 10 that he will fail to spot it, and if it is junk then there is a very high 98 per cent probability that he will recognize that it is junk. What should you think now about whether the chair is a valuable antique or just junk? If an answer to the question at the end of the example is to be the result of correct or good or strong reasoning from authority or expertise on your part, how should it proceed? On my proposal, to reason correctly, you should be using Bayes’s theorem to determine what you should
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now think about the chair based on both the expert’s opinion and your priors, that is, your own independent opinion or guess. The general format or schema for this reasoning will be this: Schema 1 Premiss 1: The prior probability of q given all information available before or independently of the expert’s opinion is prob(q/prior_ information) and of ~q is prob(~q/prior_information) = 1-prob(q/ prior_ information). Premiss 2: The true positive rate for what the expert or authority says about q is prob(expert_‘q+’/q) and the false positive rate for what the expert or authority says about ~q is prob(expert_‘q+’/~q). Premiss 3: The expert or authority says ‘q+’ or ‘q–’ Premiss 4: What is wanted is the posterior or post-expert probability of q based on what the expert or authority says and the prior or pre-expert information. This is prob(q/[expert_‘q+’ & prior_ information]) or prob(q/[expert_‘q–’ & prior_information]) depending on what the expert says. Premiss 5: By Bayes’s theorem, prob(q/[expert_‘q+’ & prior_information]) = [prob(q/prior_information)*prob(expert_‘q+’/q)]/ [[prob(q/prior_information)*prob(expert_‘q+’/q)]+[prob(~q/ prior_information)*prob(expert_‘q+’/~q)]] and prob(q/ [expert_‘q–’ & prior_information]) = [prob(q/prior_information)* prob(expert_‘q–’/q)]/[[prob(q/prior_information)*prob(expert_ ‘q–’/q)]+[prob(~q/prior_information)*prob(expert_‘q–’/~q)]] Conclusion: The chance or probability wanted is ___ (from (4)) depending on whether the expert says ‘q+’ or ‘q–’. So your reasoning in the chair example, when it is correct, should be representable as Chair Example Analysis Premiss 1: The prior probability that the chair is an antique using your own abilities to recognize one is prob(antique/prior_ information) = 0.80. Premiss 2: The true positive rate for the store owner’s ability to correctly and accurately identify antiques is prob(owner_‘antique+’/ antique) = 0.90 and his false positive rate is prob(owner_‘antique+’/~antique) = 0.02. Premiss 3: The store owner says that the chair isn’t an antique.
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Premiss 4: What you want is the posterior probability prob(antique/ [owner_‘antique-’ & prior_information]). Premiss 5: By Bayes’s theorem, prob(antique/[owner_‘antique-’ & prior-information]) = [prob(antique/prior_information)* prob(owner_‘antique-‘/antique)]/ [[prob(antique/prior_information)*prob(owner_‘antique-’/antique)]+[prob(~antique/ prior_information)*prob(owner_‘antique-’/~antique)]]= [0.80*0.10]/[[0.80*0.10]+[0.20*0.98]] | 0.30 Conclusion: The chance the chair is a real antique is about 30 per cent. Now this particular example certainly looks like it is well analysed by my schema. It certainly looks like an example where the correct conclusion is the result of deductively sound reasoning using Bayes’ theorem to calculate a posterior probability based on the new information provided by an expert’s opinion together with independent information available prior to or independent of that opinion or testimony. But the example may seem a little unusual, not a standard or ordinary textbook example of reasoning from authority or expertise. So perhaps not everyone will be convinced that this is the right way to analyse correct or good or strong reasoning from authority or expertise in general. First, the standard or paradigm textbook examples of reasoning or arguing from authority or expertise typically do not mention or include prior or pre-expert testimony probabilities. Second, information about the accuracy or reliability of the expert or authority is rarely formulated as the kind of conditional probability needed to do Bayes’ theorem calculations. Maybe the correct conclusion in reasoning from the testimony of an expert can be arrived at using Bayes’ theorem to calculate a posterior probability if these elements are present. But they typically aren’t. So although this particular example can be well analysed by my schema, most examples of correct reasoning from authority or expertise cannot be. So here is a second and perhaps more standard or paradigmatic example of arguing from expertise or authority: De Bono Example Edward de Bono responded recently in the Globe and Mail to a criticism of his work in an earlier article in the newspaper by Jan Wong. Apparently Wong had written that de Bono was incorrect in thinking that bathwater goes down the drain in a different direction in the north and south hemispheres. De Bono responded: ‘[T]he chief scientist of Australia (who should know) tells me that
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the University of Melbourne carried out serious experiments and shows that it does rotate differently in the two hemispheres. [So bathwater drains in a different direction in the north and south hemispheres.]’2 My claim must be that this more familiar kind of example, at least if it is correct or good reasoning from authority, should be able to be analysed using Schema 1. Here is the analysis: De Bono Example Analysis Premiss 1: The prior probability that bathwater drains in a different direction in the north and south hemispheres is prob(different/ prior_information) = ?. Premiss 2: The true positive rate for the chief scientist’s testimony concerning the direction of rotation of draining bathwater is prob(chief_scientist_‘different+’/different) = ? and his false positive rate is prob(chief_scientist_‘different+’/~different) = ?. Premiss 3: The chief scientist of Australia says bathwater goes down the drain in a different direction in the north and south hemispheres. Premiss 4: What is wanted by de Bono is the posterior probability prob(different/[chief_scientist_‘different+’ & prior_information). Premiss 5: By Bayes’s theorem, prob(different/[chief_scientist_‘different+’ & prior_information]) = [prob(different/prior_information)*prob(chief_scientist_’different+’/different)] /[[prob (different/prior_information)*prob(chief_scientist_’different+’/ different)]+[prob(~different/prior_information)*prob(chief_ scientist_‘different-’/~different)]] = ?. Conclusion: The probability that bathwater drains in a different direction in the north and south hemispheres is ?. We are told nothing explicitly in this example about the value of the prior probability that bathwater drains in a different direction in the north and south. And we are not given much information about the accuracy or reliability of the expert, the chief scientist of Australia. This means that if this is an example of correct or good reasoning from authority there seem to be a lot of gaps in the premisses of the reasoning and perhaps in the conclusion as well. On the other hand, I think these gaps can be filled. I assume that the prior probability that bathwater drains in a different direction is at least not 0. It might be very small (e.g., 0.001 or smaller) or it might be that de
338 Leslie Burkholder TABLE 18.1 De Bono example analysis Prior prob prob (d/prior)
True positive prob (‘d+’/d)
False positive prob (‘d+’/–d)
Posterior prob prob (d/[‘d+’&prior])
0.00001 0.00001 0.00001 0.001 0.001 0.001 0.1 0.1 0.1 0.5 0.5 0.5
0.9 0.99 1 0.9 0.99 1 0.9 0.99 1 0.9 0.99 1
0.1 0.01 0 0.1 0.01 0 0.1 0.01 0 0.1 0.01 0
0.00 0.00 1.00 0.01 0.09 1.00 0.50 0.92 1.00 0.90 0.99 1.00
Bono thinks it should have a value reflecting complete ignorance of any information about the matter, that is, 0.5. And there is what de Bono says about the accuracy or reliability of the expert, the chief scientist of Australia, on matters like this one. According to de Bono he is someone ‘who should know’ about this kind of matter. I assume this means that prob(chief_scientist_‘different+’/different) is, for example, at least 0.90 and perhaps even nearly 1 and that prob(chief_scientist_‘different+’/ ~different) is, for example, no more than 0.10 and perhaps even as little as 0. The range of posterior probability values these considerations give is in the last column of table 18.1. De Bono clearly thinks that based on what the expert says it cannot be doubted that bathwater drains in a different direction in the north and south. That is, according to my analysis, he thinks that prob(different/[chief_scientist_‘different+’ & prior_information]) | 1 is proved by the reasoning. While not every row in table 1 has that value for the posterior probability, some easily do. So my conclusion is that the gaps are not real and they can be filled in quite sensible ways, ways dictated by the proposed analysis of correct or good reasoning from authority. 2 Of course, my proposal is not the only idea about how correct or good or strong reasoning from authority or expertise should proceed.
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Woods and his colleagues in various places do us a service by reviewing many ideas for understanding how correct or good or strong reasoning from authority or expertise works.3 For instance, one idea they review (but don’t advocate) is that correct or strong reasoning from authority has really the following format: Schema 2 Premiss 1: Everything the expert or authority says concerning every topic is true. Premiss 2: The expert or authority says ‘q’. Conclusion: q But this won’t work, as they remind us. It would mean that any answer you produced based on the expert’s opinion in your thinking about the chair in the Chair Example would be unsoundly reasoned. It is a given that the store owner is not perfect even at telling antiques from junk, so the first premiss isn’t true if the reasoning in that example is correct reasoning from authority and is analysed following this schema. This case is surely not an exception. Experts aren’t infallible or omniscient, so all reasoning from authority or expertise will be deductively unsound on this analysis. Moreover, if my general hypothesis that reasoning from medical and other diagnostic test results is just a special instance of reasoning from authority is correct, since medical and other diagnostic (for example, lie detector) tests aren’t perfect, all of these examples of reasoning would be deductively unsound on this analysis too. Another idea Woods and his colleagues review (and again do not advocate) is the more sensible one that correct or good or strong reasoning from authority or expertise has the following form: Schema 3 Premiss 1: Almost everything the expert or authority says concerning topic s is true. Premiss 2: The expert or authority says ‘q’. Conclusion: q Again, Woods and his colleagues are quick to point out the problems. The difficulty with this analysis is not that it makes correct or good or strong reasoning from authority or expertise impossible as does Schema 2. The difficulty is that it is incomplete. In the terminology of Bayes’s theorem applications, it does not tell us what to do with our priors. Consider again the Chair Example. In that case, you initially thought
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that the chair in the store window was a valuable antique, or at least you thought there was a pretty good chance that it was. Then you learned what the store owner, an antiques expert, thought of it. He thought it was junk. On Schema 3, what should you now think? One possibility is that Schema 3 means that your initial inclination to think that the chair in the store window is a valuable antique should make no difference to the reasoning; it should be ignored or thrown away as irrelevant. The only relevant fact is what the expert says. Thus Chair Example Analysis Premiss 1: Almost everything the store owner says about furniture being an antique or not is true. Premiss 2: The store owner said the chair is junk, i.e., not an antique. Conclusion: The chair is junk. and Chair Example Analysis Premiss 1: Almost everything the store owner says about furniture being an antique or not is true. Premiss 2: The store owner said the chair is junk, i.e., not an antique. Premiss 3: You said the chair is a valuable antique. Conclusion: The chair is junk. are equally correct or strong (or incorrect and weak). The addition of premiss 3 makes no logical difference. The other possibility is that Schema 3 does not tell us what to do when we have extra information like that stated in premiss 3 above. It only tells us that if all that we have is what an expert or authority has to say about a subject and no priors, to again use terminology from Bayes’s theorem, then correct reasoning based on that should proceed as outlined by the schema. In the Chair Example it seems that no obvious error results from ignoring your priors and throwing away everything but what the expert says. But it is pretty easy to show that in general this deliberate neglect of priors or other information is a recipe for disaster. Here’s an example: Pregnancy Example 2 Typical pregnancy tests work by detecting elevated levels of a hormone, as explained in Pregnancy Example 1. The hormone is human chorionic gonadotropin (hCG). It’s a hormone produced by the placenta during pregnancy, as explained in the example. Sometimes
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women take this hormone as part of a fertility treatment. Women taking this treatment could test positive but not really be pregnant. Imagine that John Woods, a man, has been taking this hormone (under medical supervision). John Woods submits to a pregnancy test in a doctor’s office and unsurprisingly tests positive. The doctor’s judgement in this office always tracks the test result. So he says that John Woods is pregnant. Should we conclude that John Woods is pregnant? Our first alternative for Schema 3 says that Pregnancy Example 2 Analysis Premiss 1: Almost everything the doctor says concerning a patient’s pregnancy is true. Premiss 2: The doctor said that John Woods is pregnant Conclusion: John Woods is pregnant. and Pregnancy Example 2 Analysis Premiss 1: Almost everything the doctor says concerning a patient’s pregnancy is true. Premiss 2: The doctor said that John Woods is pregnant. Premiss 3: John Woods is a man. Conclusion: John Woods is pregnant. are equivalent because all that matters is what the expert says. They are equally correct or strong (or incorrect and weak), and the addition of premiss 3 makes no logical difference. But it obviously does. The correct conclusion, when that information is added, is that John Woods isn’t pregnant despite the expert’s testimony. Our second alternative for Schema 3 doesn’t get things wrong in this case. On the other hand, as Woods and his colleagues suggest with other examples, it doesn’t tell us what would be the right thing to conclude either. It just leaves unresolved what to believe in examples of reasoning from authority or expertise when prior independent information like that in premiss 3 is added. 3 As well as reviewing faulty analyses of correct or good or strong reasoning from authority or expertise, Woods and his colleagues tell us
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how to properly understand good or strong or correct arguments from authority or expertise. Of course, the analysis is not the one I proposed in an earlier section of this paper using Bayes’s theorem to calculate the posterior probabilities in the conclusion of the reasoning. They tell us that it requires the use of what Nicholas Rescher calls ‘plausibility indexing’ or what Woods and his colleagues call ‘plausibility screening.’4 The simplest format or schema for arguments from authority is this: Schema 4 Premiss 1: The expert or authority says ‘q’ about topic s. Premiss 2: {q} is a maximally consistent subset of available claims about s. Premiss 3: The authority or expert has reliability rating Rx about subject s. Subconclusion: plaus(q) = Rx Premiss 4: Rx is high. Conclusion: q In this schema, ‘plaus(q) = R’ is not just another way of stating an absolute or conditional probability; it is not really ‘prob(q) = R’ or ‘prob (q/_) = R’ in disguise. It states a plausibility value or index for q, and plausibility values behave differently than probabilities. In addition, ‘The authority or expert has reliability rating Rx about subject s’ is not just a disguised way of saying ‘R per cent of everything the authority or expert says concerning subject s is true,’ nor is it a conditional probability of the kind used in Bayes’s theorem calculations providing the true or false positive rate for the expert’s testimony on subject s. Now this schema won’t help much with examples like the Chair Example or Pregnancy Example 2. In these cases we have more than one source of information to consider. There’s not only what the expert or authority has to say but also other information which seems to be relevant. Recall that it was this other information which created a problem for Schema 3. So how do Woods and his colleagues propose this kind of case be analysed? Here is a slightly more complicated schema to do this: Schema 5 Premiss 1: The authority or expert X says ‘q+’ about topic s. The Information source Y (perhaps another authority or expert or per-
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haps another kind of information source) says ‘q–‘ about topic s. Subconclusion: plaus(q) = Rx and plaus(~q) = Ry. Premiss 2: {q} is a maximally consistent subset of available claims about s. {~q} is a maximally consistent subset of available claims about s. There are no others. Premiss 3: Authority X has reliability rating Rx about subject s. Information source Y has reliability rating Ry about subject s. Subconclusion: plaus(q) = Rx and plaus(~q) = Ry. Premiss 4: Rx is high and Rx > Ry. Conclusion: q Now let’s see what this tells us about how to proceed in examples like the Chair Example or the Woods pregnancy case (Pregnancy Example 2). According to this schema the correct way for you to reason about that chair you thought might well be an antique (in the Chair Example) is this: Premiss 1: The store owner said that the chair is junk. You thought it was a valuable antique. Premiss 2: {~antique} is a maximally consistent subset of available claims about the value of the chair. {antique} is a maximally consistent subset of available claims about the value of the chair. There are no others. Premiss 3: The store owner has reliability rating Rx when it comes to discriminating antiques from chairs. You have a reliability rating Ry on this same subject. Subconclusion: plaus(~antique) = Rx and plaus(antique) = Ry. Premiss 4: Rx is high and Rx > Ry. Conclusion: The chair is junk, not an antique. The correct way for us to reason about Woods’s possible pregnancy is: Premiss 1: The fact that Woods is a man indicates that he isn’t pregnant. The doctor administering the pregancy test said he is. Premiss 2: {~pregnant} is a maximally consistent subset of available claims about Woods. {pregnant} is a maximally consistent subset of available claims about Woods. There are no others. Premiss 3: The fact that Woods is a man has reliability rating Rx when it comes to his being pregnant. The doctor administering the
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pregancy test has a reliability rating Ry on this same subject. Subconclusion: plaus(~pregnant) = Rx and plaus(pregnant) = Ry. Premiss 4: Rx is high and Rx > Ry. Conclusion: Woods isn’t pregnant. But I think there are some problems. Some of these problems concern the arithmetic of plausibilities, at least as this is explained or defined by Woods and his colleagues following Rescher. Some of the problems concern the particular use of plausibility screening or indexing to analyse correct or strong reasoning from authority. Plausibilities are, of course, supposed to be much different than probabilities. In particular, for example, since prob(A & B) = prob(A) * prob(B) when A and B are independent and the maximum value a probability can have is 1, it follows that prob(A & B) d min {prob(A), prob(B)} Plausibilities are supposed to work differently, however: plaus(A & B) t min {plaus(A), plaus(B)} But it isn’t obvious that they should work this way. Consider the plausibility of (fair coin A turns up heads & fair coin B turns up heads & fair coin C turns up heads & ...). This joint plausibility should be lower than the plausibility of each conjunct. But the arithmetic for plausibilities says otherwise. It is easy to multiply examples like this: The plausibility of my left eye being brown is fairly high. The plausibility of my right eye being blue is also fairly high. But the joint plausibility is low. The lesson here is not that there is a particular problem with this or that rule for the arithmetic of plausibilities. It is that plausibilities don’t seem to have a satisfactorily defined arithmetic, and if that is so then it doesn’t seem like we are further ahead to use plausibilities to analyse what makes some instances of reasoning from authority correct or strong or good and others weak or bad. This first point concerned the arithmetic of plausibilities in general. It had nothing to do with the use of plausibility indexing or screening to analyse reasoning or arguments from authority. What about problems associated in particular with that? Let me mention two. One of these might be repaired but I don’t see how the other can be. We all know about the Lottery Paradox. We know that the lesson of
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this paradox is that the inference from prob(I win the lottery) = very low, nearly 0 to I don’t win the lottery is a fallacious or unsound inference. It is fallacious or unsound because if it were correct or sound or valid to reason this way then the logical thing to conclude would be that no one would win the lottery. Notice that the offered analysis for arguments from authority commits a parallel mistake in reasoning involving plausibilities. In every instance the analysis ends with plaus(q) = high. So, q or plaus(q) > plaus(~q). So, q But from plaus(I win the lottery) = very low, nearly 0 to I don’t win the lottery seems a bad or fallacious step of reasoning, just as bad as the parallel for probabilities. This problem might be repaired. According to my analysis, arguments or reasoning from authority end with a posterior probability. No further unsound or fallacious inference is made from this to a non-probability claim. Woods and his colleagues obviously might claim something parallel for a slightly revised version of their analysis. Correct or sound or strong arguments or reasoning from authority ends with a statement of plausibility and no further nonplausibility claim is to be inferred from this. I don’t see an easy fix for the final problem I’ll mention for the analysis of correct or sound or strong arguments or reasoning from author-
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ity favoured by Woods and his colleagues. That problem can easily be illustrated with two examples: HIV Example 2 The ELISA test for an HIV infection looks for antibodies to the virus, as explained in HIV Example 1. The test is very reliable or accurate but it can produce false positives and false negatives. Robert is in a very high-risk group for HIV infection. Suppose the rate at which members of this population are infection-free is 25 per cent. (In some African countries, this is the infection rate in the general population.) He has an ELISA test and the result is unfortunately positive. Suppose Robert’s doctor’s judgement always tracks the test result. So he says that Robert is infected. The doctor is exactly as reliable or accurate as the ELISA test. Should we conclude that it is highly plausible that Robert is infected? HIV Example 3 Sara is in a lower-risk group for HIV infection than Robert. Suppose the rate of infection for members in her subpopulation is 1 per cent. She has an ELISA test and the result is unfortunately positive. Suppose her doctor’s judgement always tracks the test result. So he says that Sara is infected. The doctor is exactly as reliable or accurate as the ELISA test. Should we conclude that it is highly plausible that she is infected? On the analysis proposed by Woods and his colleagues these two examples will end with the same result, namely, that it is highly plausible that the patient is infected. There is nothing wrong with that. What seems incorrect is that they will end with the conclusion that it is just as plausible that Sara is infected as Robert is. They will do that because for each the plausibility of being infected is greater than the plausibility of not being infected, given the reliability or accuracy of the ELISA test (or that of the doctors whose judgements just mirror the test results). In effect, the prior independent information about their pre-test risk of being infected is in a way thrown away or ignored. I think it is easy to see that it is wrong to conclude that it is just as plausible that Sara is infected as Robert is if we ask that plausibilities be turned into or related to something probabilities often are, betting rates. If I were to ask you to place bets on these two being infected or not, would you be willing to place the same size wager on Robert being free of infection as Sara? I think not.
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4 I’ve offered a positive analysis for arguments or reasoning from authority. According to this analysis reasoning or arguing from authority is good or correct or strong when it is deductively sound reasoning using probabilities. In particular, it is good or correct or strong when it is deductively sound reasoning using Bayes’s theorem to calculate a posterior probability based on the new information provided by the authority or expert together with independent old information available prior to the authority or expert’s testimony. This analysis isn’t the same as that offered by Woods and his colleagues in various places; it improves on that analysis. But surely, even if the analysis they offered isn’t perfect, they are to be thanked for producing the detailed studies of this kind of reasoning that prompted this one. notes 1 John Woods, Andrew Irvine, and Douglas Walton, Argument: Critical Thinking, Logic and the Fallacies (Toronto: Prentice Hall, 2000). 2 Edward de Bono, ‘Edward de Bono Responds to Jan Wong’s Column,’ Globe and Mail, 8 September 2001. 3 Woods, Irvine, and Walton, Argument: Critical Thinking, Logic and the Fallacies; John Woods and Douglas Walton, Fallacies: Selected Papers 1972–1982 (Providence, RI: Foris, 1989). See also Merrilee H. Salmon, Introduction to Logic and Critical Thinking, 3rd ed. (Orlando, FL: Harcourt Brace, 1995), and Howard Kahane and Nancy Cavender, Logic and Contemporary Rhetoric, 9th ed. (Belmont, CA: Wadsworth, 2002), for other thorough treatments. 4 Nicholas Rescher, Plausible Reasoning (Assen: Van Gorcum, 1976); Woods, Irvine, and Walton, Argument: Critical Thinking, Logic and the Fallacies.
19 Premiss Acceptability and Truth JAMES B. FREEMAN
1. Truth versus Acceptability As is well known, formal deductive logic proposes soundness as the criterion of argument cogency, and to be sound the premisses of an argument must be true.1 A hallmark of the informal logic movement has been replacing the formal logic criterion with premiss acceptability, relevance, and ground adequacy. Acceptability is not the same as truth. Arguing that being true is neither a necessary nor a sufficient condition for acceptability is straightforward. It is not necessary because the preponderance of evidence at one’s disposal might favour some statement which is, in fact, false. In such circumstances, it seems that statement would be acceptable, even though false. It is not sufficient, for a statement may in fact be true, yet one might possess no evidence for it. Indeed, the preponderance of evidence might even be against it. In such cases, the statement would not be acceptable, even though true. Johnson and Blair2 cite Problematic Premiss as a basic fallacy. A premiss is problematic just in case it is not defended but should have been. There are situations where an arguer is exempt from defending a particular premiss, for example where the premiss is self-evidently true, a matter of common knowledge, or the arguer has expert credentials to put forward the statement.3 But if no excepting condition holds for an undefended premiss, it is problematic and the argument flawed. If the opposite of being problematic is being acceptable, then to be cogent the premisses of an argument must be acceptable. Clearly, acceptability does not amount to truth under this understanding. If common knowledge means commonly or widely held beliefs within a certain group at a certain time, then presumably common knowledge can change over
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time and can at some time include claims which are actually false. A recognized expert can still render a false expert opinion. Hence, premisses may be acceptable which are not true. On the other hand, a statement may be true but not self-evidently true or a matter of common knowledge or vouched for by suitably qualified experts. Govier argues forcefully that acceptability is the appropriate criterion if one wants ‘standards of argument appraisal that are intended to give real practical guidance.’4 For Govier, the truth criterion sets an impossibly high standard, for practical purposes. If we stipulate that people should be convinced only by those arguments that have true premisses, we would in effect be stipulating that in many times and places, people should not be convinced by arguments at all. In fact, it would be a very tough epistemological task to show that we ourselves should often be convinced by arguments on this kind of model.5
I believe that Govier here is assuming that if truth is a criterion of premiss adequacy, it is not sufficient that the premisses of an argument be true; they must be recognized as true by the person to whom the argument is addressed for that person to be properly convinced by the argument. Govier makes this point explicitly: ‘It is ... incorrect in fact, to see the truth of an argument’s premisses as a sufficient condition for their epistemic adequacy. What counts is not whether they are true, but whether the audience has good reasons to believe them true.’6 Given this understanding, appreciating Govier’s point is straightforward. Consider some historical report for which there is significant evidence. Does this evidence guarantee that the report is true? Can we, given our knowledge, say that the report is true? If not, then the report cannot be a legitimate premiss on the truth criterion. By contrast, an audience can in principle recognize when premisses are acceptable to it. However, ‘Acceptability is not acceptance.’7 It is a normative concept, which should be explicated as a relation of justification between some claim and ‘the beliefs, knowledge, and inferential capacities of the audience’8 to which that claim is addressed. Govier is urging replacing a logical or ontological criterion of premiss adequacy with an epistemological criterion. Feldman has also urged an epistemological criterion for evaluating arguments. Indeed, he emphasizes that it is necessary to recognize that argument evaluation has an epistemological dimension. As he sees it, ‘The point of arguments, or at least the point of thinking about arguments, has something to do with
350 James B. Freeman
figuring out what to believe.’9 Good arguments ‘provide people with reason to believe their conclusions,’10 that is, a justifying reason from the point of view of the person confronted with the argument. ‘Thus, a good argument is, roughly, one with premisses that are justified and with a justified premiss/conclusion connection.’11 It is easy to construct pairs of arguments both of which are valid, one of which has all true premisses, yet neither of which justifies our accepting the conclusion, given the evidence available to us. Suppose I know that exactly one of A or B has happened, but have no evidence which one. Consider the following pair of arguments: A or B not-A ?B
A or B not-B ?A
One or the other of these arguments is sound, but neither gives me a justified belief in its conclusion.12 If the purpose of argumentation is to provide justification for holding certain beliefs, neither is a good argument.13 Soundness is not a sufficient condition for argument goodness. Arguments, rather, are good for a person at a given time relative to the evidence the person possesses at that time.14 A premiss of an argument then is adequate for that person at that time if and only if the person is justified in believing that premiss at that time. 2. Two Criteria Approaches Not all philosophers are convinced that a purely epistemic account of argument goodness is adequate or that the epistemic aspect of argument goodness should be given the pride of place it has received in Govier’s and Feldman’s accounts. Johnson15 has called for recognizing both truth and acceptability as criteria of premiss adequacy. In line with Feldman, Johnson sees the purpose of an argument as rational persuasion. A good argument achieves that purpose. He agrees that to be good, the premisses of an argument must be acceptable to the intended audience, else they will not be rationally persuaded. This means that each premiss either is defended by a good argument or is one which it is rational for the audience to accept without support.16 In either case, the premiss is acceptable to the audience. Johnson also concedes that there are some good reasons for questioning the truth requirement.17 In particular, he points out that should truth be a requirement for good
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argument, the canons of logic will not be sufficient to determine which arguments are good. Determining whether a factual premiss is true depends on the field of knowledge to which that premiss belongs. It is not a matter of logic. On the other hand, Johnson argues that there are good reasons to maintain the truth requirement, the strongest being that those who claim not to require truth for premiss adequacy are still covertly relying on the requirement. In particular, Johnson claims this happens when such theorists use terms such as ‘inconsistency, contradiction, assumption, validity’ in evaluating arguments.18 For example, to say that an argument is fallacious because its premisses are inconsistent, where a set of premisses is inconsistent if they cannot all be true together, is to rely ‘heavily on the truth-requirement ... There is an expectation that the premisses be true.’19 Johnson’s argument here, however, does not hit the mark. To say that an argument with inconsistent premisses cannot rationally persuade because to recognize inconsistency is to recognize that not all the premisses are true does not commit one to affirming that without recognizing that all the premisses of an argument are true, one cannot judge whether the argument is good. Recognizing non-truth of a premiss as a flaw, indeed in many cases as a fallacy in an argument, does not entail that non-recognizing the truth of each premiss is to recognize the argument as flawed or fallacious (or at least that one must suspend judgment as to the cogency of the argument), as affirming the truth requirement would seem to require. One may allow that acceptability and truth are related without affirming truth as a criterion of premiss adequacy. This same confusion also surfaces when Johnson argues that ‘It is natural enough for the Other to criticize an argument by claiming that a premiss is false. If the premiss is false, it would not be rational to accept it as a basis for accepting the conclusion.’20 What Johnson says here can be explicated in several ways. I understand him to be saying that should the person or more generally the audience to whom an argument is addressed know or suspect that a premiss is false, they should not accept it. But this does not entail endorsing the truth criterion. Again, to allow that recognition of falsity means it is not rational to accept a premiss does not entail that non-recognition of truth also means that accepting the premiss is not rational. Johnson also claims that those rejecting the truth requirement nonetheless covertly accept it in their meta-linguistic explication of certain concepts. For example, they explicate relevance by asking in the meta-
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language whether the truth of P entails a truth-value for Q or they explicate fallaciousness by pointing out that the truth of P does not make Q true. However, neither move commits one to saying that to be rationally persuasive, the premisses of an argument must be true. We cannot examine Johnson’s argument further here. Suffice it to say that although his case for requiring both acceptability and truth as criteria for premiss adequacy is problematic, he is one of those calling for both criteria. Allen21 distinguishes logically good arguments from cogent arguments, holding that we should not speak of good arguments simpliciter. To be logically good an argument’s premisses must support its conclusion. Allen holds that false premisses give no support to the conclusion.22 Thus, as long as the premisses of an argument are statements which are either true or false, the premisses of a logically good argument must be true.23 Acceptability of premisses will not be sufficient for their adequacy as premisses in a logically good argument, since acceptable but false premisses will fail to support their conclusions. By contrast, ‘an argument is cogent for an audience if the audience would be epistemically justified in believing the argument to be logically good.’24 Since audiences differ in what they are epistemically justified in believing, an argument which is cogent for one audience need not be cogent for another. But if an audience is epistemically justified in believing the premisses of an argument, then those premisses are acceptable for the audience. Acceptability, then, is a necessary condition for cogency: ‘The basic premisses of an argument must be acceptable if the argument is cogent for its audience.’25 Allen26 similarly argues that we should make room for both truth and acceptability in a theory of argument, distinguishing a logical conception of a good argument from an epistemological conception. His theory thus accords both premiss truth and acceptability a place in the criteria for good arguments, although they would be criteria for different senses of good argument. Allen argues that there is good reason for a theory of argument to incorporate both criteria. Agreeing with Govier, he says that ‘The social institution of argument has as its typical function or purpose the rational persuasion of a rationally critical audience,’27 which entails for Allen that the member or members of the intended rationally critical audience need to consider whether the argument is good from their point of view, and the arguer and any external assessor also need to consider whether the argument is good from the intended audience’s point of view. ‘It is precisely such a conception of argument goodness
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that an epistemological conception of good argument provides.’28 Given just the epistemological conception, however, ‘we will not have a criterion of argument goodness relative to which an argument is defective if it has false premisses.’29 But the logical conception constitutes just such a criterion. Since we want to say both that a function of argument is to rationally persuade a rationally critical audience and that having false premisses constitutes a defect in an argument, we would do well to have our theory of argument embrace both the logical and epistemological conceptions of argument goodness. 3. The Integration Problem and Goldman’s Reliabilist Solution By holding that a theory of argument should embrace both of these conceptions and thus both truth and acceptability as criteria of premiss adequacy, Allen raises a problem. Johnson refers to it as the Integration Problem: these two criteria may conflict. Logically good arguments may not be epistemologically good, and epistemologically good arguments may not be logically good. ‘A premiss may be true but not acceptable; a premiss may be false but acceptable.’30 However the problem should be defined not as how to deal with this particular tension but more broadly as how to relate these two requirements to each other. A proper answer to that question should determine whether the divergence of these criteria is actually problematic. Thus understood, the integration problem has already been addressed. Goldman31 contrasts ‘argument’ with ‘argumentation.’ ‘Argument’ he understands in the logic textbook sense, where the paradigm of a good argument is a sound argument. Argumentation is dialectical or social. One or more speakers present and defend theses to a hearer or audience. The goodness of an argument in the logical sense is not sufficient for the goodness of the corresponding argumentation. Even if a premiss is true, if one’s interlocutor in a dialectical situation has rejected it, one simply cannot reiterate that statement ‘as a premiss (even if [one] believes it justifiably).’32 Goldman however holds that the norms of good argumentation ‘are best seen as a social quest for true belief and error avoidance.’33 Indeed, ‘the core purpose of argumentation is to persuade audiences or interlocutors of truths.’34 This goal then motivates rules which specify various duties for the speaker S, concerning in particular the premisses he may put forward. Ideally, the hearer or audience H should find the asserted premisses acceptable. This is necessary for the argument to be epistemologically good from H’s perspective.
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But it is not enough to satisfy the primary goal of argumentation for S to present an argument which is epistemologically good from H’s perspective, since it is logically possible that the argument would not be good from S’s perspective. Here S would be creating justification, not transmitting it. Rather, S must assert only premisses which are believed and justifiably believed by S. How does this requirement further the goal of promoting true belief? If one adopts a reliabilist theory of justification, then if speaker S is justified in believing some proposition p as a matter of the evidence S possesses35, then ‘there is a substantial probability that this proposition is true.’36 Hence, if by presenting argumentation to H, S is transferring justification for p to H, S is justified in believing each premiss is true. But this means that the probability is substantial that each premiss is true. Understanding a premiss to be acceptable at a given time only if the speaker would be justified in believing the premiss at that time, and understanding justification in Goldman’s reliabilist sense, truth and acceptability are thus clearly integrated on Goldman’s account. Notice that this integration assumes a reliabilist theory of justification, where one’s being justified in accepting p implies that there is a substantial probability that p is true. In this way we get truth and justification together. This may mean that objections to Goldman’s reliabilism are ipso facto objections to this account of how truth and justification or acceptability are integrated. Why on Goldman’s reliability theory do justified beliefs have a high probability of being true? In (1979) he says, The justificational status of a belief is a function of the reliability of the process or processes that cause it, where (as a first approximation) reliability consists in the tendency of a process to produce beliefs that are true rather than false.37
Justified beliefs have a high probability of being true because a belief’s justification consists in its being caused by reliable processes and a process is reliable if the probability is high that the beliefs it produces are true. Objections to such reliabilism are straightforward and well known. Many would claim that one is justified in believing that p only if one had access to evidence sufficient to indicate that p was true or give one good reason to think p true and one’s belief was based on this evidence. But need one possess such evidence on a reliabilist account? Suppose you want to inquire into the truth of some claim. I present
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you with an argument for that claim. The premisses of my argument contain a number of factual claims for which you have no evidence beyond my word asserting them. Let us also suppose that these claims concern not matters about which I could give personal testimony but technical issues. But suppose also that I have no expert credentials in the areas these claims concern and you do not regard me as being an expert in these areas either. But suppose also, unknown to you, I am scrupulous over the truth of what I assert. Hence, barring minor misstatements or inaccuracies, the percentage of truths in the statements I utter is significantly high. So the probability is significantly high for any statement that p that if I assert that p, p is true. Whenever I assert that p, I have overwhelming evidence that p. Hence I possess overwhelming evidence that each of the premisses in the argument I am presenting to you is true. Suppose also that I have a strange psychological effect on you. Unique among human beings, I cause you to have complete trust in my word. No other human being has this effect on you. Let us suppose also that the premisses and conclusion of my argument are properly related and that both you and I recognize this. Now there is no question that I am personally justified in accepting the conclusion of this argument because I see that it follows from (in a suitable sense of following) the premisses, where my belief is justified by the evidence I possess for each premiss. By giving you this argument, do I transfer this justification? I think not, just because I have transferred no suitable evidence for the premisses to you. But you do accept the conclusion of the argument because you accept all the premisses and see that this conclusion follows from those premisses. And you accept the premisses because I say so. Since I am scrupulous, your taking my word is a reliable belief-generating mechanism. But it does not seem that your accepting the premisses is justified, and thus this very conception of justification is not acceptable.38 4. An Alternative Solution to the Integration Problem Can we identify an alternative concept of justification avoiding reliabilist objections through which we can address the integration problem? We have argued that the following, adapted from Alston,39 is especially appropriate for a definition of acceptability:40 S is justified in believing that p iff S’s belief that p is based on pre-
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sumptively adequate grounds from S’s point of view and S lacks sufficient overriding reason to the contrary. Let us consider an example of what constitutes a justified belief on this definition. Suppose I perceive a tree outside my office window and on this basis form a belief that there is a tree outside my office window. My perceptual experience, my being appeared to in this way, is the ground of my belief. This ground will be adequate just in case it is a reliable indicator of the truth of that belief – that is, given that ground, the probability that the belief is true will be high. This probability would ordinarily be determined by the overall reliability of my (visual) perceptual mechanism. That mechanism will be presumptively reliable just in case there is a presumption for its reliability, that is, just in case the burden of proof would be on someone to show that it is not reliable. But there is a presumption for sense perception. As Rescher puts it, ‘Theses based on observation ... are to have the benefit of doubt, a presumption of truth in their favor – they are to stand unless significant counterindications are forthcoming.’41 Hence my perceptual experience is a presumptively adequate ground of my belief that there is a tree outside my office window. Now this presumption of adequacy can be defeated. There are perceptual illusions. My perceptual mechanism may possibly malfunction. But let us suppose that I am aware of no defeaters or counter-indications in this case. Hence I am justified in believing that there is a tree outside my office window. This example illustrates what our definition of justified belief amounts to. If I am aware of grounds for a belief, if there is a presumption of reliability or trustworthiness for the source of these grounds, and if I am aware of no defeaters of this presumptive reliability, then the belief is justified. But if a premiss in an argument is thus justified from my point of view, why should I not find it acceptable? Should I believe that claim, my belief would be justified. What would be wrong in my taking that justified belief as a premiss in an argument and reasoning from that belief, should it be relevant to the conclusion I am trying to establish? Accordingly, we offer this definition of premiss acceptability: A premiss that p is acceptable for S if and only if should S believe that p, S would be justified in believing that p,42 where being justified is explicated according to our definition.
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What is the relation between truth and justification on this view? Are they so related that we may construct a solution to the integration problem? The relation seems plain. Grounds for p from S’s point of view are grounds that p is true. If these grounds were adequate, then, all things being equal, there would be a high objective probability that p is true. Lack of evidence that things were not equal would then indicate an even higher degree of probability. If S then had presumptively adequate grounds for p and no awareness of sufficient overriding conditions, S would be justified in believing that p is true. In reasoning about what to believe or do, S can take p as true, until or unless S receives sufficient evidence to the contrary. In light of this, we may straightforwardly solve the integration problem using this analysis of justification. Following Allen, we may speak of logical and epistemological criteria of premiss adequacy and indeed of argument adequacy. Truth is the logical criterion of premiss adequacy. Acceptability is the epistemological criterion, where acceptability is understood in terms of justification. Our epistemic goal is the truth, or to increase our stock of truths and avoid error. But if we possess evidence that p is true, if the source of this evidence is presumptively reliable, and if we possess no evidence to undercut that presumption, taking p as true would accord with our epistemic goal. The epistemological criterion is satisfied for S just in case S is justified in believing that the logical criterion is satisfied. In this way, truth and acceptability are related and thus integrated. It is still logically possible that ‘a premiss may be true but not acceptable, [or] a premiss may be false but acceptable,’43 the consideration that led Johnson to raise the integration problem. But in light of our discussion, is it right to say that these criteria pull in opposite directions? It would seem that the only criterion that would do any pulling would be the acceptability criterion. Surely to pull in opposite directions in this context is to pull someone in opposite directions. How could we apply the logical criterion of truth or falsehood independently of the acceptability criterion? Suppose that p is true, but S is not justified in believing that p. Is S in some conflict here? The statement p is not an acceptable premiss for S until and unless S comes to have sufficient evidence for p. But then S will be justified in believing that p and p will be acceptable for S. S will at no time be confronted with conflicting criteria of premiss adequacy. Likewise, if p is false, but S possesses adequate evidence for p, p is acceptable from S’s point of view. But will S be under a conflict here?
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S’s evidence justifies S in accepting that p. Should S become aware of evidence that p is false, that evidence would constitute reason to the contrary against p and sufficient evidence would constitute sufficient overriding reason to the contrary. But then S would no longer be justified in believing that p. That these two criteria may conflict simply reflects the fact that human beings are not omniscient. But since one is not omniscient, one will have to make do with the evidence available to one. But if S’s epistemic goal is to increase his or her stock of truths and avoid errors, and that p is based on presumptively adequate grounds from S’s point of view, and S lacks sufficient overriding reasons to the contrary, how can S be neglecting this epistemic goal by accepting p? We may develop an analogy between the logical and epistemological concepts of argument goodness and the distinction in moral theory between an act’s objective and subjective rightness, or whether an act is an objective or subjective duty. Whether an act is objectively right or an objective duty depends on certain features of the act. Philosophers may disagree on what exactly these features are. On Ross’s intuitionist analysis, for example, if by virtue of satisfying certain objective conditions an act is prima facie right and if by virtue of satisfying others it is prima facie wrong, yet the right-making conditions outweigh or trump the wrong-making conditions, the act is right or a duty simpliciter. This is an objective property of the act. In a genuinely complex conflict of prima facie duties, one involving vexed issues, however, I may be honestly mistaken about what is my objective duty. Am I guilty necessarily or blameworthy if I perform an act which is against my objective duty? We would say no. As Plantinga points out, we ‘think that someone who has done no more than what she nonculpably thinks duty permits or requires, is not culpable or guilty in doing what she does, even if we think that what she has done is wrong.’44 The act, although objectively wrong, is subjectively right or permissible. Even an act which is objectively right may be subjectively wrong. If upon conscientiously considering the morally relevant features of a situation I conclude that a certain act is wrong but yet I commit that act anyway, my act is subjectively wrong, even though the action is objectively right. Notice then that it is possible that the objective and subjective criteria of an act’s rightness may diverge in a given case. But this is not thought to raise an integration problem for the objective and subjective rightness of an act or its being an objective or subjective duty. Analogously, we may speak of the objective and subjective correct-
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ness of arguments. An argument is objectively correct if it is logically correct. An argument is subjectively correct for S if and only if it is epistemically correct for S – that is, all the premisses are acceptable to S and S is justified in believing that the argument is valid or inductively strong. Are ‘objective’ and ‘subjective’ appropriate adjectives here? Whether the premisses of an argument are true and whether the argument is deductively valid or inductively strong are objective features of the argument. What we call the objective rightness of an argument is thus an objective feature of arguments. On our analysis of acceptability, for S to be justified in believing a premiss, S must be aware of presumptively adequate grounds on which the premiss is based and not aware of sufficient overriding reasons to the contrary. Whether or not S is justified in believing a premiss p (or that a set of premisses P is properly connected to a conclusion c) is a matter of S’s awareness. Recognizing premiss acceptability and connection adequacy is a matter of internal awareness and in this sense is subjective. As an act is subjectively right for S, just in case on conscientiously considering the evidence S judges the act to be objectively right, so an argument is subjectively correct for S just in case in light of the evidence S judges the argument to be objectively correct. But if there is no anomaly or paradox in distinguishing the objective and subjective rightness of an act, so there should be no anomaly in distinguishing the logical and epistemological senses of argument correctness. notes 1 By the term ‘argument’ in this discussion, we understand a set of premisses put forward to support one and only one conclusion. Hence we are not concerned here with arguments involving serial or divergent structure. 2 Ralph Johnson and J. Anthony Blair, Logical Self-Defense (Toronto: McGrawHill Ryerson, 1977), 22–9. 3 Compare ibid., 23–6. 4 Trudy Govier, Problems in Argument Analysis and Evaluation (Dordrecht, Holland/Providence, RI: Foris Publications, 1987), 280. 5 Ibid., 280. 6 Trudy Govier, The Philosophy of Argument (Newport News, VA: Vale Press, 1999), 109. 7 Govier, Problems in Argument Analysis and Evaluation, 284. 8 Ibid., 280. 9 Richard Feldman, ‘Good Arguments,’ in Socializing Epistemology: The Social
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10 11 12 13 14 15 16
17 18 19 20 21
22
Dimensions of Knowledge, F.F. Schmitt, ed. (Lanham, MD: Rowman and Littlefield, 1994), 159–88, 166. Ibid., 165. Ibid., 176. Compare ibid., 165. Ibid., 166. Ibid., 176. Ralph Johnson, Manifest Rationality: A Pragmatic Theory of Argument (Mahwah, NJ: Lawrence Erlbaum Associates, 2000). Ibid., 194. Here I am parsing what Johnson has said about unsupported premisses in a particular way. He says, ‘it is rational for the arguer to believe that the audience will accept the premiss and that it is rational for them to do so without support’ (194). The scope of ‘it is rational for the arguer to believe that’ is unclear. Is it the first or is it both conjuncts? If the latter, then a premiss could be acceptable even if it were not rational for the audience to accept it, as long as it was rational for the proponent of the argument to believe that it was rational for the audience to accept it. Clearly, given what the proponent of an argument knows about the evidence available to the intended audience, it may be rational for him to believe that it is rational for them to accept a premiss. But unknown to the proponent, the audience might be aware of evidence which would bring the premiss into question and thus it would not be rational for them to accept it. This second parsing is not how we understand Johnson here. Ibid., 196–7. Ibid., 197. Ibid., 198. Ibid., 191. Derek Allen, ‘Assessing Basic Premises,’ in Analysis and Evaluation: Proceedings of the Third ISSA Conference on Argumentation, vol. 2, Frans H. van Eemeren, Rob Grootendorst, J. Anthony Blair, and Charles A. Willard, eds. (Amsterdam: Sic Sat, 1995), 218–25. Allen carries out his discussion presupposing the framework of direct arguments. Indirect arguments, such as reductio ad absurdum, require certain refinements or conditions which we need not consider here. I believe we also need to distinguish at least two senses of ‘support.’ In one sense, ‘support’ concerns exclusively the connection between premisses and conclusion in an argument. If the premisses either entail the conclusion or render it highly probable, they support the conclusion. In this sense the premisses might support the conclusion, even if they were false. But in
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23
24 25 26 27 28 29 30
31 32 33 34
35 36 37
38
another sense, it would be incorrect to say that false premisses support a conclusion, even if they deductively entail it. Intuitively in assessing whether an argument is logically cogent, if one recognized that a premiss were false, the question of how much weight it lent to the conclusion, either separately or in conjunction with other premisses, is rendered moot. In this sense we can agree with Allen that it gives no support to the conclusion. Allen does not want to engage meta-ethical non-cognitivists at this point. Hence he allows it possible for an argument to be logically good and not all of its premisses to be true, if some of them are of a sort to be neither true nor false. However, in ‘Should We Assess the Basic Premises of an Argument for Truth or Acceptability?’ in Argumentation and Rhetoric CD-ROM, Hans V. Hansen, Christopher W. Tindale, and Athena V. Coleman, eds. (St Catharines, ON: OSSA, 1998), he specifically argues against this non-cognitivist thesis. Derek Allen, ‘Assessing Basic Premises,’ 222. Ibid. Derek Allen, ‘Should We Assess the Basic Premises of an Argument for Truth or Acceptability?’ 6. Ibid. Ibid., 6–7. Ibid., 7. Ralph Johnson, ‘Commentary on Allen’s “Should We Assess the Basic Premises of an Argument for Truth or Acceptability?”’ in Argumentation and Rhetoric CD-ROM, Hans V. Hansen, Christopher W. Tindale, and Athena V. Coleman, eds. (St Catharines, ON: OSSA, 1998), 1. Alvin I. Goldman, ‘Argumentation and Social Epistemology,’ Journal of Philosophy 91 (1994): 27–49. Ibid., 28. Ibid. Alvin I. Goldman, ‘Argumentation and Interpersonal Justification,’ in Perspectives and Approaches: Proceedings of the Third ISSA Conference on Argumentation, vol. 1, Frans H. van Eemeren, Rob Grootendorst, J. Anthony Blair, and Charles A. Willard, eds. (Amsterdam: Sic Sat, 1995), 60. Ibid., 58. Ibid., 61. Alvin I. Goldman, ‘What Is Justified Belief?’ in Justification and Knowledge: New Studies in Epistemology, George Pappas, ed. (Dordrecht: D. Reidel, 1979), 10, quoted in Alvin Plantinga, Warrant: The Current Debate (New York: Oxford University Press, 1993), 10. Goldman has presented refinements of his account of justification, in par-
362 James B. Freeman ticular in Epistemology and Cognition (Cambridge: Harvard University Press, 1986): A cognizer’s belief in p at time t is justified if and only if it is the final member of a finite sequence of doxastic states of the cognizer such that some (single) right J-rule system licenses the transition of each member of the sequence from some earlier state(s) [where] ... A J-rule system R is right if and only if R permits certain (basic) psychological processes, and the instantiation of these processes would result in a truth ratio of belief that meets some specified high threshold (greater than .5) (83, 106); (quoted in Plantinga, Warrant, 10) Here again, there is appeal in the definition of justification to reliable processes, or reliable instantiations of processes. But again, we may ask whether this is an adequate account of justification. Consider again your belief that p formed on the basis of your receiving my argument that p. This conclusion is properly connected to the premisses. Let us suppose that you are conveyed from believing the premisses to believing the conclusion by some physiological process which Peirce would call a leading principle. Your system of J-rules licenses this transition. But again, let us suppose that your psychology is such that you believe (just about) everything I say because of some effect I have on you. But suppose again that the only statements I make are those for which I have significant evidence. Hence, the vast majority of statements I assert are true and the vast majority of your beliefs formed by taking my word are true also. So the J-rule licensing your forming these beliefs on the basis of receiving my word would be a right Jrule. So having these two rules in your J-rule system is not a compromise or in itself mean that the system is not right. So again, on Goldman’s second view, it would seem that your belief that p is justified. But you have no evidence for the premisses of my argument other than my say-so and you have no evidence for my reliability in making these statements. It would seem that your belief in the premisses is not justified. As Plantinga points out, Goldman has considered further refinements of his view. However, I do not believe these escape the counter-examples we have offered. Pursuing these refinements is beyond our scope here. 39 William P. Alston, ‘Concepts of Epistemic Justification,’ Monist 68 (1985): 57–89. 40 See James B. Freeman, ‘Epistemic Justification and Premise Acceptability,’ Argumentation 10 (1996): 59–68. This definition represents a slight rewording of Alston’s concept of justification Jeg in ‘Concepts of Epistemic Justification,’ 77, and is equivalent to that definition. This last point corrects my
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41 42 43 44
critical comments on Alston’s definition in ‘Epistemic Justification and Premise Acceptability,’ Nicholas Rescher, Dialectics: A Controversy-Oriented Approach to the Theory of Knowledge (Albany: State University of New York Press, 1977), 37. Compare Freeman, ‘Epistemic Justification and Premise Acceptability,’ 67. Johnson, ‘Commentary,’ 1. Plantinga, Warrant, 15.
20 Emotion, Relevance, and Consolation Arguments TRUDY GOVIER
There is a kind of argument offered to console people who are sorry or depressed, to the effect that they should not feel so badly because others are even worse off. In such arguments, B tries to console A for A’s suffering on the grounds that some other person or persons, C, have suffered equally bad things or even worse. Here, A and B may be the same person: people sometimes seek to console themselves. The point is to diminish A’s grief on the grounds that he or she is not alone in feeling it. If a person is grieving from having lost a job, well, there are others who have had similar experiences or worse; they may have lost several jobs or never had a decent job in the first place. If she has been diagnosed with an illness requiring unpleasant lifestyle restrictions, well, many other people are ill and have worse diseases – terminal illnesses characterized by severe physical pain, for instance. I have often played the role of A in this scenario and that of B, and who knows, perhaps I have, without knowing it, played the role of C as well. Many in the readers could probably report the same thing, since Consolation Arguments of this type are rather common.1 A recent item circulated on the Internet and forwarded to me by David Hitchcock included the following, among others: Should you despair over a relationship gone bad, think of the person who has never known what it’s like to love and be loved in return. Should your car break down, leaving you miles away from assistance, think of the paraplegic who would love the opportunity to take that walk. Should you find yourself at a loss and pondering what is life all
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about, asking what is my purpose, be thankful. There are those who didn’t live long enough to get the opportunity. A kind of reductio can be generated from such arguments. If one person’s grief should be obviated whenever someone else has suffered equally or more, then ultimately only the worst-off sufferer in the world is entitled to feel sad. My interest in Consolation Arguments arises in part from personal experience, my experience being that I have some tendency to offer such arguments to myself and others and occasionally experience them as rather consoling, but more typically find them frustrating and rather patronizing. The latter was certainly my response when – just after I had presented an earlier version of this paper to a small audience in Lethbridge, Alberta – the philosophy editor for a major publisher used a consolation argument to dismiss my anxiety and unhappiness about a title for a new book. I had made many counter-suggestions to her desired title, which I despised. But the editor would not move. She told me that it was no matter for grief: tragedy lay not here but in the killing fields of Rwanda. In other words, I should calm down, and she was entitled to dismiss my concerns because other people had problems that were much worse. One could say, in this editor’s defence, that she did not entirely ignore me; she addressed an argument to me, even though it would appear to be a strikingly bad one. My interest in Consolation Arguments is philosophical as well as personal. I’m interested in understanding the nature and strength of emotion by considering arguments that address emotional responses. And I’m also interested in issues of relevance, which arise here in a revealing way. One might argue Irrelevance in Consolation Arguments, alleging that they commit a fallacy of relevance, given that C’s suffering is one thing and A’s another. Alternately, one could argue Positive Relevance in such cases on the grounds that A’s suffering is properly put in perspective by C’s suffering, as pointed out in the argument. A third possibility is to claim Negative Relevance on the grounds that the premisses count against the conclusion because the fact that C has suffered should invite more, not less, depression on A’s part. I gather from Richard Sorabji’s recent Gifford Lectures that in ancient times what I have called the Consolation Argument was referred to as the argument that ‘you are not the only one.’2 Democritus is said to have comforted the king of Persia in a time of bereavement by telling him he was not alone in having suffered from the loss
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of a loved one. One idea here was that it might be useful to distract oneself by thinking of somebody else’s problems. This comment about distraction suggests irrelevance of the other person’s suffering to one’s own; such irrelevance can be psychologically useful since the distracting case soothes one’s mood. Sorabji reports that Cicero referred to this kind of argument in a reflection he wrote for himself when he was desperate both about his career and about loss of a beloved daughter. Cicero commented that such arguments were much used and sometimes helpful, though often not well received. He thought they would be more helpful if accompanied by an explanation of how other suffering people had coped with their grief and loss. One can say the same about many weak arguments; they improve if you add something good. From Epictetus, we have this version of the Consolation Argument: Someone’s child is dead, or his wife. There is no one who would not say, ‘It’s the lot of a human being.’ But when one’s own dies, immediately it is, ‘Alas! Poor me!’ But we should have remembered how we feel when we hear of the same thing about others.3
In other words, we should put our sorrow in perspective by appreciating that bad things happen to other people too. 1. Argument and Emotion One might allege that there is a systematic theoretical problem arising in connection with Consolation Arguments and it arrives even before we arrive at questions about relevance. It could be argued that emotional states are not voluntary and not rationally amendable, that Consolation Arguments presuppose that people can voluntarily amend their emotional states, which is not true, and that because these arguments have a faulty presupposition, they are fundamentally flawed and can serve no real purpose. There is a sound reply to this objection, I think. Emotions are subject to indirect voluntary control even in contexts in which they are not subject to direct voluntary control. For example, a person who becomes cognitively convinced that her fear of flying is both an irrational reaction and a serious handicap to her career might seek therapeutic assistance or other relevant experience with a view to ridding herself of that fear. She cannot eliminate her fear by fiat but she can work to overcome it over time. Arguments to the effect that the fear was irrational
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and constituted a handicap in life could play a central role in her decision to do that. And similar things can be said about other emotions. In Appeals to Pity and Fear, which are, of course, acknowledged to be fallacious arguments, expressions of or descriptions of emotion or emotion-warranting situations are found in the premisses. There is an attempt to exploit those emotional elements in order to gain assent to a conclusion. Although emotion plays a role in the Consolation Argument, its role is quite different in that context from its role in Appeals to Pity or Fear; the intended effect is not to exploit emotion but rather to diminish it. B argues that X, which depresses A, should be deemed by A to be less depressing on the grounds that Y, which is just as bad as X or worse, has been experienced by another person, C. B is attempting to console A by reminding him of another case, that of C. In their use of analogy between A’s suffering and that of C, Consolation Arguments seem to bear some similarity to analogical arguments of the ‘Two-Wrongs’ type. The arguer invites the inference that the case at hand is better than it seems because some other case is worse. Douglas Walton explores such argumentative devices as Appeals to Pity and Fear, describing these as arguments in which there is an attempt to shift the balance of evidence toward a conclusion by appeal to the emotional state of the audience. Walton claims, ‘There is nothing wrong or fallacious per se with appeals to emotion in argument. Emotion should not be (categorically) opposed to reason, even though emotion can go wrong or be exploited in some cases.’ Walton says that when emotional appeals are fallacious ‘the powerful appeal to an emotion’ is ‘used to get an audience to accept a prejudiced, one-sided, or biased point of view without looking too carefully at relevant evidence in a more balanced way.’4 The emotion is exploited to achieve a more dogmatic attitude (toward the conclusion, presumably) in which bias is hardened, and emotion has a greater role than it should have. Nevertheless, Walton advises, emotional appeals may be ‘reasonable and nonfallacious’ and may have positive value in demonstrating empathy or expressing solidarity. These comments seem quite sensible but they do nothing to resolve the issue of whether Consolation Arguments should be regarded as reasonable or as fallacious. One might say all such arguments are out of place and useless because there is no point in arguing about emotions: people just feel things or they do not, and reasons, evidence, and arguments have nothing to do with the matter. On this view, all Consolation Arguments would be faulty because all would be based on a false presumption. But
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this objection does not hold: there are connections between emotion and belief that allow for the amending of emotional states by means of arguments. Many emotions are directed toward things or events that are characterized in some particular way and, accordingly, presuppose beliefs or judgments. For example, a person who is sad about his daughter’s miscarriage believes she has had a miscarriage. If not, he cannot be sad about this, though he may be sad for some other reason. Irving Thalberg distinguished emotions with objects from emotions without objects; generalized depression or free-floating anxiety would fall in the latter category. Of emotions with objects, Thalberg noted that many have propositional commitments.5 Thalberg carefully distinguished the object of an emotion from its cause, offering the example of a man who fears that a rhinoceros will chase him across Central Park. If there is no rhinoceros around, this man’s fear is unfounded – which is not to say that it is uncaused. Whatever the cause of his emotion might be, a rhino is not it. When an emotion has an object, it is typically tied logically to thoughts and beliefs about that object. In such a case, if someone refutes the relevant presupposition, the emotion will disappear. The case of the rumoured miscarriage is a simple example of a presupposed belief being refuted. But evaluations are also relevant to emotions, and those emotions may shift if relevant judgments of value are amended. The man might discover, for instance, that although his daughter did have a miscarriage, her miscarriage could be deemed a good thing overall, because she has, in undergoing medical treatment, discovered an underlying condition such that continuing the pregnancy would have posed a substantial risk to her life. All this is simply to say that emotions characteristically presuppose beliefs and evaluative judgments and, since beliefs and evaluative judgments are amendable on the basis of argument, such emotions are also amendable on the basis of argument.6 In fact, there can be many questions and arguments about emotions that we feel and express. We can ask whether an emotion, E, is fitted to its object. An emotion E is fitted to its object if that object is the kind of thing in response to which E would be an appropriate emotion. (If one finds a nuclear attack fearsome, that is fitting.) We can also ask whether it is prudent for a person to feel E, in the circumstances in which he does feel that way. We can ask whether it is morally good, or right, for him to feel E, given that he does. In a recent article Justin D’Arms and Daniel Jacobson describe such questions and argue that there is a tendency for moral considerations to take up all the evaluative space when they are
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raised. D’Arms and Jacobson claim that people tend to give too much weight to moral considerations in commenting on emotions. They speak of a Moralistic Fallacy in this connection and argue persuasively that whether E fits its object is different from the question of whether it is prudent, or moral to feel, express, or act upon the emotion E. For example, a joke could be funny even though it is imprudent or even perhaps immoral to laugh at it. When I was preparing the first draft of this paper in the spring of 2002, the Pope issued a statement on sexual abuse by some members of the Catholic clergy. Among other things, he said that the offenders were infected by a mysterium iniquitatis. I found this comment hilarious and was amused – but given the suffering of victims and the seriousness of the overall problem, many would argue that my response was immoral. In most contexts it would have been imprudent to laugh. We can also usefully distinguish between feeling an emotion, expressing that emotion verbally, and acting on it. For example, one might feel sad without expressing that sadness verbally or otherwise acting on it; one might feel sad and verbally complain but not take any further action; or one might feel sad, complain verbally, and take some further action based on one’s feelings. Correspondingly, we might want to distinguish versions of the Consolation Argument. We could imagine B seeking to console A in the sense of actually trying to say things that will lead A to feel better. Alternately, B might really be getting tired of listening to A and want A to stop complaining about how awful he feels, whether or not A actually feels better. Or B might be using the argument to dissuade A of some further course of action he is about to take because of the sadness or depression he feels. It is the first sort of context I am concentrating on here, the context in which B is seeking to amend A’s feelings by offering him a Consolation Argument. The distinctions drawn by D’Arms and Jacobson allow for the possibility that grief might be fitting as a response to some situation even though on some occasions it might be imprudent or immoral or unpatriotic to feel and express it. Writing about ‘Melancholic Epistemology,’ George Graham maintains that although depression and pessimistic thinking may be illogical, they are not necessarily so.7 One may be in a situation to which sorrow and depression are entirely fitting responses. Graham considers several such cases. One is that of a person in a concentration camp who expects to die painfully and has already seen several family members meet such deaths; another, is that of a dancer who has met with a serious accident and is paralyzed from the neck down. While such per-
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sons might fare better in the world if they could rally from their depression or grief and ‘snap out of it’ – or at least refrain from complaining or bothering others about their problems – there is no denying that their sadness or depression fits their situations. When people feel sad or depressed in such cases, it is not because they have made mistakes in logic or factual or evaluative errors about themselves and their circumstances. Applying the terminology of D’Arms and Jacobson here, we might say that insisting that people should snap out of it when they feel sorrow or depression is to mistakenly conflate the issue of the usefulness or prudence of an emotional response with that of its fittingness to its object. (Such a response is often objectionable as a lack of acknowledgement of the feelings of others, but that is another point.) Such a conflation between what fits and what is prudentially or morally right may underlie some Consolation Arguments. As Hume famously said in explaining sympathy, ‘the minds of men are mirrors to one another.’ We tend to pick up moods from other people. Some of our efforts to get other people to cheer up are no doubt motivated partly by selfishness: when we have to be around other people who feel awful and express that, we begin to feel rather awful ourselves. But none of this tells us whether Consolation Arguments are fallacious due to errors of relevance. 2. Contexts In several recent works, Douglas Walton has suggested that the interpretation and evaluation of an argument should vary, depending on the context in which that argument is used. In a recent work on argument structure, for example, Walton lists the following sorts of contexts, or ‘dialogues’: (a) the critical discussion; (b) the negotiation; (c) the inquiry; (d) the deliberation; (e) the quarrel; (f) information-seeking; (g) a context in which an expert is being interviewed; and (h) a pedagogical context.8 In an earlier work on emotion in argument, Walton offered a similar list.9 One might question aspects of these lists. For example, why is information-seeking distinguished from consulting an expert? Why could the latter not be a subcase of the former? Similarly, why did Walton at one point distinguish scientific inquiry from other types of inquiry? Again, might it not be a subcase? Walton himself seems to allow that there may be questions about his distinctions. He mentions that the boundaries between these different sorts of context are not always def-
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inite – people can cross from one to another, often confusing themselves and others along the way – and that people are not always clear what sorts of discussions they are having. My sense is that Consolation Arguments might be used in several of the contexts Walton seeks to distinguish. Suppose that A is sorrowful or depressed and B is trying to lessen A’s grief. It seems slightly implausible to think of such occasions as contexts of critical discussion in which B is trying to rationally persuade A of the truth of some proposition, and where A and B are in adversarial roles in that discussion. One might defend such an interpretation by saying that the adversariality of critical discussions need not be pronounced. Alternately, one might regard A and B as conducting a kind of joint inquiry as to whether other people (C) have indeed suffered more than A and whether and how such suffering might serve to diminish A’s grief. Another possibility, within the framework Walton puts forward, is that one might imagine the Consolation Argument occurring in a context of expert consultation: the sad or depressed person, A, is consulting the expert, B – a context that suggests itself if one is willing to categorize therapists as experts. Or, considering that people sometimes direct Consolation Arguments to themselves when reflecting on their circumstances and attitudes, one might say that such arguments are used in contexts of deliberation. Deliberatively, we might suppose, a depressed or sad person begins to pose to herself some form of Consolation Argument and reflects on its merits, asking herself how, if at all, the suffering of other people might serve to diminish her own. It seems, then, that Consolation Arguments may be used in various of the contexts or ‘dialogues’ distinguished by Douglas Walton. But if our question is one about the relevance of premisses to conclusions in such arguments, it is not clear why such considerations would resolve the question about relevance in Consolation Arguments. Suppose that A, who has lost a child in a custody case, consults a therapist, B, who with the very best of intentions tells A that she should calm down about losing custody because after all, B has another client, C, who is not only being divorced by her husband and unable to care for her children but is, in addition, paralyzed from the neck down. How, if at all, is C’s sorry situation relevant to A’s problem? We have seen that the context could be one of critical discussion, expert consultation, joint inquiry, or deliberation: we know not which. But that does not settle the relevance issue, which would remain whatever interpretive decision we made about the context.
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3. Relevance I turn to some recent accounts of relevance to see whether they can offer useful guidance at this point. 3.1 Pragma-Dialectics In the pragma-dialectical account developed by Frans van Eemeren and the late Rob Grootendorst, fallacies are understood as violations of rules that are elements of a code of conduct for critical discussions. Within the framework of this theory, if Consolation Arguments were based on fallacious irrelevance, that would be because the move from premiss to conclusion would constitute a hindrance to the reasonable resolution of a disagreement between the parties A and B in their critical discussion. To use the language of pragma-dialectics, were B’s reference to C’s suffering in this context to constitute a ‘speech act that prejudices or frustrates efforts to resolve a difference of opinion, and a violation of a rule for the proper conduct of critical discussions,’ Consolation Arguments would amount to fallacious argumentation. To apply the pragma-dialectical model here, we have to assume that when B is trying to console A, A and B are engaged in a critical discussion, and within that discussion, they have agreed that what is at issue between them is a particular standpoint to the effect that A’s grief should be less than it is. We are advised by the pragma-dialectical theory that the standpoint can be defended only by advancing argumentation that is ‘related’ to it by means of an ‘appropriate’ and ‘correctly applied’ argumentation scheme. Thus the questions of relevance that have been raised here resurface as questions about whether points about C’s suffering are related to the issue of A’s suffering, whether an appropriate argumentation scheme is used, and whether that argumentation scheme is correctly applied in the case. It is by no means clear that such shifts constitute progress.10 More detail is given in van Eemeren and Grootendorst’s 1992 book Argumentation, Communication, and Fallacies. Here rule 4 for the conduct of discussions states that ‘A party may defend the standpoint only by advancing argumentation relating to that standpoint.’ In the explanation of this rule, the authors claim that the argumentation must not be irrelevant to the standpoint; nor may the argument employ means of persuasion that play on the emotions of the audience or seek to parade the qualities of the arguer. Such comments do not resolve our puzzle
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about relevance in Consolation Arguments. Rule 7 says that ‘A party may not regard a standpoint as conclusively defended if the defense does not take place by means of an appropriate argumentation scheme that is correctly applied.’ We might (recalling the fact that Consolation Arguments resemble Two Wrongs Arguments, which are commonly deemed fallacious) speculate that Consolation Arguments inappropriately apply reasoning from similarity or analogy, but there is nothing in the van Eemeren–Grootendorst account to guide us on the point. Rule 8 says, ‘In his argumentation, a party may only use arguments that are logically valid or capable of being validated by making explicit one or more unexpressed premisses.’ This rule raises the hoary problem of missing premisses and again gives no substantive guidance. One might wish to amend Consolation Arguments with some proposed ‘missing premiss,’ said to be implicit in the original version. Such a premiss might be of the type, ‘If a person suffers less than some other people do, that person should not feel sad about his or her suffering.’ Or perhaps, ‘If a person suffers less than some other people do, that person should feel less sad than he does about his or her suffering.’ Then the Consolation Argument would contain no flaw of relevance but, as in other cases when the ‘missing-premiss’ device is used, we would face instead the task of determining whether the reconstruction was appropriate and whether the inserted ‘missing premiss’ was true or rationally acceptable at this point in the discussion. The explanatory comments for rule 8 in Argumentation, Communication, and Fallacies refer to necessary and sufficient conditions and part-whole issues and do not seem to connect usefully with our problems about consolation. Even if one accepts the pragma-dialectical rules for conducting a discussion, they do not seem helpful at this point. 3.2 Walton In a work entitled A Pragmatic Theory of Fallacy, Walton comments that there are various different argumentation schemes and that to these, various critical questions are attached. A claim – call it X – is relevant to a contested conclusion being defended by one of those schemes if X answers one of those attached critical questions. On this approach, if we knew what sort of argumentation scheme was being used in Consolation Arguments and we also knew what critical questions were attached to it, we might be able to answer our query. But the problem is, we don’t know such things.
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As elsewhere, in this work Walton construes arguments in contexts of dialogue. He says, ‘Any argument or other move in an argumentation is relevant to the extent that it fits into that type of dialogue as an appropriate move.’11 A move may be dialectically relevant either by being locally relevant (one claim, X, supports some other claim, Y, within the dialogue) or by being globally relevant (this would seem to mean claim X supports a claim Z, which is the main conclusion at stake in the dialogue). Furthermore, making the claim X must be a type of move that is appropriate to the sort of dialogue in which the participants are engaged. Discussing relevance, Walton considers a truly wonderful example from the Canadian House of Commons, in which Sheila Copps, at that time a member of the Liberal opposition, criticized the (Conservative) Mulroney government’s bill to eliminate Family Allowance payments. Copps based many of her comments on a scandal about incompetent inspections that had allowed cans of rotting tuna to be available for sale in Canadian supermarkets. Commenting on her rhetorical shifts, Walton offers the observation that the tuna fish issue was not dialectically relevant to the family allowance debate because ‘the particular issue of the tuna fish, even if it could be resolved, would seem to carry little or no weight in influencing anyone reasonably to vote for or against the Family Allowance Act.’12 Copps alleged that tainted tuna and family allowance cuts were connected because government callousness was the cause of both; and besides, tuna was connected with poverty, because poor families had to survive by eating tuna. Thus, Copps said, there was no red herring in the tuna reference. Commenting on this delicious example, Walton claimed that such connections were not substantial enough to make the issue of the tuna relevant to the House discussion of family allowance policy. In his view, Copps was merely seeking to exploit the scandal of the rotting tuna to attack the Mulroney government. These comments by Walton strike me as entirely sensible. Still, there seems to be a problem. Walton does not seem to need, or to use, his own theoretical apparatus in order to generate this account of the tuna comments. The context of the House debate was clearly an adversarial one, but Walton does not advise whether the turbulent discussion should count as a critical discussion or amounts to a quarrel. (It strikes me as containing elements of both.) Nor does Walton tell us what rules should apply to this ‘dialogue,’ apart from rules of the House of Commons itself. He seems to base his comments on his own good sense
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rather than on his theories about dialogue contexts, argumentation schemes with attached critical questions, and different types of relevance. By the end of the discussion, Walton has mentioned not only local, global, and dialectical relevance, but indirect, topic, and probative relevance as well, and I begin to lose myself in the web of possibilities. Nor have I found any systematic guidance with reference to the Consolation Arguments. 4. Another Approach In order to think about anything we need to rely on our sense of relevance and irrelevance, and that fact, I suspect, explains why it is so very difficult to say anything that is both general and useful about relevance itself. In my text A Practical Study of Argument,13 I comment that relevance is such a basic concept in thinking that it is difficult to pin it down with an exact definition. Though some have regarded that view as an escape from responsibility, it may, perhaps, acquire some added plausibility from the relative unhelpfulness of the accounts just considered. I first define positive relevance of claim X to claim Y as obtaining if and only if the truth of X counts in favour of the truth or rational acceptability of Y. I then define negative relevance of claim X to claim Y as obtaining if and only if the truth of X counts against the truth or rational acceptability of Y. And I say that claim X is irrelevant to claim Y if and only if it is neither positively relevant nor negatively relevant to Y. Clearly the explanatory weight here shifts to the notion of ‘counting in favour of,’ which is not further defined. I comment that context can make a difference to judgments about relevance and offer further examples to illustrate that point. But I do not handle contextual issues in quite the way Walton does. For me, the point is not so much a matter of distinguishing types of context, contrasting inquiry with deliberation or expert consultation, or a critical discussion with a quarrel. Rather, it is to consider specific examples and their content in the light of some specific fact or presumption, Z, about a context. My account is couched in relatively simple terms because it was originally developed for a textbook. One might deem it primitive on the grounds that it attempts to address relevance in a proposition-toproposition framework, without reference to different types of context or dialogue or rules for the conduct of discussions. It should be noted, however, that those considerations have not proven their usefulness with relation to our puzzle about relevance in Consolation Arguments.
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One might allege that such an account as mine will, in the end, leave us appealing to various ad hoc considerations or even simply to our ‘intuitions.’ To the latter criticism, it would appear that I could reply ‘tu quoque.’ But it would be nice to do better. Since time and space are running out, let me come to the point. One who argues from a premiss to a conclusion is committed to the claim that that premiss is positively relevant to that conclusion. If that claim is contested, the arguer faces the challenge of showing that, and how, the premisses are relevant to the conclusion. Those not convinced by the argument can dispute specific points. A critic who alleges irrelevance in the argument faces the challenge of showing how and why the premisses are irrelevant. Disputes about relevance cannot be answered by a purely intuitive appeal. By themselves, intuitions are insufficient – and in this case as elsewhere, different people are likely to have different ones. When issues of relevance are explored, the arguer and the critic should defend their judgments in further arguments. In such arguments, there is room for premisses drawn from theories about pragmatics, dialectics, dialogues, contexts, argument reconstruction, and implicit premisses. But there is also room for substantive considerations about the relation this premiss, or this sort of premiss, bears on this conclusion or this sort of conclusion. On the basis of such considerations, my inclination is to find irrelevance in Consolation Arguments. Essentially the reason is that A’s emotional state depends on A’s beliefs, evaluations, and circumstances, not on those of C. When A is feeling sad, if B wants to console him or her, B should attend to the case at hand and really address what A is feeling. That is to say, if B is going to consider facts and use arguments in his efforts to console A, those facts and arguments should be such that they tend to confirm or to disconfirm the relevant beliefs and evaluative judgments made by A. Typically, those beliefs and judgments are about the object of A’s own emotion and not about the circumstances and feelings of some other person, C. (Typically it is not C who is the object of A’s emotions, so facts about C will not show whether those emotions are fitting or not.) If B is trying to console A, B should attend specifically to the content and intensity of A’s own emotion. Putting this point in more psychological terms, we might say that if B shifts to considering the miseries of C, he is in so doing not acknowledging A’s own particular feelings, thereby failing to communicate empathy for A and suggesting that A is not entitled to as feel sad as he does. This irrelevance is likely to be felt by A as lack of acknowledge-
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ment, and even as insult. The shift from A’s problems to those of C suggest that C’s case is more legitimate or important than A’s – which is likely to be frustrating to A. B suggests that A’s own feelings are unfitting or inappropriate or unworthy of attention in their own right because of their comparative significance with regard to C’s concerns. This is a shift of topic – and because it is, it is fair to say that the Consolation Argument involves a mistake of relevance. Its premiss is not even about A or the object of A’s emotion; it is about C, another person entirely. There is a qualification to be made here, though, and I think it is an interesting and important one. What I have just said depends on A’s feelings not being implicitly comparative in the first place. I have presumed that, typically, C and others were not the objects of A’s emotion – that A was feeling depressed about his own situation in particular, and in itself. But that is not always the case, and when it is not, the judgment of irrelevance may be altered. Often the judgments on which our emotions depend are implicitly comparative because they have a decided ‘why me?’ element. The notion that we have been unfairly singled out for bad treatment or bad luck is characteristic of many of our emotions – of which resentment is an outstanding and particularly interesting example. A’s grief may presuppose beliefs that are comparative in crucial ways. A may feel singled out by fate, bad luck, or human persecutors to undergo miseries not suffered by others, miseries that she did not deserve. A woman may feel, for example, that she has been unlucky to suffer a chronic disease that imposes considerable dietary and lifestyle restrictions; she may believe this situation to be unfair, focusing on the fact that there are millions of other people who do not face such restrictions, all of them seeming to be, in this respect, far more fortunate than she is herself. Or a man may think there is an unfairness in the fact that his children have grown up to be obnoxious and unsuccessful, whereas other people’s children have become polite young people advancing in professional careers. When A’s feelings are based on the conviction that he or she is unfairly disadvantaged, as compared with other people, there is room for relevance of the premisses of the Consolation Argument. In such cases, A’s feelings are explicitly or implicitly comparative; for this reason, the suffering of others is relevant to A’s emotions. In such cases, by bringing C into the picture, B offers to A the prospect of gaining perspective on his situation and questioning his presumptions about expectations, proportion, and fairness. A may come
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to recognize that he had tacitly assumed a comparative judgment that can be questioned in the light of B’s claims about what C has been going through. Thus, in some Consolation Arguments, allegations of irrelevance can be rebutted. But note where this exception comes from. It certainly does not emerge from the application of formal or general rules; nor does it depend on whether A and B are conducting a critical discussion, an inquiry, a deliberation, or a therapeutic session deemed to be one of expert consultation. It is not a matter of whether there is a rule of discussion that has been violated or whether an inappropriate argumentation scheme has been used – or whether the issue in question concerns topical, local, or global relevance. Rather, the relevance of C’s case can be established here because of the nature, and presuppositional structure, of A’s feelings. Fundamentally, relevance is a matter of ‘aboutness,’ and A’s feelings are what the Consolation Arguments are about. notes Thanks to John Woods, David Hitchcock, and Timothy Schroeder for comments on an earlier version of this paper. 1 Many different arguments might be put forward to console people. For purposes of easy exposition here, I use the term ‘Consolation Argument’ to refer to an attempt to conclude that one person’s suffering should be diminished or discounted because others have suffered more. 2 Richard Sorabji, Emotion and Peace of Mind: From Stoic Agitation to Christian Temptation. The Gifford Lectures (Oxford: Oxford University Press, 2000), 177–8. 3 The Handbook of Epictetus, translated and annotated by Nicholas White (Indianapolis: Hackett, 1983), Epigram 26. 4 Douglas Walton, The Place of Emotions in Argument (University Park: Pennsylvania State University Press, 1992), 257. 5 Irving Thalberg, ‘Emotion and Thought,’ American Philosophical Quarterly (1964): 45–65. 6 Justin D’Arms and Daniel Jacobson, ‘The Moralistic Fallacy: On the Appropriateness of Emotion,’ Philosophy and Phenomenological Research (2000): 65– 90. 7 George Graham, ‘Melancholic Epistemology,’ Synthese (1990): 399–422. 8 Douglas Walton, Argument Structure: A Pragmatic Theory (Toronto: University of Toronto Press, 1996).
Emotion, Relevance, and Consolation Arguments 379 9 Walton, The Place of Emotion in Argument, 19–23. 10 Frans H. van Eemeren, ed., Crucial Concepts in Argumentation Theory (Amsterdam: University of Amsterdam Press, SicSat, chaps 1 and 6. A longer discussion may be found in Frans H. van Eemeren and Rob Grootendorst, Argumentation, Communication, and Fallacies: A Pragma-Dialectical Perspective (Hillsdale, NJ: Lawrence Erlbaum Associates, 1992). 11 Douglas Walton, A Pragmatic Theory of Fallacy (Tuscaloosa: University of Alabama Press, 1995), 163. 12 Ibid., 166. 13 Trudy Govier, The Practical Study of Argument, 6th ed. (Belmont, CA: Wadsworth, 2005). See chapter 6.
21 Temporal Agents JIM CUNNINGHAM
1. Appreciating Psychologism By embracing psychologism in their ‘New Logic,’ Gabbay and Woods1 admit inference that is fallacious in traditional logic. This is justified by arguments such as the usefulness of shortcuts for achieving effective response from an agent with limited computational time and space. The cited paper stands as an informal prelude to a logic of wider ambit, including abduction and discovery, an unconventional, groundbreaking logic whose consequences are to be induced from inference steps which include the seemingly fallacious. Although inspired by the new logic, this paper is more concerned with processes and their logical presentation: the computational processes of a bounded agent in an unbounded environment, processes to be composed from primitive steps, and how the way in which processes are composed ultimately constitutes the mental architecture of the agent. If these processes are to operationalize the inferences of a more conventional logic, its consequences will perhaps be perceived as epistemic states of the agent. We take the view that in modelling an agent the processes are no more ephemeral than the states they may produce, that the observational behaviour of an agent depends on rich, dynamic, and communicative ingredients that stative concepts alone do not capture, and that the reasoning processes of agent, through being temporally bounded, can be distinguished from the atemporal stative inference which may be proper for an institutional agent. From this perspective, unwelcome psychologism may be the misplaced adoption of temporal reasoning processes when not appropriate. Despite the encapsulated rationality of traditional mathematics and
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logic, and the hunger of computational theorists for guidance, these disciplines have not been overly successful in characterizing real computation. One can argue that a faithful description of computational engines, and of operational agents, depends on a temporal sense of causality and consequence that can be hidden by atemporal interpretation when rationality is expressed in traditional logics. The discipline of symbolic logic may come as inherited background in the drive to represent computational processes, and it has been elegantly exploited by causal rules in logic programming styles, but all too often the underlying logic of a computer program is obscured by cumbersome notation with implicit temporal order, whether expressed as Prolog, C, or Java code. At least for reasoning agents we wish to re-examine some of the links between logic and process. Implicit in our discussion, a (practical) agent is assumed finite in its initial knowledge, in its processing speed, in the duration of its previous lifetime, and thus in its current knowledge and beliefs. When we allow an agent the processing capacity for learning, abstraction, and introspection, we would argue that a primary resource limitation comes from its temporal nature, not just processing time, but time for gaining experience of its environs, and for temporal reasoning itself, which we wish to address. An agent’s reaction, and its planning, are limited by its incomplete, finite knowledge. Its best decisions may be incorrect, when viewed from an atemporal global perspective, yet rather than being an imperfect reasoner, it may even be locally optimal by some measure, although prone to error because of imperfect knowledge. Of course, part of an agent’s survival strategy, or perhaps an inherited strategy for survival of its group, may well be for the individual to acquire new beliefs through introspection and abstraction when recovering from mistakes, and to validate these beliefs or seek new knowledge to enhance its control of the environment. At issue is the coherent design of such an agent. The Beliefs, Desires and Intents (BDI) of Bratman’s agents2 have proved insightful and durable – a structurally coarse model for the states of a computational agent’s ‘mind.’ But they are a weak explanation for the computational processes themselves. Deliberation without learning and inquiry ensures a witless agent. But learning and inquiry are processes too, albeit abstract ones, hidden but salient and dynamic traits of rational agency. Whether inquiry is driven by introspection, and so could be a basis for a conscious agent, is more contentious, but a rational agent must act and react on imperfect knowledge, decide how to recover from
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apparent inconsistency, and learn from experience. A coherent process architecture for such behaviour should be part of the explication for mistakes of reason. 2. Windows on the Mind It is our privilege to live in an age when formal reasoning cannot only be described, but simulated by machines at speeds so surpassing our human capacity that it also becomes evident that algorithmic formalizations of arithmetic, logical, and even chess-playing processes are mere trinkets in comparison with the richness of human reasoning. However, if we are indeed reasoning agents, we cannot be smug, for there are already challenging forms of computational agent, including realizations of the BDI agents alluded to above. These models can be quite general in their domain of applicability. Nor should the scientific advance which computer technology provides be underestimated, even if it may seem merely smart technology. Just as the adoption of the toy magnifying glass in Kepler’s day provided new sights on the heavens, we too have a toy for the first time: a toy for investigating models of reasoning. The subjectivity of introspective metaphysics can become at last the basis of observation and experiment. Kant saw that Copernicus had distinguished movement of the observer from movement of the heavenly bodies and sought similar separation between the mind and logical reasoning itself.3 If we are, in some sense, to readmit psychologism to formal reasoning, we can still require their integration to be objectively assessed. For mathematical logicians it is perhaps a fear of subjectivity that devalues evidence from the arguably ephemeral and synthetic phenomena of linguistics, even relatively benign taxonomical data. But to ignore linguistic evidence as contaminated by psychologism unless reinterpreted by a more orthodox dogma introduces an unnecessary barrier to mental insight, for language, too, can be the basis of experiment with artificial agents. For example, although tense logics are inspired by natural language, a conventional tense logic with monadic past, and/or future, possibility operators (represented, say, as áPñ, áFñ) cannot distinguish distinct times when a proposition is true. So an intended temporal sequence of propositions like áPñp, áFñp, áFñ áFñp ... can be satisfied by, and hence confused with, a sequence that includes replicated or spurious occurrences. We do not know whether this is a mistake which occurs with human communication or human memory, but it is an expressive limitation for simple tense logics.4
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In adopting some notions of tense and aspect we do not presume that our suggestions for an interpretation of temporal language are without need of validation, but we believe the approach provides insight. For a start, we adopt an interval model of time, not only because it is well attested as more faithful in the semantic literature,5 enabling aspect as well as tense to be represented, but, more contentiously, because by doing so we can represent a progressive sense of process and thereby get a more expressive handle on temporality, and perhaps consciousness, as an inner perception of activity and belief. 2.1 An Interval Tense Logic For the purposes of exposition we adopt and extend a propositional variant of an interval tense logic introduced by Halpern and Shoham, which we call HS.6 The expressions of HS are constructed from syntactic atoms using Boolean and modal connectives. The latter incorporate Allen’s binary relations on temporal intervals, first introduced as a basis for qualitative reasoning.7 The HS relations are illustrated in figure 1. Each relation R is incorporated in the logic as a modality with dual forms for the possible, áRñ, and the necessary, [R]. The modalities are, as usual, evaluated with respect to a current interval, or ‘world,’ which may loosely be regarded as the period we call ‘now.’ The fifteen binary relations illustrated in figure 21.1 are not all independent. Thus the later relation L can be defined by the modal statement áLñp l áAñ áAñp, while a definition for the beginning point of the current interval is [[BP]]p l áBñ(p [B]A). So a beginning point is defined as a beginning interval on which all (further) beginning intervals are impossible. The operators áLñ and áLñ and can be used in place of more conventional past and future tense operators to provide a rendering of the simple past tense, such as John ran, and the future tense, John will run, on an interval time domain, although this does not remove the expressiveness problem of simple tense logics mentioned above. However the conventional rendering of tense is also inadequate if we hold to Reichenbach’s use of an auxiliary reference point to distinguish the perfectives.8 A fragment of HS was extended with a focus operator by Leith and Cunningham9 to provide a computational representation of perfective tense and aspect, with supporting decision procedures for linguistic inferences. Although more expressive than traditional point-based tense logics, the HS logic was also shown by Venema10 to have both obscure expressive limitations and computational complexity, so its illustrative uses for qualitative reasoning appear not to have been pursued. We can
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Figure 21.1 Interval relations of the HS logic
construct an extension of the HS logic that is actually a more expressive, hybrid logic, which we will call HSN, by including nominals for naming intervals. Nominals are atoms which are disjoint from the usual propositional variables.11 While HSN is still a tense logic in its style, such nominals enable us to overcome the above expressive limitations of simpler tense logics; they allow a more direct presentation of the perfective aspect and a scalable representation of knowledge through named intervals, similar to the reification of events with predicate logics.12 A logic like HSN will have an operator, which we denote by @, for binding a proposition to a nominal for the interval on which it holds. This is an internalization of Allen’s metalogical holds predicate.13 In this way we can bridge a gap between the less-expressive but linguistically faithful tense logics and the thematic relations of eventbased sentential semantics. Even more subtle issues such as temporal linearity can also be ensured by the elaboration of more conventional temporal axioms. As an example of the additional expressiveness in a logic like HSN, we find that a tense operator may apply not just to a sentence, but to a sentence and its temporal reference. Thus the proposition John ran may be better represented as áLñ(John_runs’ @. r), where John_runs’ is a sub-
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formula representing the sentence John runs and r is the nominal for the interval on which the subformula John_runs’ holds. The nominal can then serve as a reference for further discourse. The sense of immediate past in the English present perfect John has run is difficult to capture without interval analysis. It is at least tempting to consider the present perfective as explained by use of the interval operator áAñ rather than áLñ, without modifying the temporal reference, so that the past perfect John had run is indeed the composition of a past and perfective operators, as in áLñ (áAñ John_runs’ @ r). This appears to be compatible with Reichenbach’s analysis. 2.2 Process-Oriented Language Aspect is as important as tense for the semantic analysis of the English verbs. Indeed, the English verbs are divided by grammarians such as Quirk et al.14 into the dynamic and the stative depending on whether the progressive aspect – as in is making, is running – can be accepted. Normal verbs like make and run are dynamic, but a minority, such as the verbs be, know, and desire, are classified as stative, indicative that the expressions *is being, *is knowing, and *is desiring are not accepted as progressives in the same sense. The progressive epitomizes ongoing process or activity. Informal analyses of verb aspect by Vendler,15 and more recently by Moens and Steedman,16 consider the durative character of events they describe, distinguishing processes (to run), culminated processes (to run home), and points and culmination (to win). These distinctions are difficult to capture by atemporal expressions of formal events such as those by Davidson, by similarly motivated event-based calculi, or by purely point-based temporal or action logics. This is not to deny a proper place for events and action in the conceptual frame of an agent, but to emphasize a gap between process and event models. It appears that this gap may be bridged by using the interval nominals of HSN instead of the reified events of Allen. In the HS logic a formula may be true on an interval without being homogeneously true on its subintervals. To enforce homogeneity on stative conditions we can use the modal operator [D] directly or indirectly through a similarly defined hom operator. Indeed, such an operator is a candidate for the underlying representation of the verb to be, once we have decided our representation for the condition indicated by other sentential components like Mary and happy. This use of a temporal modal operator to provide the homogeneity associated with the
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most basic stative verb has an interesting parallel in the well-established use of atemporal modalities to represent stative verbs of propositional attitude such as knows, believes, and desires. With an intervalbased temporal frame these also become homogeneous. Similar verbs of perception like sees, feels, and hears have been less successfully treated without time. Here a sense of process makes homogeneity more contentious. Although in general an activity may be interruptible and decomposable, the progressive aspect is an important case in which an activity is perceived as proceeding or ongoing. For example, is running, the present progressive of run, can be said to require not only an interval model, but either the condition that the activity of running is also true on each subinterval of some interval embracing the current period (in which case the activity is being treated as temporally homogeneous), or else that running can be composed of subactivities such as a sequence of running, resting or walking. So the progressive form of an activity can perhaps be treated by a suitably defined operator, for illustration, prog, where prog p l. áDñ([D]p . p = r x s (áBñ prog r áEñ prog s)). Here ‘x’ is an associative operator denoting sequential, alternative, or parallel composition (and where the last two modes of composition are commutative). Thus this prog operator forces a process to be ongoing, and either homogeneous or else composed of subprocesses, at least one of which is progressing in a beginning and one in an ending subinterval. As the dynamic state of an agent must change at the start and stop points of a homogeneous activity, these become control points for execution. For example, an activity start point operator is defined by Leith17 as [[SP]] p l (áAñ[D] ¬p áAñ[D] p). More sophistication for linguistic and operational purposes can be introduced by defining points where an activity break, resumption, or finish occurs. It thus becomes evident that the use of interval models of time provides a very rich basis for the analysis of processes, since not only can familiar temporal connectives, like after, before, while, and others be given plausible representation, but, rather like the decompo-
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sition of an activity allowed by prog above, noninterval-based models of process composition can be given an interval interpretation. A hazard of such a linguistic perspective of process is the difficulty of generality in language itself, because our own fluid interpretation will depend on context. The works of Moens and Steedman,18 and of Verkuyl,19 systematize much of this fluidity. We have addressed what Quirk et al.20 call the dominant, temporal use of tense, rather than the many literary devices for fictional, presumptive, and anaphoric use. In doing so, we get a glimpse of a formal language sufficiently rich in temporal descriptive power to offer possibilities for multi-agent coordination with controlled semantics. Less obviously, but also pertinent to the logical role of process, is that when treating a declarative sentence as true on an interval, for instance a sentence like John will be puffing after he has been running, we also find that a verifying temporal model can be computed by attention to the way the bounding pairs of points for each interval interleave and associate with logical atoms.21 The sentence may be represented in the HS logic as:
áLñ(prog john_puffs’ áAñ prog john_runs’), where the progressive modality is defined as above. A temporal model is illustrated by figure 21.2. 2.3 Ontological Issues In introducing an interval model of time for tense and aspect, we have used a propositional tense logic rather than a first-order or predicative one. This is not only for presentational reasons, but also because it seems that a multi-modal propositional logic, at least a hybrid one, can represent much more of conventional language than is commonly realized, thereby simplifying the reasoning processes by avoiding unnecessary and sometimes undecidable first-order issues. But in so doing
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we perhaps are introducing ‘mistakes’ of conventional reasoning. A case for hybrid multi-modal logics and correspondences with description logics has been presented elsewhere by Blackburn.22 Here we sketch some of the issues more specific to our concerns. It can be taken as self-evident that a major issue for agent reasoning is the structure of an agent’s knowledge, in particular, the distinctions between the self, the environment, and other agents. However, it is only in recent years that computational agents have been used experimentally and the problems of agent communication addressed in practice. In doing so, the researchers concerned have adopted and extended many concepts from philosophical discourse, including speech-act models of communication and ontological classifications of knowledge. These endeavours have not been without setbacks. In particular the most appropriate ways of classifying knowledge appear to require some revision of analyses such as Parsons’, notably to distinguish persons, or agents with intelligence, from objective and instrumental bodies.23 This in turn has impact on the logical representation, because theories of an agent’s state of knowledge, its perception of the environment, its capacity for action and communication, and its beliefs about other agents can depend on an underlying ontological classification. Thus the way of extending a logical representation like HSN to deal with inter-agent communication and environmental perception is a matter for debate. By way of illustration, we can suppose that a suitable extension of a logic like HSN to deal with a multi-agent environment will include appropriate modalities of propositional attitude and of perception for each agent. Furthermore, since the possible states associated with each agent’s reasoning and the identities of each agent are just as extensional as the intervals of our time domain, we expect to require nominals of each sort for expressiveness. In a classification of an agent’s knowledge, these are instances of special kinds of things, of which there will be many in an environment: man, message, and so forth. In a multi-modal representation a class like man can be treated as determined by relation between the holder of knowledge and a percept of the holder, while a proposition like happy or runs can be associated with an instance of the class determined by some relation. The consequences for logical representation are that we can choose to represent the mortality of all men by a modal necessity like [man]mortal, and the happiness of the nominal Mary of class man by a nominal possibility such as mary.happy rather than their usual predicative forms. While
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mary.happy may seem like a trivial variation of applicative form, [man]mortal is a reminder that natural quantification is relative to the accessible instances of a class. In an executable model one may use such local quantification but have unsound global results. 3. Mentalistic Processes 3.1 How Temporal Are Processes? Although it is a basic working concept of computer science, the idea of a process seems to be less of a concern for metaphysical discourse. However, there are similarities with the supposedly fundamental but unreal temporal A series of McTaggart, at least as discussed in Mellor’s relatively recent treatment.24 Statistical processes have been proposed as a weak basis for causation, for example by Salmon,25 but neither the analysis of time nor that of causation hints at the widespread use of process concepts in engineering and other applied sciences. A cynical explanation would be that while many philosophers are aware of fundamental discoveries in pure science, the more sophisticated constructions of applied science have not been considered to be of explanatory significance unless couched as deeper meta-mathematical discovery. While it is fair to say that the notion of process may not have been illuminating for nineteenth-century mechanical engines, it is central today in fields like control systems and chemical engineering, as well as in computer systems. Process notions are also of growing importance for analysing the metabolic pathways of biological systems. In each case process is inextricably linked with the passage of time, to the extent that linear systems theory is largely built on the use of differential equations in which the rate of change of measurable state with time is central. Standard notations for the symbolic processes of computer science suppress the explicit representation of time through the use of an operator for sequentiality (;), originating in the programming notation of the 1960s. This becomes the implicit temporal interpretation in process logics like dynamic logic26 and the P-calculus.27 These are not intervalbased logics. When an interval-based analysis became appropriate in artificial intelligence, the dominant framework was metalogical use of standard first-order logic with quantified temporal variables, ignoring process logics. An alternative which is possible in HSN is to embed a sequentiality operator:
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p;q l áBñ(p @ np . áAñnq) áEñ(q @ nq . áAñnp), where np and np are nominals. Without nominals the ambiguity of repeated occurrences arises. We are not aware of a notion of a process that is not intrinsically linked either with the passage of time or with physical causality, which itself seems to be an alternative metaphysical foundation for the passage of time. For both human and computational agents the temporal interpretation of a process provides for physical causality, both for internal reasoning processes and for more obviously perceived mechanical actions. Even some details of their interpretations are similar since both brain and computer require energy to run and electrical signals to process information. We may not yet know the details of internal mental processes, but they are certainly not naive sequential processes, and neither, any longer, are computer systems. In each case one signal can initiate another process and through the motor cortex and human muscles, or through integrating amplifiers and transducers, convert the low-energy signals that initiate a process into an effective physical action. For each, a symbolic process logic can provide a fair description of reality, with a chosen degree of temporal granularity, always subject to the constraint that processes are of finite positive duration and bounded energy. 3.2 Tensed Knowledge Representation and Reasoning The use of logics for knowledge representation is well established and computationally effective, where logic programming representations can be exploited, but it is not without issues of contention. Where representations which would be more faithful to epistemic criteria for introspection and communication with belief are desired, modal-style logics can be used, although, as mentioned above, more standard firstorder logics have been used metalogically for temporal interval representations and in order to incorporate ontological structure. None of these considerations addresses the non-ideal recall and memory of the human agent, even though its episodic character is well attested and can be represented through the use of reified events or interval nominals for temporal occurrences. The linguistic evidence for the episodic character of human memory and recall is implicit in the validity of thematic roles in case grammars,28 as exhibited by a reified event style of representation:
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puffingEvent(e) agent(e, John) cause(e, e0) . ... Here the dots indicate a need for more detail on the causal event e0 and perhaps its temporal relation to e. This is made more complicated by an interval analysis. While the case-style analysis and thematic roles seem to be essential for a scalable form of knowledge representation, it is not supported by conventional tense logics. The HSN logic, when syntactically enhanced to represent ontological roles, appears to provide the required scalability though named intervals rather than events. This is supported by the intuition that an event is an occurrence when something happens, a dynamic manifestation of change, for which an interval nominal can provide as much temporal information as is needed. There is much more to say on this topic, and particularly on the distinctions between representations of episodic and non-episodic memory in a human agent. Perhaps the key point is that the representation of episodic recall needs to be dynamic, a resource-limited process. That this may appear in the guise of a belief state about the past is yet another manifestation of the known capacity of dynamic logics to emulate traditional modal logics of propositional attitude. But this is not just a formal phenomenon. In the case of human and computational agents, recovering information does require processing, with its concomitant delays. Unlike episodic knowledge, the relatively unproblematic feature of non-episodic knowledge, even knowledge about the effects of action, is that it is stative. More temporally acute variants of familiar knowledge representation techniques are facilitated by logics like HSN. For example, conventional knowledge about an action a in terms of its preconditions P and post-conditions Q can be expressed temporally by an axiom such as: a l (áOñP l. áOñQ). This is a stronger condition than is possible without intervals. Whether such a representation can better bound the consequences of a (the traditional frame problem), whether known reasoning techniques for intervals can support an effective calculus, and how such a calculus would compare with human reasoning about actions are topics for further study. An interval calculus of events will not be restricted to sequential reasoning; the concurrent activities and communications of many temporal agents should lie within its ambit. 3.3 Higher-Level Mental Processes One of the motivations for this paper was a desire to improve a rather limited but perhaps insightful account of sentient consciousness as a
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progressive form of introspective awareness.29 This can be given logical form in the sort of interval logic we have been discussing, through axioms such as awarej p l. perceivesj p perceivesj perceivesj p consciousj l p. prog awarej p Sentient consciousness is only a fragment of the range of consciousness issues, but it seems to be compatible with both Baars’ global workspace theory30 and some readings of Dennett’s views.31 At present the desire to improve this earlier account remains unfulfilled, but it can serve as a reminder that we can expect mental processes to have tense and aspect too. We can remember previous thought processes and recover from interruption. Perhaps significantly, interruption is more difficult for those processes that we consider logical, as when we are following patterns of calculation associated with formal skills like school arithmetic. Hitherto unique to the human, but now being investigated for artificial agents too, is interaction with the irrational. Nevertheless, as Baars and others have recognized, the mind has features which superficially correspond to the operating systems processes and supervisory workspaces of a computer system, even though its behaviour can be changed emotively to accord with environmental pressures. Today one might claim that the greatest challenges for knowledge processing are not knowledge representation, or reasoning with it (where there has been much computational progress), but the acquisition, discovery, and communication of knowledge. Each gives rise to mental processes. We briefly consider acquisition here, for to the extent that acquisition is learning, it continues to be an active research area of its own. But learning is relevant to temporality in agents because learning is so expensive, both in processing capacity and in the time required. Explicitly logical approaches to learning, like inductive logic programming,32 are exciting because they offer the possibility of acquiring knowledge in a logical form, which enables us to reason about it. But learning itself is a process in which the form of the goal ability – the explication of knowledge or the capability for action – can often be described in advance. The learning process can then become a form of successive approximation, improving detail. So perhaps learning as a mental process can be described by a least-fixed-point recursion of Kozen’s P-calculus, or with a more explicitly temporal extension, where
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the actual training process will vary with the goal. One particular facet of learning is the improvement of existing mental processes. We will have achieved something in the logic of temporal processes when such learning can be better explicated! 4. Integrating Temporal Processes A simple, imperfect, but reactive behaviour in a temporal world may well assist survival if it is quick to compute. But rationality in a human agent appears to need the coherent and potentially costly integration of many processes. A simple sequential integration of a few processes in a cycle seems insufficient. Yet the BDI agents mentioned in section 1 are normally portrayed in just such a manner – as a cyclic sequence: generate plan options to achieve goals, deliberate and filter to choose the intended plan for the current subgoal, partially execute through action, then update beliefs and goals for the next cycle. One can summarize the BDI model as a process to achieve persistent goals, or desires, by deliberation rather than reaction, in particular, by treating previous intentions as partial plans that can more economically be revised than generated afresh when beliefs are updated. Whenever a goal has not already been attained, and persists through time, reasoning about options and plans can determine whether the agent believes the goal can be attained in the future, and by which actions. A now-well-known modal logic of branching time has been evolved to provide a theory for discrete execution.33 In application the simple sequential BDI architecture has been shown to be a robust, although not trivially extendable, scheme. Goals may be meta-goals like survival, or updated or communicated subgoals. Even integration with knowledge acquisition, discovery, and communication is less radical in principle than it might seem to be in practice. For example, learning and discovery can be accommodated by introducing an abductive goal of explicating the environment, with rational introspection to account for unexpected behaviour, and exploratory learning a consequence of a subgoal to remove uncertainty. In an environment with other agents, communicative acts enhance capacity to acquire knowledge and achieve goals. Although no explicit mention of time or interval is needed for this sequential architecture, when we allow durative actions, or indirect ramifications of actions, or concurrent processes in general, it becomes more difficult to distinguish cause from effect because effects can over-
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lap in time with sensor updates to beliefs. Thus a need for temporal reference, coordination, and calculation creeps in. A simple, imperfect, reactive way of reasoning in a temporal world may itself assist survival if it is quick to compute. Only when we admit other agents does it become evident that an agent needs to accommodate concurrent and potentially durative actions to perceive the temporal world, and in order to coordinate through communication. Then it seems we may need interval logics, and names, like dates for the periods we address, and make mistakes in our need for haste. But some of the unresolved issues in realizing the BDI model already have a more principled resolution with an interval-based model. For a start, there is an underlying dilemma in the treatment of intention. Although recognizing intention in action, Bratman avoids issues such as volition, and the impredicative treatment of action in conventional logics, preferring to consider intention as a revocable commitment to a future state.34 But as a result, the status of any commitment to a plan of action becomes problematic in attempts to realize a BDI-style agent. Once processes become first-class predicative entities of our meta-logic for agent design we can hope to avoid the underlying dilemma, indeed, to rethink the BDI model to properly reflect a mental model of concurrent temporal activity. 5. Conclusion By treating language as a window on the mind, and not just as a communication mechanism, and by acknowledging other mentalistic behaviour as processes subject to architectural limitations, such as our episodic and error-prone memory, we provide a challenge for logic and for agent models. Some confusing aspects of human rationality are explicated by the consideration of different logical systems. But can a logic plausibly describe an integrated collection of non-ideal reasoning and non-reasoning processes? Can logic help explain evolutionary phenomena, and design artificial agents, and if so, will it still be logic? The tradition of simplification to an elegant essence, perhaps some classical or non-classical logic, or to a single-minded view of agenthood in terms of beliefs, desires and intents, has analytic attraction that we may wish to retain but an unreality that we cannot ignore. Not only the human mind but also modern systems of information technology exhibit a complex rationality that coexists with a real, temporal world. And yet in virtually no case do we see any representation of the temporal by any logical system. Perhaps the brutal truth is that it
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is difficult to incorporate pyschologistic elements into the meaning of a logic, even temporal elements, and we have settled for less psychology rather than emasculated or ineffective logics and models. But if the traditions of logic and of philosophy are to have more impact on the technological world, this may need to change so that scalable logical systems of great complexity can explain and guide a future world inhabited by rational and temporally aware agents. notes The author wishes to thank the organizers of the John Woods commemorative event for the invitation to participate, and Lloyd Kamara for his critical comments on an early draft of this paper. 1 Dov Gabbay and John Woods, ‘The New Logic,’ Logic Journal of the IGPL 9, no. 2 (2001): 157–90. 2 See Michael Bratman, Intention, Plans, and Practical Reason (Cambridge: Harvard University Press 1987). 3 Immanuel Kant, Critique of Pure Reason, preface to the 2nd ed. trans. Norman Kemp Smith (New York: St Martin’s Press, 1965). Originally Published in 1781. 4 For a more formal treatment of such expressive limitations, see D. Gabbay, I. Hodkinson, and M. Reynolds, Temporal Logic: Mathematical Foundations and Computational Aspects (Oxford: Oxford University Press, 1994). 5 Consider Z. Vendler, ‘Verbs and Times,’ in Linguistics in Philosophy (Ithaca: Cornell University Press, 1967), 97–121; D.R. Dowty, Word Meaning and Montague Grammar (Dordrecht: Kluwer, 1979); P. Blackburn, ‘Tense, Temporal Reference and Tense Logic,’ Journal of Semantics 11 (1994): 83–101. 6 J. Halpern and Y. Shoham, ‘A Propositional Modal Logic of Time Intervals,’ in Proceedings of the Symposium on Logic in Computer Science (New York: I((( Computer Society, 1986), 279–92. 7 J.F. Allen, ‘Maintaining Knowledge about Temporal Intervals,’ Communications of the ACM 26, no. 11 (1983): 832–43. 8 H. Reichenbach, Elements of Symbolic Logic (London: Macmillan, 1967). 9 M. Leith and J. Cunningham, ‘Aspect and Interval Tense Logic,’ Linguistics and Philosophy 24, no. 3 (2001): 331–81. 10 Y. Venema, ‘Expressiveness and Completeness of an Interval Temporal Logic,’ Notre Dame Journal of Formal Logic 31, no. 4 (1990): 529–47. 11 Hybrid Logics are derived from A. Prior, in Past, Present and Future (Oxford: Clarendon Press, 1967). For an introduction, see P. Blackburn,
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13 14 15 16 17 18 19 20 21 22 23 24 25
26 27 28 29 30
‘Representation, Reasoning, and Relational Structures: A Hybrid Logic Manifesto,’ Logic Journal of the IGPL 8, no. 3 (2000): 339–625. D. Davidson, Essays on Actions and Events (Oxford: Oxford University Press, 1980). T. Parsons, Events in the Semantics of English: A Study in Subatomic Semantics (Cambridge, MA: MIT Press, 1990). J.F. Allen, and G. Ferguson, ‘Actions and Events in Interval Temporal Logic,’ Journal of Logic and Computation 4, no. 5 (1994): 531–79. R. Quirk, S. Greenbaum, G. Leech, J. Svartvik, A Comprehensive Grammar of the English Language (London and New York: Longman Group, 1985). Z. Vendler, ‘Verbs and Times.’ M. Moens and M. Steedman, ‘Temporal Ontology and Temporal Reference,’ Computational Linguistics 14 (1988): 15–28. M.F. Leith, ‘Modelling Linguistic Events’ (PhD thesis, Imperial College, University of London, 1997). Moens and Steedman, ‘Temporal Ontology and Temporal Reference.’ H. Verkuyl, A Theory of Aspectuality (Cambridge: Cambridge University Press, 1993). R. Quirk, S. Greenbaum, G. Leech, J. Svartvik, A Comprehensive Grammar of the English Language (London and New York: Longman Group, 1985). M.F. Leith and J. Cunningham, ‘Aspect and Interval Tense Logic,’ Linguistics and Philosophy 24, no. 3 (2001): 331–81. P. Blackburn, ‘Representation, Reasoning, and Relational Structures: A Hybrid Logic Manifesto,’ Logic Journal of the IGPL 8, no. 3 (2000): 339–625. See L.N. Schneider, ‘Naive Metaphysics’ (MSc thesis, Imperial College, University of London, 2001). D.H. Mellor, Real Time (Cambridge and New York: Cambridge University Press, 1981, 1985). W.C. Salmon, ‘Causality: Production and Propagation,’ in Causation, E. Sosa and M. Tooley eds. (Oxford and New York: Oxford University Press, 1993). D. Harel, ‘Dynamic Logic,’ Handbook of Philosophical Logic, vol. 2 (Dordrecht and Boston: Reidel, 1984), 497–604. D. Kozen, ‘Results on the Propositional µ-calculus,’ Theoretical Computer Science 27 (1983): 333–54. C.J. Fillmore, ‘The Case for Case,’ in Universals in Linguistic Theory, E. Bach and R.T. Harms, eds. (New York: Holt, Rhinehart and Winston, 1968), 1–88. J. Cunningham, ‘Towards an Axiomatic Theory of Consciousness,’ Logic Journal of the IGPL 9, no. 2 (2001): 341–7. B.J. Baars, In the Theatre of Consciousness (Oxford and New York: Oxford University Press, 1997).
Temporal Agents 397 31 D. Dennett, Consciousness Explained (New York: Penguin, 1991). 32 S. Muggleton, ‘Inductive Logic Programming,’ New Generation Computing 8, no. 4 (1991): 295–318. 33 A.S. Rao and M.P. Georgeff, ‘BDI Agents: From Theory to Practice,’ Proceedings of ICMAS-95 (San Francisco, CA: ICMAS, 1995), 312–19. 34 Bratman, Intention, Plans, and Practical Reason.
22 Filtration Structures and the Cut Down Problem for Abduction DOV M. GABBAY AND JOHN WOODS
It is a primary hypothesis underlying all abduction that the human mind is akin to the truth in the sense that in a finite number of guesses it will light upon the correct hypothesis. Peirce, Collected Works
The surprising fact C is observed. But if A were true, C would be a matter of course. Hence there is reason to suspect that A is true. Peirce, Collected Works
1. Abduction In its most basic intuitive sense, abduction is a process of reasoning in which a target which cannot be hit with existing information is judged hittable upon the assumption of additional facts. It is then concluded that this is reason to take that assumption itself as at least a candidate for the status of fact. We want to say something here about the logical structure of such reasoning. We want, more particularly, to expose something of the logical structure of abductive reasoning when it is transacted by a practical agent. The present chapter draws upon our recent book, The Reach of Abduction: Insight and Trial.1 This chapter is a comparatively early attempt to acquaint the relevant research communities with our approach to abductive reasoning. As befits such an enterprise, this work is in a number of respects programmatic, indeed something of a promissory note. We have tried to execute the program and redeem the promise in The Reach of Abduction.
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Before we turn to the main task, some preliminaries are required. In the section to follow we say something about practical agency, and in the section after we shall say what we take to be the basic structure of an abduction problem. 2. PLCS The Reach of Abduction is part of a multi-volume project entitled A Practical Logic of Cognitive Systems (PLCS), to which it might be helpful to orient the reader. In the interest of space, we offer in this section the briefest of sketches.2 A practical logic of cognitive systems can be conceived of in the following way: – a logic is a principled description of (various aspects) of the behaviour of a cognitive system – a cognitive system is a triple of an agent C, cognitive resources R, and cognitive tasks T performed in real time – a cognitive agent is an information-processing device capable of belief, inference, and decision – a cognitive agent is an agent of a certain type, depending on where he, she, or it sits under the partial order ‘commands greater resources than’ – such resources include information, time, and computational capacity – a cognitive agent is a practical agent to the extent that it ranks low in this partial ordering – accordingly, practical reasoning is the reasoning of a practical agent – a cognitive agent is a theoretical agent to the extent that it sits high in this same partial ordering – accordingly, theoretical reasoning is the reasoning done by theoretical agents – practical agents include individuals – theoretical agents include institutions – it cannot in general be supposed that practical and theoretical reasoning are subject to the same performance standards – from the point of view of theoretical agency, practical reasoners operate with scarce-resources – accordingly, we postulate for practical reasoners scarce-resource compensation strategies
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– these include or involve (among other things) • a propensity for hasty generalization • a facility with generic inference • easy discernment of natural kinds • a propensity for default reasoning • a disposition toward belief-update and discourse economies • unconscious cognition The last of these is especially interesting. There is considerable evidence that a good deal of our cognitive effort is discharged unconsciously or, as we shall say, down below. Down below reasoning need not be linear in the way that reasoning up above is. It is natural to suppose that down below reasoning is highly efficient; it thus presents itself as an advantage in the cognitive economies of practical agents. This leaves us with a pressing question. Should we expect reasoning down below to be the proper province of a logic such as PLCS? Here, too, we state our position without argument (and with appropriate tentativeness), but not before taking note of an interesting point. It is that there is psychological evidence to suggest that the following pairs of contrasts are not strictly equivalent.3 1. 2. 3. 4. 5. 6. 7.
conscious versus unconscious processing controlled versus automatic processing attentive versus inattentive processing voluntary versus involuntary processing linguistic versus nonlinguistic processing semantic versus non-semantic processing surface versus depth processing
Why is this interesting? It is because certain aspects of reasoning seem to be unconscious, inattentive, non-linguistic, involuntary, and automatic. To the extent that these descriptions are not equivalent, reasoning of this sort has a rich and complex nature. What, then, might a logic of such reasoning be like? Here are two possibilities to consider: 1. Connectionist logics. Prototypical reasoning is modelled in the patterning of weights in the subject’s neural network. Thus inference to the best explanation is simply activation of the most appropriate prototype vector. This is made possible by the fact that we possess
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an organized library of internal representations of situations to which prototypical behaviours are the computed output of a welltrained neural network. This kind of reasoning is a matter of prototype activation by way of vector coding and vector-to-vector transformation. It is not a matter of linguistic representation governed by standardly logical reasoning.4 2. The Representation Without Rules (RWR) Approach to Cognitive Modelling. Cognitive systems employ representational structures that admit of semantic interpretation; yet there are no representationlevel rules that govern the processing of these representations. Connective nets are amenable to this approach.5 We want to close this prefatory section with a methodological point. Consider, first, what we call the Can Do Principle: If an investigator is working on a problem in a domain of enquiry D, and if results that are achievable in another domain D* bear favourably on the first task, the D-theorist would do well to help achieve those results in D*. It is a good principle, liberally and manifestly at work in much of our best science. It is a fallible principle. We cannot always know in advance that the work done in D* will indeed facilitate the work to be done in D. On occasion, it will become clear to the investigator that his results in D* don’t help in D. For example, an economist may discover that the mathematics of D* is indiscernible in the behaviour of the actual economic entities that are the subject of D’s investigation. Or a decision theorist may discover that theorems of the probability calculus are little evident in the business of real-life probabilistic reasoning. Consider now a certain range of such cases. They are cases in which the theorist doesn’t know how the results that he has very successfully produced in D* facilitate the production of results that actually get the work in D done. In such cases, he may behave in ways that conform (perhaps tacitly) to what we call the Make Do Principle, which is a corruption of the Can Do Principle. It provides that the inquirer persist with his efforts in D* simply because it is work that he knows how to do and that he attempts to defend his doing such work in one or other of three ways: 1. Brazen it out. Insist without any attempt at demonstration on the applicability of the D*-work to his D-tasks.
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2. Adopt a variation of Keynes’ bluff.6 Insist that the D* results aren’t really required to conform to or elucidate the phenomena of D. 3. Seek refuge in ideal normative models.7 Our purpose in raising these matters here is purely cautionary. A PLCS is not a mainstream approach to logic, even though any logician working on a PLCS will have cut his teeth on the chewy bark of mainstream logic. There are ways of proceeding that are intrinsic to mainstream logic. They are attractive in two ways. One is that they work for mainstream logic. The other is that we know how they work – they are familiar. The admonition we sound is that in the unreflective application of these familiar procedures to a PLCS there is risk of Make Do. We also note in passing that if the perspective of traditional fallacy theory held true, then practical agency would be shot through with – indeed awash in – fallaciousness. Such a consequence is too Calvinist for our tastes, and we propose that a piece of reasoning or an episode of argumentative exchange is or is not fallacious according to the agent’s degree of command of the requisite cognitive resources. So it is far from obvious that the scarce-resource compensation strategies of practical agents are fallacious in a significant way. 8 3. The Basic Structure of an Abduction Problem In its most basic form, abductive reasoning instantiates the following form, in which T is a target, so is a consequence relation, H a hypothesis, and K background knowledge: Abductive Schema 1. T 2. (K so T) 3. K {H} so T 4. ? H It is easy to see that the Abductive Schema charts a characteristic course in the broader context of belief dynamics.9 For present purposes, a simplified sketch will suffice. In some accounts of abduction it is also required that * (H so T)
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While * is a condition that is frequently satisfied in actual abductive practice, we judge it inadvisable to make it a necessary condition. It is also commonly proposed that T is always a statement that records the existence of some or other phenomenon or state of affairs. In such cases, the abducer seeks for a hypothesis which will assist in accounting for or explaining that phenomenon or state of affairs. This is accomplished by finding a sentence H and a consequence relation so such that the sentence T reporting that phenomenon or states of affairs is in the counter-domain of the so -relation. In other cases, the state of affairs that triggers the abduction is not the state of affairs that the abducer seeks to prove. This happens when the triggering state of affairs is the absence of something the abducer would like to see realized. A case in point is Planck’s quantal conjecture. The trigger was the fact that black body radiation lacked unified laws, but Planck’s target was not to infer that this lack exists but rather to produce a state of affairs in which it would be eliminated. The T in Planck’s abduction therefore was the complex of sentences that expressed the new (and tentative) laws of black body radiation. The Planck example motivates us to distinguish between the state of affairs that triggers an abduction and the state of affairs toward which the abduction is targeted. Often the two coincide, but it is far from necessary that they do. It should also be noted that the habit of reading the so -symbol as ‘proves,’ common in the computer science and AI literature, is something of an idiosyncrasy. In some cases, what the abducer aims at is indeed a hypothesis H that will assist in a strict proof of a target T; but there are cases galore in which the notion of consequence is less strict. This is true for certain conceptions of explanation and prediction, for example, in which what the abducer aims for is an explanation of T from K {H}, or a prediction of T from it. Even so, there is no harm in reading so as ‘proves’ provided that this variability is borne in mind. In a similar spirit, we must also recognize a like variability for the conclusion-symbol ?. In some cases, the abducer undertakes to find an H and a so -relation such that H is a strict consequence of the schema’s premisses. But in other cases, it is abductively sufficient that the derivation of H be one that satisfies less strict standards of probability or plausibility and the like. Here, too, it is permissible to construe the terminal line of our schema as a derivation of it from prior lines, so long as the variability of the conclusion-operator is not forgotten. For those minded to think that abduction is the sort of reasoning for which the idea of a logic of abduction is a defensible idea, it is entirely
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straightforward that the abductive logician will have at least the following two tasks to perform: (a) He must give an account, in all its variability, of the consequence relation so. (b) He must give an account, in all its variability, of the conclusion operator ?. There is an example of a particular issue that these sublogics should take heed of. It would be helpful to know whether there exists a deduction theorem for so and ?. If so, whenever ) so < holds, so does )?< hold for some suitable interpretation of ?. The question is whether this interpretation of ? always is the interpretation to be accorded to the ?-sign in line (4) of the Abductive Schema. Clearly the answer is No. It also seems reasonable to suppose that in making his abductions a cognitive agent selects his H from a space of possibilities å. It is also rather plausible that when H has been correctly or judiciously or defensibly selected from å, H will stand to å is one or more relations that fall within the logician’s ambit to describe. If, as we ourselves believe, this is so, then in addition to a sublogic (a) of consequence and a sublogic (b) of derivation, the abductive logician has obligations in (c) a sublogic of hypothesis-selection, which itself would appear to subsume a process of hypothesis-generation and a process of hypothesis-deployment or engagement. Here is an example. In large numbers of actual cases, there exist indeterminately many candidates for eventual selection as the abducer’s hypothesis H. Let us call these possible hypotheses for the abduction problem at hand.10 Typically å, the space of possible hypotheses, has a daunting cardinality. In pinning his hopes on some particular member H of å the abducer is in effect rejecting up to indefinitely many rival hypotheses. We shall say that in selecting H from å, the abducer solves a Cut Down Problem. For large ranges of cases, it is reasonable to suppose that the abducer is aided in these exclusions, that is, in his solution of the Cut Down Problem, by considerations of irrelevance and implausibility. We may take it, then, that a task of the sublogic of hypothesis selection is to give an account of the byplay of such factors in spaces of such possibilities. Virtually everyone who writes about abduction recognizes that abductive reasoning involves in a fundamental way the generation and engagement of hypotheses. What is not as uniformly acknowledged is, again for large ranges of cases, that the deployment of hypotheses is
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typically a two-phase process. In phase one, H is forwarded hypothetically rather than categorically. This factor should not be confused with the fallibility that attends all empirical knowledge. A proposition forwarded hypothetically is one assumed to be the case; a proposition forwarded categorically is one asserted to be the case. If fallibilism is true, it is true across the grain of this distinction; for even though I assert a proposition (and am justified in doing so) I may consistently recognize that future developments could require me to withdraw it. Abducers frequently categorialize propositions to which they formerly gave hypothetical expression. When this happens we will say that the abducer’s hypothesis H has been discharged. This may suggest a fourth sublogic, at least for those inclined to think that abduction has a logic. We may take it as given that when an abducer discharges a hypothesis H, then there are conditions C that H used to satisfy but no longer does, and conditions C* that H used not to satisfy but now does. If it might be supposed that C and C* include conditions whose description lie within the ambit of a logician’s interest and competence, the postulation of a sublogic of hypothesis discharge is entirely reasonable. Even when a substantial consensus about the identity of the abductive theorist’s fourfold task is assumed, there is considerable disagreement as to how much of it actually does fall within the province of logic. Task (3) is especially contentious. If thought of as appropriate matters for the logician, the sublogics of generation and engagement would constitute a demand for a logic of discovery, which many a commentator, from Reichenbach11 on, regard as indefensible poaching in the proper domain of psychology. We will not here take up this debate, but will declare without argument our sympathy for the following view of the matter. The contrast between logic and psychology is neither exclusive nor precise.12 Since both logic and psychology are concerned with how we think, the two disciplines share a common ground. What makes their respective approaches to the investigation of that common ground appropriately different is that the theorist of the respective disciplines brings to the inquiry what interests him and what he is good at. A philosophically trained logician will bring to the fray a facility with conceptual analysis. A mathematically trained logician will deploy a capacity for formal modelling. A cognitive psychologist will offer his talents in the design of experiments. So, once the salient connections with psychology are given due attention and weight, we accept on sufferance the suggestion that the obligations of what we have been calling a sublogic of hypothesis-selection is dischargeable in
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a logic of discovery. In fact, this chapter is offered as a modest contribution to just such a logic. We briefly revisit the question of the logic of discovery in section 6 below.13 4. The Cut Down Problem It is perhaps only meet and just that the logic of abduction is itself a highly abductive enterprise. As we have said, it may be that, in reaching an abductive target, the abductive reasoner solves what we call the Cut Down Problem. But this is conjecture. It is a case of hypothesis selection with respect to the fact that abductions are actually made. What the abductive agent aims for in the selection sublogic is specification of a space å of possible hypotheses. Let ) be a wff in å. Then ) plays the role of an H. That is, ) is such that there is a requisite so and ? for which the Abductive Schema holds. It is this fact that makes ) a possible candidate for actual deployment. We see in this the necessity to characterize with due care the notion of hypothesis and conjecture as they operate in the generation logic. Rather than say that when ) meets the present conditions that it is conjectured outright as a fully fledged hypothesis by the abductive agent, better to say that the agent has identified as a candidate for such deployment, that is, as a possible hypothesis for possible conjecture. The space å of such possibilities can be seen, on the present assumption, as input to the next step in the abduction process. Selection is a way of activating (some of) the possibilities in å. Here we might suppose that the activation condition is one of relevance. Relevance we can think of as a filter that takes å into a proper subset R of relevant possibilities. Thus the relevance filter seems to cut down the space å to the smaller space R. The relevance filter plays a role similar to Harman’s Clutter Avoidance Principle, which bids the reasoner not to clutter up his mind with trivialities.14 In its role here, the relevance filter enables the abducer to concentrate on possibilities that are fewer than those found in å, and make a better claim for deployment. R, the space of relevant possibilities, advances subsets of å closer to the point of activation or engagement. A second task is to bring hypothesis-activation off, to replace possible hypotheses that are relevant conjectural possibilities with hypotheses that the abducer actually deploys by actual conjecture. This, too, involves the passing of larger sets through a contraction filter. The rel-
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evance filter took possibilities into relevant possibilities. What is wanted now is a filter that will take relevant possibilities into plausibilities (a cut down of R to P). The plausibility screen accordingly shrinks R to subset P. P in turn presents the abducer with three activation åoptions. Assuming P to be non-empty, 1. If P is a unit set of R, then select the hypothesis in P. 2. If P is a pair set of R or larger, then select all the hypotheses in P. 3. If P is a pair set of R or larger, then select the most plausible hypothesis M in P. It is well to note that the filter that takes å into a subset R of relevant possibilities can be taken either as a theory of relevance or as judgments about relevance made by the abducer. Seen the first way, R exists when conditions sanctioned by the theory are satisfied, irrespective of what judgments of relevance, if any, are made by the abducer and irrespective of whether they are correct. Taken in the second way, this independence is, of course, reversed. (We shall return to this point below.) As an aid to exposition we can now use the letters P, R as names of the presumed filters and as the names of the resultant sets. The same is true of M. It names either the order on P or the resultant set. We may now say that the triple áå, R, Pñ represents a filtration structure on an initial space of possibilities in which the succeeding spaces are cut downs on their predecessors. It is possible that there also exists a further filter M on the space of plausibilities to a unit subset of (intuitively) the most plausible candidate for engagement/discharge. We leave the further discussion of the logic of discharge for another time. It is important to emphasize that áå, R, Pñ and, where it exists, áå, R, P, Mñ, exists independently of whether any abductive agent has ever thought of it. The filters that cut sets down to smaller sets at each stage do so independently of whether anyone has actually tried to deploy those filters. In the speculation that we have been entertaining these past few pages we have supposed that in reaching his desired H, the abductive reasoner proceeds by solving the cut down problem by constructing (or reconstructing) the requisite filtration structure. In so supposing we have it that the would-be engager of H proceeds in a topdown fashion, first by entertaining mere possibilities, then cutting to relevant possibilities, and finally to the most plausible of these. Let us be clear about the present point. å is a structure of propositions. We already have it that å satisfies the consequence constraint;
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that is, each element in å is part of the domain of the so -relation with regard to some abductive target T. We assume that there is a theory of relevance that cuts å down to R, a proper subset of those propositions. We suppose further that there is a theory of plausibility that cuts R down to P, a proper subset of plausible and relevant propositions. And we suppose, finally, that there is an order on P that, sometimes, picks M the most plausible candidate in P. There is reason to think that for any given å, such a filtration structure exists. If so, for any å there is at worst a subset of most plausible members of it. Sometimes, this is a unit set M. Given that what an abductive agent seeks to do is to choose the most plausible hypothesis H, there is reason to think that for any such H there exists a filtration structure whose component P contains H, and whose component M, if there is one, just is {H}. On the current assumptions, if the H which the abducer seeks exists, then there exists a filtration structure in which H exists and in which it has a determinate occurrence. This it owes to the relevance theory, the plausibility theory, and the partial order that takes å to M. These are the devices that locate H in the filtration structure. So why not suppose that, since the abducer’s task is also to locate H, he does so by deploying on å successively the relevant logic, the plausibility logic, and the partial order, that is, by making judgments that (he hopes) would be sanctioned by these respective theories? If this were a tenable supposition, then the tasks of our first two sublogics of abduction – the generation and engagement sublogics – could be said to be achieved by the logics of relevance and plausibility, supplemented by the required partial order; in short, by constructing the requisite filtration structure. 5. How Is It Done? On our present conjecture, an abductive agent solves an abduction problem by solving the Cut Down Problem; and he solves the cut down problem by constructing the requisite filtration structure. How reasonable are these conjectures? This is an abduction problem for us and for anyone else who might be drawn to our question. We take it as evident that if it could be shown that our hypothesis is the best explanation of the fact that abductive agents have considerable success in solving abductive problems, we should have made considerable, perhaps decisive, headway with our abduction problem. But would it have been the case that this success was brought off by our constructing the requisite filtration structure áå, R, P, Mñ? Let us see.
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How might we suppose the abducer gets from å to R? Let the * be the set of all consistent subsets 4 of å such that for some subset ' of K, ' 4 is consistent and ' 4 so T irredundantly. Then * may be identified with R, the set of relevant possibilities in å. In so saying, we identify relevance with irredundancy. The members of *= R are those sentences or sets of sentences which, together with subsets of what is already known, irredundantly prove an abduction target T. Sets of sentences that meet these conditions may be seen as constituting an irredundancy-minimalization of å. R is clearly such a set. It contains all those wffs or sets of wffs that prove T using the fewest possible resources from å and K. If an abducer were able to construct an R from a å, we could say that he could solve the relevance phase of the Cut Down Problem. We should note that the irredundancy constraint, while it bears some resemblance to relevance in the sense of Anderson and Belnap, actually originates with Aristotle as a condition on syllogisms. We should also point out that the concept of irredundancy (or strong relevance, as we might also call it) has not yet been adequately formalized.15 How then to proceed from R, the set of relevant possibilities, to P, the set of relevant plausibilities? Here we must also offer a promissory note. Plausibility logics are hardly thick on the ground, despite some promising pioneering work by Rescher.16 For present purposes it suffices to conjecture as follows. The desired set P stands in relation to the larger set R at least some of which would be captured by a satisfactory plausibility logic PL. Since the abductive agent manages to get P somehow, some will think that he does so by implementing the appropriate parts of PL. Since full PL has been only promised rather than offered, we might conjecture further that PL is part of the logic of ‘down below,’ briefly discussed in section 2. Where P is not a unit set, it might also be proposed that the abductive agent imposes the partial order that gives M, often in the same tacit way. 5.1 Empirical Discouragements Well, is this how it is done? It would appear not. Everything that is known empirically of actual human agency suggests that this is not the structure of the abducer’s reasoning; for one thing, it would involve the searches of spaces that are computationally beyond the reach of such agents. This can hardly be a surprising finding, given what is known of the fast and frugal character of practical reasoning. So, while it is perfectly reasonable to suppose that hypothesis-engagement
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involves considerations of relevant plausibility, how it does is not all that clear. Of course, it could be true that for every H that solves an abduction problem there is a filtration structure in which H occupies a unique point or region. If this were so, every winning H would satisfy requisite conditions on possibility, relevance, and plausibility. It could be supposed that it is precisely in this sense that factors of possibility, relevance, and plausibility are intrinsic to the solutions of abduction problems. Since sets that are structured in the way that filtrations provide are indeed sequences of cut downs of prior structures, it may also be said that intrinsic to the solution of abduction problems is the existence of structures that produce the requisite cut downs. But it is worth repeating that it needn’t also be true (and evidently isn’t) that the abductive reasoner homes in on his final H by making the corresponding judgments of possibility, relevance, plausibility, and (when possible) greatest plausibility; still less that they be judgments made in that order. What this suggests is that successful reasoning is reasoning that occurs when certain conditions are met, but that when reasoning successfully the reasoner is not (always) behaving in ways that meet them. This raises an extremely interesting question. We take it to be a question of fundamental importance to the logic of discovery. Given that any such H will occupy a unique point or region in a filtration structure, given that any such H will satisfy requisite conditions on the possibility, relevance, and plausibility of hypotheses, how is it that a reasoner in homing in on such an H does do without having made judgments of possibility, relevance, and plausibility, and (possibly) in utter ignorance of the three factors that are intrinsic to any proposition that the abducer is justified in selecting as his abduction-solving H? This is not a question that is unique to abductive reasoning. Causal reasoning is another such case, and so is probabilistic reasoning. In each case, the reasoner may proceed successfully without having any idea of, hence without making any judgments about, the conditions that are intrinsic to his success. This bears in an attractive way on the very idea of a logic of discovery. Do we think that because of the disparity between what the causal logician is able to determine as conditions on correct causal reasoning and the actual reasoner’s own unawareness of those conditions, there can be no such thing as a causal logic and that all of causal reasoning must be handed over to cognitive psychology? If not, why is the same not true for the logic of discovery? This is a rhetorical question, of course. In pressing it we mean to be
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saying that the fact that, for the wide range of cases we have been considering, every winning H has a unique location in a filtration structure makes a contribution to the logic of discovery. For it allows us to say that, however it is done in fine, if an actual abducer picks an H that lacks such a place in a filtration structure, he will have picked the wrong H. But, in saying this, we don’t mean to suggest that this exhausts the logician’s contribution to the logic of discovery. Clearly a further necessary task is to elucidate with some rigour the very ideas of possibility, relevance, and plausibility which give rise to the required filters. Once this is done, the abductive theorist may have occasion for further conjectures as to how the abducer manages to hit the possibility, relevance, and plausibility targets without being aware of doing so.17 6. Discovery Logics Revisited We here return to the question of whether it is justified to suppose that logic has something illuminating to say about the logics of generation and engagement. But first a brief word about the relationship between the hypotheses generated for a given abductive problem and the hypotheses that abducers actually engage. Since it is often the case that more hypotheses are generated than engaged – that is, engaged hypotheses are often proper subsets of generated ones – it might seem reasonable to endow the subset relation with chronological significance. When we examine the empirical record (such as it is, and such as we can), there is little reason to doubt that sometimes this chronological assumption is correct; that is, for lots of cases of abductive resolution, hypotheses are entertained before (and more numerously) than they are adopted. But it is also a fact of actual practice that sometimes the abducer goes directly to the hypotheses that he or she engages, and often automatically, inattentively, and without effort. This, we might say, is cut-to-the-chase abduction. It will also be helpful to remind ourselves that our approach to logic is preoccupied with the kinds and quantities of cognitive resources available to agents of the type in question, and that the concept of rationality that we assume for practical or individual agents is that of bounded rationality. Seen our way, logic is a disciplined description of how, with regard to certain aspects of their cognitive agendas, beings who are boundedly rational employ their limited cognitive resources. As we have said, important examples of such resources are information, time, and computational expectation. To these we now add a
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fourth: modes and methods of inquiry. What we are going to try to show in this section is that for certain ranges of cases, conceiving a theoretical entity is often a function of what the agents’ mode and method of inquiry predispose him or her to think of. One particularly impressive example is the attempt to model discovery with machine-learning programs such as BACON, which attempts to extract scientific laws inductively from data.18 But on another approach, discovery is a function of methods that produce and process data, rather than the data themselves. (This pair of approaches to discovery is loosely congruent with the split between inductivists and non-inductivists concerning the theory-ladenness of observational terms.) One example of the modes and methods route to discovery has to do with the role of statistical inference and theory testing that has dominated the social sciences since the 1960s, applied to the study of cognitive processes. This development was a sequel to the ‘institutionalization of inferential statistics in American experimental psychology between 1940 and 1955.’19 Under the influence of such methods, theories of cognition were cleansed of terms such as restructuring and insight, and the new mind has come to be portrayed as drawing random samples from nervous fibers, computing probabilities, calculating analyses of variance, setting decision criteria, and performing utility analyses.20
As the inferential statistics became canonical for investigations in the social sciences, cognitive processes of various kinds were assumed to embody a kind of ‘intuitive statistics,’ and cognitive agents were viewed as executing its rules in the course of their cognitive engagement. Gigerenzer cites the following two examples. First Example: Beings like us have the capacity to distinguish between the presence of a signal and the presence of mere noise. According to Tanner and Swets21 this is done in the same way that an investigator employs Neyman-Pearson statistical methods to adjudicate between two hypotheses.22 Neyman-Pearson techniques eventually evolved into a more general theory of cognition, including memory recognition,23 eyewitness testimony,24 and discrimination between random and nonrandom patterns.25 Second Example: When an agent attributes a cause to an effect, it is proposed by Kelley26 that the cognitive agent (or his brain) brings this off by producing an analysis of variance and testing null hypotheses.27
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The significance of these examples lies not in the claim that the cognitive agent reacts to the cited data in the manner of intuitive statisticians, but rather in the claim that the agent as intuitive statistician reconceptualized the very data theories of cognition sought to take note of. As Gigerenzer sees it, the construction of cognitive theories based on the assumption of the cognizer as intuitive statistician ‘radically changed the kind of phenomena reported, the kind of explanation looked for, and even the kind of data that were generated.’28 What is more, researchers who adopted the methods of inferential statistics were unaware of this change, since these methods had become canonical in psychology. If Gigerenzer is right, then a change in psychological methodology from that of inferential statistics to the methods of bounded rationality should likewise change the kind of phenomena reported, the kind of explanation sought, and the kind of data generated. We find it rather striking that this bounded rationality analysis dovetails with three matters that we have already touched on in Seductions and Shortcuts: 1. First is the idea in Hanson29 and Popper,30 and among critics of inductivism generally, that observational data are theory-laden. This being so (if it is so), observational data have conceptualizations that bias what will count as candidates for the underlying theoretical explainers or other abductive targets. What is important about Gigerenzer’s analysis is that it points out that a theorist’s conception of an appropriate investigative methodology can itself conceptualize the data in particular ways. So if you think that proper investigation requires that you see a practical agent as an intuitive statistician, then there is some likelihood that you will conceptualize the agent’s behaviour accordingly. 2. A second point of connection is our own Can Do Principle and its degenerative instance, the Make Do Principle, both discussed in section 2 above. Why, we asked, do researchers sometimes persist in using methods that seem inapplicable to the targets of their main inquiry? Our suggestion is that, in lots of cases, the researcher knows how to do the work that he actually ends up doing and does not know how to do the work that he has set out to do. (So he pretends that there is a connection, or that there doesn’t need to be one. Or he resorts to normatively ideal models.) The particular importance of the bounded rationality analysis is that it gives a further
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explanation of why investigators persist with methods inappropriate to their theoretical objectives. Their answer is that these methods have become canonical for these fields of inquiry. So, if you are obliged by the field’s conception of what constitutes acceptable scientific practice to see cognitive agents as intuitive statisticians, then you can’t help seeing the behaviour of such agents in that kind of way. And how you see the data will naturally influence your choice of hypotheses that best account for them. 3. The idea that cognitive agents (or their brains) are embodiments of Neyman-Pearson statistical models or of ANOVA models is of a piece with the idea that human reasoners execute first-order logic or the calculus of probability. In all these cases, the empirical evidence amply attests that beings like us do not execute these programs; hence the funny business mentioned in the discussion of normativity in section 2 above. What is interesting about the bounded-rationality analysis is that if we get the investigative tools right, we will conceptualize the data more appropriately and, in turn, be led to more attractive hypotheses to account for them. So, then, what lessons might discovery logicians learn from this discussion? They learn the following: • Method influences the conceptualization of data. • How the data are conceptualized influences what counts as abductively adequate responses to them. notes We are grateful for helpful comments from Erik Krabbe, Theo Kuipers, Jeanne Peijnenburg, David Atkinson, Jan-Willem Romeyn, Jan Albert van Laar, Atocha Aliseda, Alexander van den Bosch, Menno Rol, Sven Ove Hansson, Hans Rott, Tom Mailbaum, Alice ter Meulen, Howard Barringer and Peter McBurney. 1 Dov M. Gabbay and John Woods, The Reach of Abduction: Insight and Trial, vol. 2 of A Practical Logic of Cognitive Systems (Amsterdam: North Holland, 2005). 2 Fuller treatments may be found in Dov M. Gabbay and John Woods, ‘The New Logic,’ Logic Journal of the IGPL 9 (2001): 157–90, and Dov M. Gabbay et al., eds., Handbook of the Logic of Argument and Inference: The Turn toward the Practical, vol. 1 of Studies in Logic and Practical Reasoning (Amsterdam: North-Holland, 2002).
Filtration Structures and the Cut Down Problem for Abduction 415 3 R.M. Shiffrin, ‘Attention, Automatism and Consciousness,’ in Scientific Approaches to Consciousness, Jonathan D. Cohen and Jonathan W. Schooler, eds. (Mahwah, NJ: Erlbaum, 1997), 49–64, 62. 4 Paul Churchland, A Neurocomputational Perspective: The Nature of Mind and the Structure of Science (Cambridge, MA: MIT Press, 1989), 189–207. 5 T. Horgan and J. Tienson, ‘Representations without Rules,’ Philosophical Topics 17 (1989): 147–74. 6 ‘Progress in economics consists almost entirely in a progressive improvement in the choice of models ... But it is of the essence of a model that one does not fill in real values for the variable functions. To do so would make it useless as a model ... The object of statistical study is not so much to fill in missing variables with a view to prediction, as to test the relevance and validity of the model.’ John Maynard Keynes, ‘Letter to R.F. Harrod, 4 July 1938,’ Collected Writings of John Maynard Keynes: The General Theory and After: Part II: Defense and Development, D.E. Moggridge ed. (London: Macmillan, 1973), 296. 7 A manoeuvre we seek to discourage in Dov M. Gabbay and John Woods, ‘Normative Models of Rational Agency,’ Logic Journal of the IGPL 11 (2003): 597–613. 8 This proposal is worked out in greater detail in Gabbay et al., Handbook, ch. 1, and is the principal motivation of our Seductions and Shortcuts: Fallacies in the Cognitive Economy, vol. 3 of A Practical Logic of Cognitive Systems (Amsterdam: North-Holland, forthcoming). 9 A full account of the structure of abduction would describe in requisite detail its belief-dynamical character, concerning the latter of which see Sven Ove Hansson, A Textbook of Belief Dynamics: Theory Change and Database Updating, vol. 11 of Applied Logic Series (Dordrecht: Kluwer, 1999); Hans Rott, Change, Choice and Inference: A Study of Belief Revisions and Nonmonotonic Reasoning, vol. 42 of Oxford Logic Guides (Oxford: Oxford University Press, 2001); and Dov M. Gabbay, Gabriella Pigozzi, and John Woods, ‘Controlled Revision,’ Journal of Logic and Computation 13 (2003): 5–27. For an account of abduction that makes some of these connections, see Atocha Aliseda-Llera, ‘Seeking Explanations: Abduction in Logic, Philosophy of Science and Artificial Intelligence’ (PhD dissertation, Amsterdam, Institute for Logic, Language and Computation, 1997). 10 Alan Newell and Herbert Simon, Human Problem Solving (Englewood Cliffs, NJ: Prentice-Hall, 1972). 11 Hans Reichenbach, Experience and Prediction (Chicago: University of Chicago Press, 1938). 12 See, e.g., Paul Thagard, Computational Philosophy of Science (Princeton, NJ: Princeton University Press, 1988).
416 Dov M. Gabbay and John Woods 13 Our position is developed at greater length in Gabbay and Woods, The Reach of Abduction; Gabbay and Woods, Seductions and Shortcuts; Gabbay et al., Handbook; and Gabbay and Woods, ‘The New Logic.’ 14 Gilbert Harman, Change in View: Principles of Reasoning (Cambridge, MA: MIT Press, 1986). 15 Strong relevance is discussed but not formalized in Dov M. Gabbay and John Woods, Agenda Relevance: A Study in Formal Pragmatics vol. 1 of A Practical Logic of Cognitive Systems (Amsterdam: North-Holland, 2003), and is the subject of a forthcoming paper, ‘Strong Relevance.’ 16 Nicolas Rescher, Plausible Reasoning: An Introduction to the Theory and Practice of Plausible Inference (Assen: Van Gorcum, 1976). 17 We have attempted to tell this further story in Gabbay and Woods, The Reach of Abduction. 18 P. Langley, H.A. Simon, G.L. Bradshaw, and J.M. Zytkow, Scientific Discovery. (Cambridge, MA: MIT Press, 1987); cf. Donald Gillies, ‘A Rapprochement between Deductive and Inductive Logic,’ Bulletin of the IGPL 2 (1994): 149–66. 19 Gerd Gigerenzer, ‘From Tools to Theories,’ in Historical Dimensions of Psychological Discourse, Carl Graumann and Kenneth J. Gergen, eds. (Cambridge: Cambridge University Press, 1996), 336–59, 338. See also Gerd Gigerenzer, ‘Probabilistic Thinking and the Fight against Subjectivity,’ in Ideas in the Sciences, L. Kruger, G. Gigerenzer, and M.S. Morgan, eds. (Cambridge, MA: MIT Press, 1987), 11–73, and Stephen Toulmin and D.E. Leary, ‘The Cult of Empiricism in Psychology, and Beyond,’ in A Century of Psychology as a Science, Sigmund Koch and David E. Leary, eds. (New York: McGraw-Hill, 1985). 20 Gigerenzer, ‘From Tools to Theories,’ 339. 21 W.P. Tanner and J.A. Swets, ‘A Decision-Making Theory of Visual Detection,’ Psychological Review 61 (1954): 401–9. 22 In Neyman-Pearson statistics, two hypotheses H and H9 are forwarded; these are construed as sampling distributions. And a decision rule is defined; this is construed as a likelihood ratio. Thus in our present example the agent computes two sampling distributions for ‘noise’ (H) and ‘signal plus noise’ (H9 ). He (or it) then fixes a decision criterion which takes into account the cost of the two possible decision errors. The brain takes sensory input which is transduced in such a way that enables it to calculate its likelihood ratio, which is then judged against the criterion, giving ‘signal’ or ‘signal plus noise’ as answer, depending on whether the ratio is smaller or larger than the criterion (Gigerenzer, ‘From Tools to Theories,’ 340).
Filtration Structures and the Cut Down Problem for Abduction 417 23 B.B. Murdock, Jr, ‘A Theory for the Storage and Retrieval of Item and Associative Information,’ Psychological Review 89 (1982): 609–25, and W.A. Wicklegreen and D.A. Norman, ‘Strength Models and Serial Position in ShortTerm Recognition Memory,’ Journal of Mathematical Psychology 3 (1966): 316–47. 24 M.H. Birbaum, ‘Base Rates in Bayesian Inference: Signal Detection Analysis of the Cab Problem,’ American Journal of Psychology, 96 (1983): 85–94. 25 L.L. Lopes, ‘Doing the Impossible: A Note on Induction and the Experience of Randomness,’ Journal of Experimental Psychology: Learning, Memory and Cognition 8 (1982): 626–36. 26 H.H. Kelley, ‘Attribution Theory in Social Psychology,’ Nebraska Symposium on Motivation, D. Levine, ed. (Lincoln: University of Nebraska Press, 1967). 27 In Kelley’s ‘attribution theory,’ the experimenter infers a causal connection between two variables from computing an analysis of variance in the manner of Fischer’s (R.A. Fisher, ‘Statistical Methods and Scientific Induction,’ Journal of the Royal Statistical Society B [1955]: 69–78) analysis of variation (ANOVA) and then running an F-test. Similarly, the practical or individual agent makes causal attributions by running these same calculations unconsciously (Gigerenzer, ‘From Tools to Theories,’ 340). 28 Gigerenzer, ‘From Tools to Theories,’ 339. 29 N.R. Hanson, Patterns of Discovery (Cambridge: Cambridge University Press, 1958). 30 K. Popper, Conjectures and Refutations (New York: Basic Books, 1962).
23 Mistakes in Reasoning about Argumentation GEORGE BOGER
1. Three Mistakes in Reasoning When critical thinking theorists, pragma-dialecticians, and informal logicians use the expressions ‘theory of argument,’ or ‘argumentation theory,’ and ‘fallacy theory,’ they usually take an argument to consist in considerably more than a set of propositions. Their view varies significantly from that of formal logicians. Rather, they take an argument to consist in, but not to be restricted to, a set of premisses that allegedly support a conclusion with the intention of changing someone’s belief. For them an argument is a dynamic relationship; indeed, it is a social activity. In this way they believe themselves to study real-life, or ordinary-language, arguments and situations. And here they make their first mistake in reasoning about argumentation: they attribute agency to an argument when agency is properly a feature of an arguer. They confuse an argument with an arguer. Now, for these logicians argument assessment, or argument evaluation, is a core concern of logic. Here again they understand argument assessment in a way at variance with formalists, that is, not in respect of examining the logical or implicational relationships among propositions, but in respect of examining an argument within a dynamic context involving an audience or disputants. Accordingly, they evaluate an argument in terms of premiss acceptability, premiss weight and relevance, and in terms of the suitability of the inferential link between premisses and conclusion, all of which, moreover, they take to be relative to persons at times. In short, a ‘good argument’ is good according to the standards and beliefs of a given audience at a given time. And here they make a second mistake in reasoning about argumentation: by relativizing cogency to the dispositions of one or another audience, they subvert a principal aim of logic,
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namely, to establish objective knowledge. This subversion is most effectively accomplished by blurring the distinction between what one believes to be the case and what really is the case. Thus, they suppress an important epistemic/ontic distinction in two respects: (1) they conflate inference and implication; and (2) they conflate thinking and being. Here again they tend to confuse evaluating an argument with evaluating the various skills of an arguer. Moreover, according to these logicians, an argument or an argumentation is not something evaluated on its own internal merits, but in relation to whether it succeeds in persuading a participant to change his or her beliefs. This immediately involves a third mistake in reasoning about argumentation: they confuse ‘argumentation theory’ with ‘persuasion theory,’ a proper part of which includes argumentation but is more narrowly construed as consisting in propositions1 and their logical relationships. Notwithstanding their valuable insights into human discourse, these logicians run the risk of undermining what they ought themselves take to be a core concern of logic, namely, developing topic-neutral methods for distinguishing truth from falsity.2 2. Divergent Notions of Logic Informal logicians, critical thinking theorists, and pragma-dialecticians are not apologetic about wanting to ‘dethrone formal logic,’ that is, about criticizing what they believe to be the project, and the sin, of formal logic. They cite (1) the irrelevance of formal logic to the needs of everyday discourse whose medium is natural language, (2) the asymmetry of formal logic – its inability to formalize fallacious reasoning and even invalidity – and (3) the attention of formal logic on strictly formal, or ‘mathematical,’ matters that relegates to the extralogical ‘everything else’ important to evaluating arguments. Indeed, the distinction between matter and form is not in the forefront of an informalist’s method of argument analysis. When these logicians forcefully affirm the participant relativity of good and bad argumentation, when they place emphasis on cognitive, even intentional, aspects of argumentation, when they embrace the ‘extralogical’ within the project of logic, and when they emphasize argument context and the pragmatics of discourse, they seriously jeopardize establishing a sound fallacy theory, which they take to be an important concern. A prominent psychologistic3 orientation haunts the informalist’s consideration of fallacy and of argumentation generally. The singular problem of this approach to argument analysis con-
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cerns its insistence on the contextuality of an argument. This emphasis diverts the purpose of logic from achieving objective knowledge and steers it toward subjectivism. Particularly hurtful here is assessing an argument according to standards set by a given audience. These logicians evaluate an argument in terms of its success to persuade a participant of a belief irrespective of the truth or falsity, or even of the morality or immorality, of the belief, and irrespective of whether an argument is valid or invalid, an argumentation cogent or fallacious. By declaring that a good argument need not be valid, that fallaciousness and cogency are participant-relative, they undermine any objective criteria for knowledge and focus on an agent’s ability to manipulate language. Doing this neglects an ontic underpinning of truth and falsity, validity and invalidity, and cogency and fallaciousness. If the objective of argumentation is to convince a participant of a given proposition, then formal logic, with its emphasis on logical consequence, naturally has no role to play, save, perhaps, for encountering participants knowledgeable about validity and other formal matters. Formal validity becomes irrelevant, since an invalid argument might be as convincing as a valid argument, and so on. No longer can we say that someone is mistaken, save for some arbitrary criteria, or, perhaps, for breaking some rules established for an arena of argumentation, say, for a presidential debate or a labour–management negotiation. Logic effectively surrenders concern for objective knowledge and studies rules for regulating discourse. This, of course, is a worthy project, however circumscribed. Still, it is understandable that formalists are disturbed, as perhaps is any person of good conscience. While it is true that informal logicians do not eschew there being norms of good argument, they seem unable to provide an objective, or universal, foundation for such norms, save for those relative to a given context with a given objective. Closing the gap between the project of logic and the needs of ordinary human beings has provided a licence for unrestrained arbitrariness – save, perhaps, for that of a good will – when it comes to objectively assessing the ‘cogency’ of an argumentation. In closing one gap they widened another, one perhaps really more pernicious than the first had seemed – that gap between establishing genuine knowledge, which distinguishes truth from falsity, and holding one or another opinion in some narrow frame of reference. Now, taking logic to be a part of epistemology and thus as having a primary goal to cultivate objectivity, we might agree that logic aims to develop concepts, principles, and methods that are useful for making a
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decision according to the facts. Surely the need for logic would be obviated were humans omniscient or infallible. From a classical perspective, logic has been concerned with ‘the perfection of criteria of proof, the development of objective tests to determine of a given persuasive argumentation whether it is a genuine proof, whether it establishes the truth of its conclusion.’4 The feeling of certainty is not a criterion of truth, and persuasion is not necessarily proof. Perhaps we can agree with John Corcoran, who construes objectivity to be an important virtue: All virtues are compatible with objectivity, and most, if not all, virtues require it in order to be effectual and beneficial. Without objectivity the other virtues are either impossible or self-defeating or at least severely restricted in effectiveness.5
Surely, then, by basing human dignity and mutual respect on the universal desire for objective knowledge, we can affirm an essential role of formal logic in everyday life – to overcome ignorance and fallibility as much as possible. Taking this posture helps to avoid reducing study of argumentation to psychology, or cognitive science, or even to rhetoric and persuasion theory. 3. Woods and Walton Attempt to Bridge the Difference John Woods and Douglas Walton are aware of a ‘cognitivist’ tendency among informal logicians.6 They have taken steps to avert a collapse of informal logic into psychologism by attempting to bridge a gap between two traditions in modern logic. Their success waits in the wings, we believe, principally because of their working from a contextualist foundation.7 Walton, developing Hamblin’s thinking, works from a notion of informal logic as applied logic in the vein of ‘the pragmatics of discourse.’ For him an argument is not merely a system of propositions where one follows deductively from the others, but something that exists in the context of a logical dialogue game: ‘Argument must be seen as a dynamic relationship where one arguer’s position is challenged or attacked, and the other arguer’s move is seen as a response to this challenge.’8 Siding with informalists, he cites, as among the limitations of classical logic, a putative lack of relevance and the use of a negative method for determining invalidity. But with formalists he decidedly
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wants to avoid psychologism in assessing informal fallacies and not relegate their evaluation to a branch of psychology or another social science.9 He applies his thinking to cogent reasoning and acknowledges that classical deductive logic might play a crucial part in a theory of argument to justify criticism of arguments, but within a concept of logical dialogue. ‘Propositional logic, so conceived, is the inner core of argument. The game of dialogue is the outer shell of argument. The two combined offer a theory of argument that shows how logic can be applied to realistic argumentation.’10 Walton then proceeds to treat fallacies in a way generally similar to the usual treatment found in introductory logic textbooks. Of course, his work has surely contributed to this phenomenon. He writes: Thus the study of fallacies is indirectly linked to linguistics, psychology, rhetoric, and other empirical sciences. But that doesn’t mean that a particular instance of argument must seem valid to some particular person or audience to qualify as a fallacy. What is most important is that to be justified in claiming that an argument is fallacious, or a fallacy, it is not enough to show that the argument seems invalid to some actual persons. It is a normative claim that the argument is in some sense bad, unsound, invalid, or at least open to reasonable criticism. And that is a claim that must be justified by an appeal to the relevant theory of (good) argument.11
Walton seems to send mixed signals. Nevertheless, he suggests that there might exist a ‘formalist’ foundation underlying the work of informal logic. He continues: The effectiveness aspect of fallacies is what makes the applied logic of the fallacies ‘applied,’ but does not mean that psychological considerations play a decisive role in the logical considerations of when an argument is correct, incorrect, or even a distinct type of argument to be considered.12
Here again he seems to invite formal logic to the table. Indeed, Walton has seriously wondered to what extent formal logic is applicable to natural language conversational disputation, and he has taken this question to be fundamental to the work of informal logic. He answers: [F]ormal logic can be used by a participant in disputations to build up his own argument or to criticize his opponent’s argument. But in speaking of criticism in disputation we are importing a framework, a conception of
Mistakes in Reasoning about Argumentation 423 argument that includes more than just the semantic structure of the propositions that make up the core of the argument. It includes as well the pragmatic structure of certain conventions or rules of argument – locution rules, dialogue-rules, commitment-rules, and strategic rules.13
Walton’s theory of argumentation is firmly ensconced in an informalist framework which embraces pragmatist considerations well beyond the formal relations of propositions treated by classical logicians. This conception of argumentation accordingly affects his definitions of formal and informal fallacy: [Thus] a fallacy is a type of move in a game of dialogue that violates a certain rule of the game. Such a fallacy may be one of the kinds traditionally called an ‘informal’ fallacy. Formal fallacies are those that pertain to the formal logic element, the core of the game that has to do with relations of validity in the set of propositions advanced or withdrawn by the players. Informal fallacies have to do with rules and procedures of reasonable dialogue.14
Now, unfortunately, just in making this statement Walton reneges on his commitment to the role that formal logic has in a theory of argumentation and a theory of fallacy. He shifts focus from argument assessment to arguer assessment and thus loses focus on obtaining objective knowledge. Nevertheless, much of the work of Woods and Walton has aimed to ‘formalize’ certain aspects of reasoning in ordinary discourse. This is evident in their numerous studies of fallacies. Indeed, Woods writes that there are two distinct advantages to using formal methods: ‘One is the provision of clarity and power of representation and definition. The other is provision of verification milieux for contested claims about various fallacies.’15 Woods in particular has argued that ‘being a mathematical system is not necessarily a liability for a theory of the fallacies’ even if fallacy theory cannot fully embrace certain mathematical features.16 Still, he forcefully states that a fallacy theory need not be constructed along the lines of an axiomatic logistic system, which, in any case, he recognizes to be a virtual impossibility. However, he continues, ‘we know ... that axiomatic formalization does not exhaust formal treatment.’17 Woods writes that his and Walton’s analyses of the fallacies have considerably benefited by ‘repos[ing] the theoretical burdens of the fallacies in probability theory, acceptance theory, epistemic and doxastic logic, and rationality theory.’18 Woods then states his thesis:
424 George Boger This leads me to suggest not that the mature theory of the fallacies is a branch of logic that is essentially informal, but rather that the mature story of the fallacies is a branch of formal theory that is essentially extralogical in major respects. The formal theory of the fallacies is not (just) logic.19
Woods here, as Walton elsewhere, seems to vacillate between the two poles, which vacillation pivots on an equivocal use of ‘formal.’ The primary concern, in connection with our project as logicians, is not merely with a systematization, or formalization, of ordinary-language argumentation according to the pragmatics of discourse, but with the inherent cogency, or fallaciousness, of argumentation. And this just concerns logical consequence. Formalists have difficulty with informalists placing the mistake in reasoning in the realm of dialogue rather than in the logical relations of propositions. This problem devolves to how each understands what a theory of argumentation embraces. And this matter requires clarity on what an argument is and what an argumentation is. Informalists themselves recognize that there is a problem with determining just what a fallacy is and then constructing a typology of the fallacies. This matter is particularly focused when we consider how informalists, even formalists, assess begging the question. How can a valid argument be a fallacy when it does not violate cogency? What makes such a ‘fallacy’ an informal fallacy? Becoming clear about this putative fallacy might go far toward closing a gap between formal and informal logic. 4. The Problem: Assessing an Arguer or an Argument? Walton has insisted on taking a dialectical approach to the study of arguments and maintaining that the criticism and analysis of arguments require bringing out the question-answer context of an argument, or the challenge-response model of interactive dialogue. ‘Thus generally the theory of informal logic must be based on the concept of question-reply dialogue as a form of interaction between two participants, each representing one side of an argument, on a disputed question.’20 Walton identifies the project of the informal logician in the following way: For the most part, applying critical rules of good argument to argumentative discourse on controversial issues in natural language is an essentially pragmatic endeavor. It is a job requiring many of the traditional skills associated
Mistakes in Reasoning about Argumentation 425 with the humanities – empathy, a critical perspective, careful attention to language, the ability to deal with vagueness and ambiguity, balanced recognition of the stronger and weaker points of an argument that is less than perfectly good or perfectly bad, a careful look at the evidence behind a claim, the skill of identifying conclusions, sorting out the main line of argument from a mass of verbiage, and the critical acumen needed to question claims based on expert knowledge in specialized claims or arguments. Thus the terms ‘informal logic’ and ‘critical argumentation’ are well suited to the subject matter and methods of this handbook.21
However, in truth, this is not the project of a logician, formal or informal. Rather, this is the project of a human being, and, more specifically, of a human being committed to obtaining objective knowledge, however difficult such an undertaking is. Obtaining objective knowledge requires many skills – one of which is constructing a discourse, or argumentation, consisting in a sequence of propositions expressed in an object language. A human being assumes many roles in such a process, which frequently includes persuasion. The question for modern logicians, then, is whether an argumentation per se is subject to a systematic, objective, and context-free evaluation. Woods and Walton have aimed to rescue the project of informal logic by employing some of the theoretical apparatus of formal logic, enriched, they believe, by notions of relevance and dialogue. However, they seem not to have fully rescued cogency and extricated the analysis of an argumentation from a strict contextualism that reduces analysis to subjective relativism. A pervasive mistake in reasoning about argumentation among some logicians is to confuse the activity of arguing with the activity of persuading. Informal logicians, critical thinking theorists, and pragma-dialecticians tend to use the expression ‘argumentation theory’ to denote a ‘theory of persuasion.’ They do not consider argumentation theory to be that part of persuasion theory that treats argumentations more narrowly conceived as sets of propositions. As a consequence of this confusion, they mistake the proper object of argument assessment and lose sight of a concern with truth and falsity. In the context of argumentation-as-persuasion, they confuse the success or failure of a persuader, assessment of whom requires many disciplines, with the ‘goodness’ or ‘badness’ of an argument, which really ought to be evaluated independently. They mistakenly call an argument good or bad – or right and wrong, correct or incorrect, sound and unsound, valid and invalid, logical and illogical, convincing and unconvincing, plausible and implausible, and so
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forth.22 – when really the arguer and his or her audience is assessed. Moreover, they confuse an argument with an argumentation. We might also mention, alluding to the hypothetico-deductive method and to the aspiration toward objective knowledge, that they confuse a deduction with a demonstration. While the goal of a persuader is to convince, the goal of a logician is to assist in establishing objective knowledge. This is impossible to achieve by basing truth and falsity, validity and invalidity, and cogency and fallaciousness on the subjective dispositions of one or another audience at one or another time. These logicians have aimed to close the gap between logic and the needs of human beings, but at the cost of eliminating the difference between the process of arguing and its context, on the one hand, and the product of such a process, the argumentation itself, on the other. They commit the process/product fallacy. And, considering that an underlying philosophical tenet of informal logic relates to its contextual relativism, its seems that their closing the gap between the theory and practice of logic and formal logic’s putative irrelevance depends on their adopting a postmodern obliteration of the subject-object distinction that confuses what is known with what is, and thus they are themselves guilty of the epistemic/ontic fallacy.23 5. An Alternative Model To help make sense of the complexities of practices in the art of persuasion, often subsumed under the rubric of ‘argumentation theory,’ we invoke Aristotle’s notion of the four causes in connection with his notion of technê to construct an analogy represented in table 23.1. Accordingly, just as no saw can cut wood, but the person using the saw cuts the wood, so no argument or argumentation can persuade a participant to believe something. Rather, a speaker using an argumentation provides occasion for a participant to change his or her beliefs. It is a category mistake to attribute agency to an argument. No argument convinces anyone. Nor, in truth, does an arguer convince anyone. Rather, presented with information in various forms, a participant grasps something in his or her mind as a mental act: this person experiences a chain of reasoning and comes to an understanding. Thus, a speaker is a kind of medium, someone presenting – packaging? – information, who takes a participant through an ordered sequence of steps in such a way as to ‘get it.’24 The success of a persuader, in respect of either changing beliefs or
TABLE 23.1 Four causes relating to production The art of cabinetmaking
Cause
The art of persuasion
The production of a piece of furniture, a cabinet.
Final cause The objective or goal of a given process.
The production of human beings who act in a desired way; the more proximate end is the production of human beings with changed (or confirmed, etc.) beliefs that inform their actions.
Wood, screws, glue, etc.: objects having certain properties, in this case, properties relating to inanimate, or passive, beings.
Material cause That transformed (a patient) within a given process.
Human participants: objects having certain properties; in this case, properties relating to passive and active beings having higherorder cognition [audience].
Geometry: the set of theorems of geometry. Formal (or ideal) cause That giving ‘shape’ to the objective of production.
A set, or sets, of beliefs that shape character as expressed in action.a
The artisan: the cabinetmaker (or carpenter).
Efficient (or productive) cause That performing (an agent) a given process to effect a given end using instruments appropriate to the subject matter and process.
The artisan: the discourser or persuader or arguer [speaker, writer].
Hammers, saws, drills, jointers, etc.
Instruments The tools appropriate to a given art for its specific purpose.
Arguments and argumentations.b Rhetorical devices. Pragmatic discourse rules. Staging.
a
b
The union of formal and material causes might be construed in two ways: (1) the logic organizing a set of beliefs; (2) beliefs shaping action. The formal counterparts of these items, sciences in their own right, are: formal logic, formal rhetoric, and formal discourse rules.
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provoking action, depends on many factors, but, in general, on two: • a persuader’s expertise in various fields of knowledge, including: formal logic; rhetoric; social science (sociology, psychology, history, etc.); natural science; linguistics, semantics, language; material staging for an arena of argumentation. • the conditions and composition of an audience, including: religious, political, moral beliefs; age, sex, ethnic, and class make-up; homogeneity or heterogeneity; and so on. These factors, then, are just the items appropriate for evaluating not an argumentation per se, but a speaker and his or her audience. A successful persuader, short of accidental success, must know his or her own strengths and weaknesses in respect of the four causes. Considering the entire arena of persuasion, we recognize many points of evaluation, for example, how adept a speaker is with rhetorical devices or knowledge of language and especially with knowledge of an audience’s beliefs. Considering only the argumentation itself, the discourse, we assess it as an instrument in the hands of a persuader. An argumentation, then, might be assessed as an argumental instrument in two ways: (1) independent of a context and (2) contextually. Exactly here we can locate the divide between two different notions of logic: • Independent assessment. In one respect, whether an argumental instrument is good or bad is independent of the beliefs of an audience. The question ‘Is it a good argumentation?’ for a logician is analogous to the question ‘Is it a good saw?’ for a cabinetmaker. Being a good saw is independent of the wood it is used to cut. Of course, we are working within a domain and thus with ‘intended interpretations,’ that is, with intended uses. Nevertheless, that being said, a good saw involves being composed of the right metal and having the right temper, the right shape, the right handle, weight, balance, number of teeth, angle of teeth, sharpness, and so on. All this is distinguished from being the right tool for a function, which is relative to a task. An argumentation, then, can be assessed independently in respect of its propositional relations. A good argumentation involves: lack of ambiguity; the absence of smuggled premisses; a conclusion that is a logical consequence of the premisses; containing a chain of reasoning cogent in context (that is, in the context of the relations of propositions); and so on. Of course, assessing an
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argument involves extracting the propositions expressed by ordinary-language sentences and then checking them against the models established by formal logic. According as informal logicians approach the matter, the question of an argument’s goodness or badness really asks about a persuader’s expertise in using an argumentation and other tools. Thus: • Contextual assessment. In another respect, whether an argumental instrument is good or bad does depend on an audience. Indeed, a skillful cabinetmaker knows which tool to use for which situation. Using the ‘wrong’ but ‘good’ tool produces an undesired result, but here the wrong or right tool relates principally to its use. In the case of cabinetmaking, a skillful artisan familiar with the properties of the wood – its grain, dryness, age, hardness, and so forth. – selects a tool appropriate to the task. In this way an artisan produces a fine article of furniture (the desired end in respect of cabinetmaking) regardless of what further end such article might serve. However, a skillful artisan might effect the same result with a well-crafted, or good, tool or with a less than excellent tool. Or, an artisan might use a ‘good’ tool for the wrong task or in a wrong way. In this way the excellence of the tool per se is irrelevant and it becomes a bad tool in use or a bad tool becomes good in use. Here the end determines the value of the means, and the ‘good’ means is that whose use successfully produces the desired result. Here also evaluating an end is suspended. If, like cabinetmaking, the objective of persuasion is to produce a desired action – apart from assessing the rightness or wrongness of an action – then the right tool certainly is determined contextually. In fact, this is more the case when the material cause is the action of audience members and the formal cause is their belief-set. Just as given properties of certain wood require certain tools to produce the desired end, so certain audiences require certain argumentations to achieve the desired end. Selecting the right, good, correct, convincing tool in this framework certainly has little to do with truth and falsity, with validity and invalidity, even with cogency and fallaciousness. The bad or good tool has virtually nothing to do with the relations of propositions and ‘everything’ to do with audience sentiments. The concern now shifts to what is acceptable, warranted, or believable for a given audience at a given time to produce a given action. The emphasis is on ‘making inferences’ and not on revealing implicational relationships.
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From the vantage point of this second approach to argument assessment, the arguer takes centre-stage. The project of logic in this respect aims to assess his or her knowledge and skill to package information. In addition, the audience also takes centre-stage from this perspective. Thus, informal logicians have spent considerable attention, not on ‘good argumentation,’ but on examining empirically how different human beings make up their minds. This is a concern of psychology and sociology. It no longer is a question of logic, of whether an argument is valid or invalid, but a meta-systematic or pragmatic question of whether an argument works or does not work. Concern for establishing objective knowledge or using formal logic becomes irrelevant, even obstructive. Perhaps the motto of the informalist movement might be: ‘There are no victims; everyone is a victim.’ This is not a happy state of affairs. Thus, we have two projects: (1) to isolate argumentation as a part of persuasion theory; and (2) to apply formal logic to fallacy theory. 6. A Role for Formal Logic in Developing a Fallacy Theory Woods and Walton must surely feel an intellectual kinship with formal logicians such as Corcoran. They are equally committed to objectivity. The question is, to what extent is the realization of their commitment compromised by their equally strong commitment to assessing arguments contextually? It seems that their understanding of the scientific (systematic) aspect of logic is incompatible with their understanding of the meta-scientific (meta-systematic) aspect of logic along with their understanding of applied logic. Nevertheless, they expect that discourse on cogent and fallacious argumentation itself be cogent. Philosophers and logicians recognize different perspectives for defining truth, such as correspondence, coherence, or pragmatic perspectives. We here employ a correspondence notion along the lines of Aristotle, Tarski, and others to help assess argumentation objectively. Interestingly Aristotle considered the truth or falsity of a sentence to depend on whether a given state of affairs is the case or not, but not that a given state of affairs is dependent on the truth or falsity of a given sentence.25 Just so would he consider the validity of a given argument to have an ontic underpinning. This is because he understands the ontic nature of the law of contradiction to undergird ‘truth following being.’26 It is not that a contradiction is impossible; it is absurd: a contradiction exists between propositions. Rather the state of affairs denoted by a contradiction is impossible: such a state of affairs
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does not exist.27 For Aristotle there is an underlying ontology for truth and falsity and for validity and invalidity that makes impossible that true propositions imply a false proposition and that makes these matters participant-independent. This ontology takes argument evaluation out of relativistic and arbitrary considerations to provide for a formal assessment of implicational relationships among propositions. Let us define a premiss-conclusion (P-c) argument to be a two-part system consisting in a set of propositions called premisses (P) and a single proposition called a conclusion (c).28 In a valid argument the premiss propositions imply the conclusion proposition, and the conclusion proposition is a logical consequence of the premiss propositions. In a valid argument it is impossible that true propositions imply a false proposition, but all other combinations of truth-values for the propositions in a P-c argument are possible for both valid and invalid arguments. Another way of expressing validity is to say that in a valid argument all the information in the conclusion proposition is already contained in the premiss propositions.29 In any case, truth and falsity and validity and invalidity are ontic properties of propositions and arguments respectively. One way to establish knowledge of an argument’s validity is to find a chain of reasoning (a derivation) that is cogent in context, which thereby helps to link, in the mind of a participant, the conclusion proposition to the premiss propositions as a logical consequence by providing a sequence of propositions that are conclusions of valid elementary arguments. We might define formal derivation (deduction) as follows: A given proposition c is formally deducible from a given set of propositions P when there exists a finite sequence of propositions that ends with c and begins with P such that each proposition in the sequence from P is either a member of P or a proposition generated from earlier propositions solely by means of stipulated deduction rules. An argumentation, then, is a three-part system consisting in a set of propositions called premisses, a single proposition called a conclusion, and a sequence of elementary arguments called a chain of reasoning. Given this understanding of argumentation, we can see that a cogent chain of reasoning is a series of interlinked, valid elementary arguments. Leaving aside the cognitive aspect, the ‘getting it,’ we see that cogency must also be an ontic property of an ordered sequence of con-
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clusion propositions of elementary valid arguments. It is one thing for the sequence to be cogent; it is another thing for someone to understand that this is so.30 To deny that cogency is an ontic property of such a sequence of propositions in elementary arguments would also deny the truth of the principle of transitivity of consequence, namely: ‘every consequence of a consequence of a given proposition is again a consequence of that proposition.’31 Thus, to hold that the cogency of a derivation presupposes knowledge of the validity of the component arguments undermines the independence of cogency and threatens our obtaining objective knowledge. Let us now take cogency to be an ontic property of a ‘good’ argumentation, specifically, of a deduction, and its counterpart, fallaciousness, to be an ontic property of a ‘bad’ argument, namely, of a fallacy. This extricates both deductions and fallacies, in respect of their consisting in propositions, from participant relativity and places responsibility for their recognition squarely on participants. We can now turn to sketch how formal logic might assist in developing, or even itself develop, a theory of fallacy, a constituent part of which sharply distinguishes ontic matters from epistemic matters. One project of epistemology is to determine means for establishing knowledge of the truth or falsity of propositions relating to one or another domain. Another project is to determine a foundation for, and to discover the means by which we establish knowledge of, logical consequence. Ontology and logic are intimate companions. Thus:32 • Formal logic has articulated the law of contradiction and the law of excluded middle as providing an ontic underpinning for intelligible discourse. One ontic expression, in relation to states of affairs, holds: ‘it is impossible that a given property both belong and not belong to a given object at the same time in the same respect’; ‘either a given property belongs or does not belong to a given object.’ Another ontic expression, in relation to propositions, holds: ‘it is impossible that a given proposition is both true and not true at the same time in the same respect’; ‘either a given proposition is true or not true.’ • Formal logic has articulated the principle of consistency: ‘every true proposition is consistent with every other true proposition.’ This principle equally underlies intelligible discourse and is applicable to various notions of truth, whether a correspondence, a coherence, or a pragmatist notion. • Formal logic has defined logical consequence as an ontic property
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•
•
•
•
•
•
existing between propositions. Two statements of logical consequence are: (1) ‘c is a logical consequence of P if every true interpretation of P is a true interpretation of c’; (2) ‘c is a logical consequence of P if all the information contained in c is already contained in P.’ Again, this notion underpins intelligible discourse by which we recognize, for example, the incoherence of a paradox, that true propositions cannot imply a false proposition. Formal logic has established the principle of form: ‘every argument in the same form as a given valid argument is valid’; ‘every argument in the same form as a given invalid argument is invalid.’ An argument has one and only one form, although a given argument might fit any number of patterns (for example, every argument fits the pattern ‘P-c’). Two propositions are in the same form when there exists between the two a one-to-one correspondence of their nonlogical constants, given the same logical constants, that carries the one over into the other. Formal logic has developed the method of counter-argument and method of counter-interpretation to establish knowledge of invalidity. Formal logic has developed the notion of cogency as consisting in linking the conclusion propositions of valid elementary arguments sequentially in an argumentation, or chain of reasoning. In this connection, formal logic has articulated the principle of transitivity of consequence: ‘every consequence of a consequence of a given proposition is again a consequence of that proposition.’ Formal logic has developed the notion of universe of discourse by which one determines what is germane to a specific discourse. Perhaps relevance might be considered in this connection. Formal logic has developed the notion of precision in thinking as exemplified in, for example, the ideal of a logically perfect language. Working with this notion enables someone to take ordinarylanguage discourse and extract the propositions it expresses. Part of this process, for example, involves identifying ambiguity. Of course, the work of semantics and linguistics is important, if only for helping to make more precise the logical form of a given proposition. In this connection, they help to distinguish the forms of propositions that appear to be the same but are different because they contain different semantic categories, such as count nouns and mass nouns. Formal logic has established methods that aim at objective knowledge, two of which are the hypothetico-deductive method for dis-
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confirming a hypothesis, or proving it to be false, and the deductive method used in axiomatic discourse for proving a hypothesis to be true. In this connection, formal logic has defined deduction, refutation, demonstration, and fallacy. • Formal logic has provided methods useful for discovering hidden consequences of propositions. Of course, formal logicians consider developing models – whether of formal or natural languages, of deduction systems, or of argumentations – as part of their stock-in-trade. These models, along with all the contributions mentioned above, serve as ideals against which to assess ordinary-language discourse.33 If sketching a fallacy theory includes providing (1) a definition of fallacy and (2) a method of formal analysis, then formal logic offers the following definition. A fallacy is an argumentation in which one or more of the following occurs: (1) the conclusion is not a logical consequence of the premiss-set; (2) the chain of reasoning is not cogent, whether or not the argument bounding the chain of reasoning is valid; or (3) the chain of reasoning is cogent but not in context. These considerations are ontic features of the argumentation that is a fallacy, and thus they are independent of a participant’s recognition. A process of formal analysis of a fallacy might involve any of the familiar methods for refutation and determining invalidity. Again, this process (1) is independent of argumentational pragmatics, dialogue rules, and context, and (2) requires extracting an argumentation from a natural-language discourse and expressing it precisely with all the tools of formal logic. Using the simple model of an Aristotelian syllogism, we can show that a fallacy violates a syllogistic pattern all of whose instances are valid arguments (syllogisms). In the case of ambiguity, while a given argument with an ambiguity has one grammatical pattern, which helps to make it appear to be a syllogism, it really has two underlying logical patterns. And in the case of equivocation, while an argument with an equivocal expression has a given grammatical pattern that makes it appear to be a syllogism, it really has, with the addition of a fourth term, an underlying logical pattern different from a syllogism. Begging the question might be considered in two ways, both of which involve no fallaciousness: (1) When, among a premiss-set, a false proposition taken to be true (or one whose truth-value is undetermined) implies a true proposition, it is a mistake to believe the conclusion to have been proved. Here there is no fallacy or mistake in
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reasoning. Rather, there is only ignorance on the part of a participant about what counts as a demonstration. Knowing that every true proposition is implied by infinite false propositions might help in this situation. (2) When a proposition to be established as a conclusion is itself among the propositions in the premiss-set, there is no fallacy. Again there is ignorance on the part of a participant about demonstration. However, here there is a need for a restriction on the deduction system along the lines of Aristotle’s requirements for his syllogistic system: the conclusion must extend knowledge beyond what is immediately stated in the premiss-set.34 The fallacies of ad hominem and appeal to authority introduce, or smuggle, additional premisses that do not contribute to a conclusion following logically from premisses. The other fallacies might be addressed in a similar fashion. 7. Concluding Remarks We know all too well that an ad hominem argument can be very effective in the hands of an accomplished rhetorician. Part of its effectiveness consists in a rhetorician presenting what appear, in the minds of audience participants, to be warranted or believable or believed-to-betrue statements – which, indeed, might include ambiguity and equivocation – and which might even appear to be relevant to the conclusion a rhetorician wants to establish. However, the rhetorician’s success really rests on at least three factors, all of which pertain to the conditions of an audience: (1) participant ignorance of formal logic; (2) participant ignorance of facts and objective knowledge; (3) participants’ lack of a clear commitment to obtaining truth and a willingness to suspend judgment toward that end. Critical thinking theorists, pragma-dialecticians, and informal logicians have aimed to diminish the gap between logic and the practices of human beings. Without going into their many accomplishments, we have addressed something problematic in their thinking about argumentation. In this connection, we have aimed to reassert more forcefully the gap between ignorance and knowledge in respect of (1) truth and falsity, (2) validity and invalidity, and (3) cogency and fallaciousness. Their relativism tends to obscure this gap to the detriment of human understanding and conflict resolution. We believe that these logicians need more determinately to distinguish the presentation of an argumentation from the argumentation itself, and in this way they could recognize that an argumentation can be evaluated on its own
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terms. Logicians could then focus on developing a person’s ability to avoid mistakes in reasoning by promoting (1) knowledge and command of information and (2) knowledge of logical consequence. John Woods has been, and continues to be, a prodigious contributor to developments in argumentation theory and fallacy theory. And while he embraces many achievements of informal logicians and pragma-dialecticians, his acceptance is tempered by his own profound understanding of formal logic. Indeed, he has warned against uncritically embracing the ‘anti-formalist skepticism’ of the informal logic movement in an article aptly entitled ‘The Necessity of Formalism in Informal Logic.’35 He has hoped for a theoretical unification of the fallacies, but he has raised serious doubts about how, for example, a pragma-dialectical analysis of the fallacies might help in this connection: ‘the idea that a fallacy is just a violation of a rule of critical discussion will surely yield to something deeper and more complex.’36 He suggests that that ‘something deeper’ has to do with distinguishing the presentation of an argument from its inner logical coherence. However, perhaps Woods’ strongest statement for a role that formal logic might play in the development of a mature fallacy theory appears in his assessment of Aristotle’s logical investigations. [Aristotle’s] greater accomplishment [than classifying fallacies] was to have set the account of fallacies in the framework of an attempt to construct a fully general theory of deductive reasoning in which the technical and original notion of syllogismos would bear most of the explanatory load.37
Aristotle’s ‘idea that a fallacy is an argumentative mistake that somehow seems not to be’ one ‘to which no subsequent fallacy theorist can afford to be indifferent.’38 Woods has noted that Aristotle in Sophistical Refutations and Topics treats a fallacy as a sophistical refutation, that is, as a syllogism, that is, as a deduction of the contradictory of a given proposition. A syllogism is just a valid argument; and syllogistic reasoning chains such elementary arguments to provide for longer, or extended, deductively cogent argumentations. Woods here affirms, as we above, that the core of a sound fallacy theory compasses the heart of formal logic. This does not mean that we can axiomatize fallacious argumentation. But it does mean that we locate a fallacy, not in breaking rules for conducting a ‘dialectical’ exchange, but in the logical incoherence of non-implicational relationships among propositions.
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Aristotle remarked in Sophistical Refutations that our ability to recognize fallacious argumentations improves our philosophic understanding and our ability to obtain objective knowledge. Woods surely is an Aristotelian in this respect and in connection with developing a sound fallacy theory. Part of becoming a virtuous human being requires developing a lifelong commitment to examination and self-reflection in the pursuit of objective understanding. Perhaps now more logicians will come to recognize that formal logic has a crucial role to play in that process as it applies to everyday matters. notes 1 Without settling the ontic status of a proposition, we take an object-language sentence to express a proposition. In this respect, then, a sentence might be ambiguous where a proposition is not. 2 The literature concerning informal logic, fallacy theory, critical thinking, and argumentation more generally, is vast and vital. This diverse trend in modern ‘logic’ or argumentation promises to continue to make important contributions to human understanding and communication. Below we cite some of the works in this area to indicate the nature of modern studies and some of the sources for our discussion. C.L Hamblin’s Fallacies (Newport News, VA: Vale Press, 1993) and Stephen Toulmin’s The Uses of Argument (Cambridge: Cambridge University Press, 1958) might correctly be taken as progenitors of the movement. J. Anthony Blair and Ralph H. Johnson, who edited Informal Logic: The First International Symposium (Pt. Reyes, CA: Edgepress, 1980), consolidated some early developments. In this work, see their introduction (ix–xvi) and their article ‘The Recent Development of Informal Logic’ (3–28); also there see Michael Scriven’s ‘The Philosophical and Pragmatic Significance of Informal Logic’ (147–60). Johnson and Blair followed this collection with another entitled New Essays in Informal Logic (Windsor, ON: Informal Logic, 1994), containing the following important contributions: their summary of recent developments, ‘Informal Logic: Past and Present,’ (1–19); Derek Allen, ‘Assessing Arguments,’ (51–7); Maurice A. Finocchiaro, ‘The Positive vs. the Negative Evaluation of Arguments,’ (21–35); James B. Freeman, ‘The Place of Informal Logic in Logic,’ (36–49); Robert C. Pinto, ‘Logic, Epistemology and Argument Appraisal,’ (116–24); Christopher W. Tindale, ‘Contextual Relevance in Argumentation,’ (67–81); and Michael Wreen, ‘What Is a Fallacy?’ (93–102). This volume of articles is complemented by Ralph H. Johnson’s The Rise of Informal Logic (Newport News, VA: Vale Press, 1996) and by another collection of papers edited by
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3
4 5 6
Hans V. Hansen and Robert C. Pinto, Fallacies: Classical and Contemporary Readings (University Park, PA: Pennsylvania State University Press, 1995). In the Hansen and Pinto volume consider: J. Anthony Blair, ‘The Place of Teaching Informal Fallacies in Teaching Reasoning Skills or Critical Thinking’ (329–38); Maurice A. Finocchiaro, ‘Six Types of Fallaciousness: Toward a Realistic Theory of Logical Criticism’ (120–9); Trudy Govier, ‘Reply to Massey’ (172–80); David Hitchcock, ‘Do the Fallacies Have a Place in the Teaching of Reasoning Skills or Critical Thinking?’ (319–27); Ralph H. Johnson, ‘The Blaze of Her Splendors: Suggestions about Revitalizing Fallacy Theory’ (107–19); Gerald J. Massey, ‘The Fallacy behind Fallacies’ (159–71); and Frans H van Eemeren and Rob Grootendorst, ‘The PragmaDialectical Approach to Fallacies’ (130–44). See also Trudy Govier, Problems in Argument Analysis and Evaluation (Providence, RI, and Dordrecht: Foris Publications 1987). For work in the pragma-dialectical tradition see: Frans H. van Eemeren and Rob Grootendorst, Argumentation, Communication, and Fallacies: A Pragma-Dialectical Perspective (Hillsdale, NJ: Lawrence Erlbaum Associates, 1992). For a good textbook treatment of these matters, see Howard Kahane, Logic and Contemporary Rhetoric: The Use of Reason in Everyday Life (Belmont, CA: Wadsworth, 1992 and reissued with Nancy Cavender in 1998). There seems to be a ‘weak’ and a ‘strong’ psychologism haunting this movement. The strong psychologism is that criticized, for example, by G. Frege, E. Husserl, and J. Sukasiewicz. This psychologism aims to reduce logic to mental states – to make the laws of logic the laws of mental functioning – the study of which is an empirical science. As Frege and Husserl maintained, such an approach to logic and arithmetic undermines objective knowledge and challenges logic’s normative role; for them mathematical and logical relations are independent of psychological processes. The weak psychologism concerns in particular the participant relativity of fallacious ‘reasoning’: for an argument to be a fallacy it must appear to someone to lack cogency. The debate between psychologism and anti-psychologism involves assessing the ontic status of propositions and ‘propositional attitudes.’ The posture we take in this essay is a correspondence epistemology (see §6) in which the ideal world is just the material world reflected by the human mind and translated into forms of thought. J. Corcoran, ‘The Inseparability of Logic and Ethics,’ Free Inquiry 9 (1989): 37. Ibid., 38. Our treatment of Douglas N. Walton’s work on informal logic includes the following books by him: Informal Fallacies: Towards a Theory of Argument
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7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Criticism (Amsterdam: John Benjamins Publishing, 1987); Informal Logic: A Handbook for Critical Argumentation (Cambridge: Cambridge University Press, 1989); and Begging the Question: Circular Reasoning as a Tactic of Argumentation (New York: Greenwood Press, 1991). For works by John Woods used in our discussion, see below n. 7. In recent years John Woods has devoted considerably more attention to abduction and, with Dov Gabbay, to editing a monumental work, The Handbook of the History of Logic, than to fallacy theory. Our comments address his studies that are more directly focused on fallacies and fallacy theory. See in John Woods and Douglas Walton, Fallacies: Selected Papers 1972–1982 (Dordrecht, Holland: Foris Publications, 1989) their ‘Introduction’ (xv–xxi) and ‘On Fallacies’ (1–10). Consider also: J. Woods, ‘What is Informal Logic?’ in Blair and Johnson, Informal Logic, 57–68; ‘The Necessity of Formalism in Informal Logic,’ Argumentation 3 (1989): 149–67; ‘Sunny Prospects for Relevance?’ in Johnson and Blair, New Essays in Informal Logic, 82–92; ‘Is the Theoretical Unity of the Fallacies Possible?’ Informal Logic 16, no. 2 (1994): 77–85; ‘Fearful Symmetry,’ in Fallacies Hansen and Pinto, eds, 181–93; ‘Aristotle (384–322 B.C.),’ Argumentation 13, no. 2 (1999): 203–120; and ‘How Philosophical Is Informal Logic?’ Informal Logic 20, no. 2 (2000): 139–67. See also J. Woods, Aristotle’s Earlier Logic (Oxford: Hermes Science Publishing, 2001). Walton, Informal Fallacies, 26. Ibid., 93–6. Ibid., 77; cf. 81, 95, 291–4. All emphases in citations here and below are added unless otherwise indicated as belonging to the author cited. Ibid., 94. Ibid., 94–5. Ibid., 95. Ibid., 95–6; cf. Walton, Begging the Question, 216–24. Woods, ‘What Is Informal Logic,’ 59. Ibid., 58. Ibid., 59. Ibid., 60. Ibid., 60; emphasis in original. Walton, Informal Logic, x. Ibid., ix. See Maurice A. Finocchiaro, ‘The Positive vs. the Negative Evaluation of Arguments,’ in New Essays in Informal Logic, Johnson and Blair eds., 21–35 and Finocchiaro, ‘Six Types of Fallaciousness: Toward a Realistic Theory of Logical Criticism,’ in Fallacies, Hansen and Pinto, eds., 120–9.
440 George Boger 23 Each of these fallacies is defined, respectively, in John Corcoran, ‘Argumentations and Logic,’ Argumentations 3 (1989): 32 and 26. 24 A specific difference, of course, between wood and an audience is the consciousness/cognition of human beings. In this respect, human beings possess both active and passive moments in the process. When a human being becomes convinced about one or another belief, a change occurs within this participant. The passivity consists in receiving information; the activity consists in ‘getting it’ – both of which are mental acts. 25 See Categories 12: 14b14–22. 26 Metaphysics 933b30–1. 27 We have treated the topic within the framework of classical ontology and formal logic and have set aside considering how dialectical logic and dialetheism, along the lines developed by Graham Priest, might affect this topic. See, for example, Graham Priest, ‘Truth and Contradiction,’ Philosophical Quarterly 50 (2000): 305–19. 28 Throughout this discussion we have not distinguished ‘argument’ from ‘argumentation,’ generally following their use by informal logicians. But here and below we distinguish the two and define each, following Corcoran, ‘Argumentations and Logic.’ Consider also Corcoran, ‘InformationTheoretic Logic,’ in Truth in Perspective, C. Martinez, U. Rivas, and L. Villegas-Forero, eds., (Aldershot, UK: Ashgate, 1998): 113–35. 29 See J. Corcoran, ‘Information-Theoretic Logic.’ 30 The ideal, of course, is that there be a kind of isomorphism between the sequence of propositions ‘out there’ that are the conclusions of elementary arguments and the sequence of getting those propositions in the mind of a participant. 31 Cited in Corcoran, ‘Argumentations,’ 34–5. 32 This list is only summary. It indicates some contributions that formal logic, as a science of underlying logics, might make for evaluating natural-language discourse. 33 In this connection, one can easily recognize that the notion of a logically perfect language, which is a mathematical artefact, is useful for working with natural-language discourse. This posture takes the science of logic to study underlying logics as Alonzo Church, Introduction to Mathematical Logic (Princeton: Princeton University Press, 1956, 1996), conceives this matter. 34 Recall that in Prior Analytics Aristotle defined a syllogism with this restriction to develop a deduction system for axiomatic discourse as outlined in Posterior Analytics. One might as easily restrict a deduction system to include or exclude identity.
Mistakes in Reasoning about Argumentation 441 35 Argumentations 3 (1989): 149–67; cf. Woods, ‘How Philosophical Is Informal Logic?’ Informal Logic 20, no. 2 (2000): 139–67. 36 Woods, ‘Is the Theoretical Unity of the Fallacies Possible?’ Informal Logic 16, no. 2 (1994): 84. 37 Woods, ‘Aristotle,’ 219. 38 In this article Woods sketches an analysis of the fallacies treated in Sophistical Refutations as breaches in deductive logic. We too have aimed to accomplish much the same thing in Boger, ‘Aristotle on Fallacious Reasoning in Sophistical Refutations and Prior Analytics,’ Argumentations and Rhetoric (CDROM), 1997 OSSA Conference Proceedings, St Catharines, ON 1998), where we also treated a thesis that Aristotle’s analyses of the fallacies might have presupposed, at least conceptually, the logical theory underlying Prior Analytics. Woods, ‘Aristotle,’ 218–19.
Part Four: Respondeo JOHN WOODS
I have come to distrust the emphasis some logicians give to the notion of rationality. I have nothing against the concept intrinsically, but it shares some associational guilt with the tendentiousness with which these theorists invoke it. If I were czar for a day or so, I would place a moratorium on all talk of rationality by logicians. The fact remains that it is a deeply entrenched idea, much favoured by the establishment; it should prove difficult to dislodge. Logicians, and many other theorists of human performance – argumentation theorists, economists, and the like – are drawn to the idea that their theories have normative force precisely to the extent that they expose what it would be rational for a human agent to do in a given set of circumstances. Rationality in turn is often identified with whatever satisfies conditions laid down in a theorist’s ideal model. Rationality-on-the-ground is then identified as an approximation to these ideals. In an alternative approach, suggestive of Rawls’ idea of reflective equilibrium, rationality is a matter of what comports with conventions that adjust the tensions between the allure of settled practice and the recognition that it must sometimes be changed, that is to say, corrected. My difficulty with the ideal-models approach is that it tells no independently convincing story about what justifies the privileging of the norms that the theorist selects for placement in his model. At the end of the day, it is often assumed that the best that can be said is that they are analytic or that they are what the theorist firmly believes and firmly believes his readers to firmly believe as well. The analyticity move is dialectically problematic. With regard to any norm called into question by a doubter, a claim of analyticity cannot subdue that doubt. The sincere-and-widely-held-belief manoeuvre has much to recommend it. But inasmuch as it is just as
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accessible in direct application to human performance on the ground, there is no need to bother with ideal performances, thus making otiose the whole ideal-models approach. There is, of course, no denying that in getting his theory up and running, the analyst must make do with what he thinks is so and thinks that others would also accede to, if not insist upon. Those who favour the intuitions methodology of analytic philosophy make the same point in saying that theories of human performance, and of certain other things as well, are creative elaborations of ‘our’ intuitions. Fine as far as it goes, the intuitions methodology brings itself into disrepute in simply assuming that a theory’s founding intuitions are normatively canonical for rationality. An appeal from reflective equilibrium won’t succeed in answering this objection. The principal reason is that it over-likens rationality to grammaticality. We may agree that what is acceptable English at a time is whatever at that time abides by rules that have achieved reflective equilibrium. There is no disturbance of this conception of grammaticality in the fact that it tolerates enormous changes over time in what cuts the mustard linguistically. If reflective equilibrium also worked for what is rational, we would have to allow for the possibility that what passes for rationality two hundred years from now will bear little similarity to what passes muster now, or that what counts as rational now could have borne little resemblance to what hit the mark in 1600 a.d. In fact, when compared with grammaticality, which is a highly progressive phenomenon, rationality is rather inert. Being rational in 1600 was pretty much doing what makes for rationality today. Reflective equilibrium was tailor-made to accommodate, indeed to de-problematize, normative change. Given rationality’s comparative inertness, reflective equilibrium is the wrong story for it. For these reasons, and others whose discussion here would take us too far afield, I am a descriptivist about normativity. Let a theory provide an accurate account of how individuals on the ground actually reason, argue, and the like; then the correct default position is that this is how they should be reasoning, arguing, and the like. Normativity is defeasibly imminent in actual practice. And actual practice reveals individual agents seeking to update their beliefs and to take decisions about what to do, under conditions of cognitive resource paucity. Among these scarcities are information, time, and computational capacity, described in somewhat greater detail in Gabbay and Woods’ chapter on filtration structures, and also in their ‘The Practical Turn in Logic.’1 A further limitation under which the human cognizer operates
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is one that turns on the fugitivity of truth and knowledge, discussed in my ‘Cognitive Yeaming and Fugitive Truth,’ chapter 8 in this volume. Since believing P is believing that P is true, and since believing Q is thinking that you know that Q, being in the relevant state of belief quells the quest for truth and knowledge. Not every belief that quells the quest for truth is true. So having the relevant belief is not the same as having achieved the knowledge that had been quested for. This matters. It reminds us that even if we are optimizers with respect to our epistemic goals (‘the whole truth and nothing but the truth’), we are satisficers about their attainment. So whether a cognitive agent’s performance is subpar is a matter of the task he is faced with, what he is capable of doing, what resources are ready to hand, and what standard it is appropriate for him to hit in the circumstances. Central to it all is this question: How costly would it be in a given case for a cognitive agent to be, unawares, in a state of false belief which, as it turned out, quelled his cognitive quest? Leslie Burkholder’s elegant chapter puts considerable pressure on all these issues. For one thing, it reminds us of an answer to the idealmodels criticism that we haven’t yet considered. Suppose that your ideal model is a mathematical structure. Then your model’s embedded norms would appear to have a mathematical validation. They would appear to have as much normative force as the mathematics which validates them. To the extent to which the Bayesian articulation of probability is mathematics, albeit applied mathematics, why can’t it be said, with Burkholder, that a piece of reasoning from authority is sound to the extent that it is deductively sound reasoning using Bayes’ theorem in the ways that Burkholder details? Consider those cases in which agents on the ground reach the same or a qualitatively similar conclusion as would be got by Burkholder’s Bayesian. Suppose that there is no reason to suppose that the agent in question made any of the calculations the Bayesian made. This leaves the proponent of Bayesianism three options, all problematic by my lights. One is that the agent on the ground got the right answer for the wrong reasons. Another is that the agent on the ground did indeed make the Bayesian’s calculations, but did so automatically and unconsciously. The third is that even though he didn’t execute the Bayesian resolution device, he did something right, which, in turn, must be seen as approximating to its execution. There are cases in which the need to know must be attended by the highest available standards. Whether I have contracted AIDS is a case in point. It is a virtue of Burkholder’s chapter that it counsels me to
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seek out the opinion of, not one, but two experts, one an epidemiologist, who will make the diagnosis, and the other a Bayesian probability expert, who will crunch the numbers for me. This is much the right thing to do when the stakes are high enough. But why make this canonical for all cases? James Freeman’s chapter provides commendable support to those who invoke a distinction between conditions under which an argument is good and conditions which make it reasonable to accept it. The doctrine of fallibilism bears on this issue. Fallibilism is not merely the recognition that beings like us sometimes make mistakes, or even that sometimes these mistakes are made in earnest good faith. Rather it is the view that sometimes mistakes are made reasonably. Arising from the fact that it is sometimes reasonable to make a mistake (indeed that it is sometimes unreasonable to desist from it) is a substantial equivocation on the concept of good argument. Seen one way, an argument is good only if it is mistake-free. Seen another way, an argument is good if it would be reasonable to accept it. The rivalry between the premisstruth and premiss-acceptability criteria reflects something of this ambiguity, sometimes with unfortunate consequences for theories that attempt to adjudicate the tension between them. I part company with Freeman and the several others who see it as a condition of an argument’s goodness that it be valid or inductively strong, and of the reasonableness in accepting an argument that its acceptor have adequate reason to think that it is either valid or inductively strong. By these lights, most arguments are bad, and most argument acceptances are unreasonable. This is certainly the case for any notion of validity that guarantees truth preservation and any conception that ties inductive strength to conditional probability. There are lots of people (perhaps Freeman is among them) for whom this is not a telling objection. This is because they think that these criteria carve out ideal conditions, to which on-the-ground practice only approximates. No such refuge is available to me. I am left with no option but to regard the criteria as excessive, a view that I develop in greater detail in ‘A Resource-Based Approach to Fallacy.’2 Consolation arguments are in an interesting class by themselves. I dare say that there is no one who would judge them either as valid or inductively strong. This will cut no ice with those who hold that only the valid or the inductively strong make the argumentative grade. They will simply reject consolation arguments as subpar. It is a virtue of Trudy Govier’s chapter that it demonstrates how difficult it is to
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make the case against consolation arguments. Consolation arguments would seem to have two quite different targets, one causal and the other justificatory. When I pursue the causal target in citing the circumstances of a third party, I am trying to get you to feel less upset or aggrieved by your own situation. But if my task is justificatory, my argument is one in which I attempt to demonstrate that your turmoil or your grief is unjustified. Consider a case. You spill a drop of ketchup on your jeans. They will have to be washed. You are utterly defeated by this, and have to take to your bed. Compare this with the case in which someone else splashes a different drop in his eye and promptly and permanently loses the sight of it. He is greatly distraught. The two of you have fallen into a condition of grief and black despair. This is, for you, not a state your situation entitles you to. The relevance of the other’s situation is that it is one in which grief and despair are justified by the awful circumstances of his case, and the fact that that is so presupposes a standard of entitlement which the first case fails to achieve. The non-justification argument is thus an argument from proportionality. The grief and despair of the first party are not proportionate to the facts of the case. If anything, a causally oriented consolation argument is still more easily defended. If I offer you the consolation of some third party’s greater suffering, my argument succeeds if it chances to bring about your consolation. Pedants may insist on making something of the distinction between the goodness of an argument and its efficacy in producing desired results. So causally efficacious consolation arguments can be awful arguments. Yes, this is so. Let’s write them off without further ado. But consider a further class of cases, in which I seek to produce in you a certain outcome by providing you with certain information. Suppose I tell you that your daughter has just won a Rhodes Scholarship. I am glad to be harbinger of such good fortune because I am glad that you will be made happy by it. What if you are left wholly unmoved by this news? The short answer is that there is something wrong with you, for it was news that should have made you happy. There are norms about these things which matter for causal consolation arguments. In the case we have been discussing, when I seek to console you with an argument designed for that purpose, what I reveal of the third party’s case insinuates a standard of awfulness appropriate for his grief but not for your own. You might then be chastened by this information and, as the vulgar have it, wise up. Mission accomplished. Govier is right to point out that there are well-known accounts of
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argument which offer little guidance, if any, on this question. In one case, the account may have a theoretical or taxonomic apparatus that is not load-bearing. In another, the theory is rigged for dispute resolution, not consolation. And so on. She is also right to emphasize that on her own account of relevance, there is a problem as to whether consolation arguments meet the required conditions. Suppose, however, that relevance is taken in the manner of Gabbay’s and my Agenda Relevance. Then information is relevant for an agent with respect to some agenda or goal or disposition he has if it plays a role in advancing, attaining, or realizing it. When I press a justificatory consolation argument, I call attention to the justificatory standard that my addressee fails to meet. When I press a causal consolation argument I call attention to a causal standard that my addressee fails to meet. In each case, I present him with information relevant to his disposition not to be stupidly unhappy. In some respects, ‘Filtration Structures and the Cut Down Problem for Abduction’ and Jim Cunningham’s paper are responses to ‘The New Logic,’ which Gabbay and I published in 2001.3 We there took note of an important development in the modern history of logic. This is the pressure exerted by computer scientists, AI theorists, cognitive psychologists, and certain reform-minded logicians to reposition logic as a theory of human reasoning. This, of course, was the original conception, given the Greek equation of a theory of reasoning with a theory of inner argument. For nearly all of its 2500-year history, logic has had a laws-of-thought orientation, one that it lost in the nineteenth century, when logic completed its turn toward the mathematical. This transformation was amazingly fruitful, attaining profound results in the four core domains of mathematical logic – set theory, model theory, proof theory, and recursion theory. However, in each case, it would be the most serious nonsense to think that these disciplines reveal the structure of human thought. Even in the austere abstractions of proof theory, we see little inkling of the human; and, in systems of natural deduction (tendentiously so-called) what pass for rules can with greater accuracy be seen simply as conditions on logical consequence. This makes it easy to appreciate mathematical logic’s contempt for psychologism. The mathematical turn in logic is here to stay. But it is in process of losing some of its exclusivity, under press, I say, of developments in adjacent disciplines. In their desire to simulate the behaviour of human reasoners, computer scientists and AI theorists would soon discover
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that the classical models of mathematical logic fare poorly as simulacra. Given their interest in human reasoning on the hoof, informal logicians, dynamic and situational logicians, and others, would soon discover how little mathematical logic has to say about it. Under these influences, logic was taking a new turn. This is what Gabbay and I have dubbed ‘the practical turn in logic.’ ‘Practical,’ in our conception of it, denotes neither the content of a piece of reasoning nor the degree of rigour with which it is expected to be expressed. In our approach, practical reasoning is reasoning transacted in psychological space, in real time, and with the assistance of limited assets, such as information, time, and computational capacity. The human reasoner is, as Cunningham reminds us, a temporal agent who does his thinking in his head. The head that does the thinking is attached to an actor who operates in the world. This fact alone requires the human reasoner to be a belief-updater. The supposition that the reasoning of such beings is properly in the ambit of logic runs into two difficulties straightaway, as we also see from Cunningham. One is logic’s residual contempt for psychologism. The other is that, to date, it has proved inordinately difficult to model such behaviour accurately and non-trivially. It might be allowed that a virtue of Gabbay and Woods’ piece on filtration structures in abductive reasoning is the (possibly inadvertent) urgency it gives to the practical logician’s task of showing the psychological relevance of his logic. Suppose we agree that in the case of abductive reasoning, the abducer seeks to hit a target which cannot then and there be hit on the basis of what he presently knows. His task is to select from an indefinitely large set of hypotheses, one which, together with his knowledge-base, does hit the target, and does so in ways that make it reasonable to accept that hypothesis provisionally. Part of the abducer’s task is to find a particular hypothesis in an arbitrarily large plenitude of them. How is this done? The winning hypothesis (or hypotheses, in case of a tie) has a determinate place in a filtration structure, which is a structure got by the interplay of relevance conditions and plausibility conditions in spaces of possibilities. This is a fact that resonates attractively, since it would appear that the abducer’s choice of hypothesis must in turn be sensitive to considerations of possibility, relevance, and plausibility. This gives rise to some interesting convergences. The on-theground abducer picks the hypothesis that has a unique place (possibly with ties) in a filtration structure. Filtration structures turn on conditions of possibility, relevance, and plausibility. Abduction on the ground
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answers to considerations of possibility, relevance, and plausibility. All the same, abducers on the ground do not, in the process of hypothesis selection, actually construct filtration structures. What, then, is the logician’s contribution? It is in part a negative one, that of displaying an imaginary process of hypothesis selection that the practical agent doesn’t use. This is the juncture at which the old psychologism issue takes a new twist. It twists around the obvious question: If not by constructing filtration structures, then by what means does the abducer hit his selection targets? Some will say that here logic has no proper contribution to make. Others are minded to think that logic does indeed have further contribution to make – a contribution over and above what the experimental psychologist can offer – but that the further contribution has yet to be worked out in detail. It is a promissory note, whose redemption will be at least as difficult as Cunningham’s sympathetic overview suggests. It is easy to understand a logician’s reluctance to invest heavily in this redemption. But, for good or ill, I am not in that number, nor is Gabbay. Readers can judge the prudence of our optimism by inspecting Agenda Relevance and The Reach of Abduction, the first volume of our omnibus work, A Practical Logic of Cognitive Systems.4 George Boger’s chapter also takes up the issue of psychologism, which he regards as a threat to the objectivity of knowledge. He notes with disapproval the uncritical disposition towards psychologism on the part of various informal logicians and fallacy theorists. He sees in the Woods-Walton approach to fallacy theory a not altogether successful attempt to avert psychologism, a failing that he here seeks to correct. If he reads the preceding pages of this commentary, I can only think that Boger will now despair of me, and that a nodding acquaintance with ‘Cognitive Yearning and Fugitive Truth’ (this volume) would only deepen his gloom. But Boger is certainly right to see in the joint work by Douglas Walton and me in the period 1972–85 a prolonged effort to do as much as possible of fallacy theory within the bounds of mainstream logic. Indeed, the dominant idea of the WoodsWalton Approach was that much, if not all, that is worth saying about the fallacies can be captured in systems or adaptations of systems that had already won their spurs in the province of logic. Much of Boger’s case against the Woods-Walton Approach is actually directed to Walton’s later work, in which he widens his theoretical embrace to incorporate aspects of the pragma-dialectical analysis of critical discussions. While Boger’s criticisms here miss their intended mark, there is no
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doubt that the Woods-Walton Approach in various respects does fail his conception of a formal logic of the fallacies, and Boger shows this to be so with considerable success. I suggested above that my latter-day espousal of psychologism in logic would be bleakly received by Boger. There is, I fear, no assurance I can give him on this score, except possibly a consolation argument to the effect that some mistakes are even stupider than mine. However, perhaps he will find it ironic that in my new-found affection for psychologism there is no diminution of my respect for formal models in logic, even in practical logic. To that extent I count the logic that Gabbay and I are working out in A Practical Logic of Cognitive Systems to be formal logic. But I don’t expect that this will make Boger any the less dissatisfied with the psychologism that it openly embeds. notes 1 Dov M. Gabbay and John Woods, ‘The Practical Turn in Logic,’ in Handbook of Philosophical Logic, 2nd rev. ed., Dov M. Gabbay and F. Guenthner, eds. (Dordrecht and Boston: Kluwer, 2005). 2 J. Anthony Blair, Daniel Farr, Hans V. Hansen, and Ralph H. Johnson eds., Informal Logic at 25: Proceedings of the Windsor Conference (Windsor, ON: OSSA, 2003), 1–9. 3 The Logic Journal of the IGPL 9 (2001): 157–90. 4 (Amsterdam: North-Holland, 2003).
Part V Values
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24 Engineered Death and the (Il)logic of Social Change MICHAEL STINGL
The focal point of this paper is John Woods’ argument in ‘Privatizing Death: Metaphysical Discouragements of Ethical Thinking.’1 I first encountered this argument, or at least its core, in one of my first years at the University of Lethbridge, at the Great Debate in 1991 between John Woods and Dr Henry Morgentaler, who was by then the triumphant icon of the pro-choice side of the abortion controversy in Canada. The debate was held in the university gymnasium, which seats nearly 1300. The seats were packed full for the debate, not just with university people, but with people from across southern Alberta. Southern Alberta, for those who don’t know it, is the Bible Belt of Canada; and while Henry Morgentaler had just won over the majority of Supreme Court judges to the justice of his cause – free-standing abortion clinics where women could exercise their choice to abort their pregnancies without the oversight of a three-physician panel to determine whether there was sufficient threat to the women’s life or psychological well-being to warrant an abortion – the good folk of southern Alberta were not so easily swayed. The crowd was itching for a good argument, and Woods was the hometown hero who would set things right when it came to Dr Henry Morgentaler, free-standing abortion clinics, and judicial activism regarding the alleged right of women to terminate their pregnancies by their own individual choices. Woods, I’m sorry to say, disappointed. He began in a jocular manner, conceding to Morgentaler what appeared to be the main point of the debate: Morgentaler thought abortion was morally permissible, and he, John Woods, was utterly powerless to persuade him otherwise. Woods then proceeded to discuss, at some length, an article he had been reading in the Times Literary Supplement, which was tangential to
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the evening’s debate but nonetheless of some passing interest. Woods, of course, was going somewhere, but by the time he got there most of the crowd was no longer with him, and those who were did not appear to understand the significance of what he then said. What he then said forms the core argument of ‘Privatizing Death,’ which I want to examine in this chapter. Woods had the time for convivial chinwagging, he thought, because he had an argument against Morgentaler for which there was no reasonable reply. And indeed, Morgentaler gave the argument no reasonable reply, either because he didn’t see the force of the argument, or because he was more interested (and more successful) than Woods in playing to the crowd, and he realized that in that regard, certainly, the argument was best left alone. One couldn’t tell, since Morgentaler simply ignored the argument Woods had finished with and started in on his own argument for why freestanding abortion clinics were a matter of necessity to save women’s lives. What was clear, or so I thought at the time, was that Woods couldn’t read a crowd to save his life. But I was a younger philosopher then, and although I didn’t know it and would have argued the point, I still had a lot to learn, a good deal of it, it has since turned out, from John Woods. Like Socrates, Woods wasn’t arguing before a crowd to save his life. He was arguing toward what he had good reason to believe was the truth, and he was prepared to be convinced otherwise, if anyone in the room had the logical mettle to do it. I doubt that I did, at least at the time; in any case, as the evening continued, it became clear that certainly no one else did. Woods didn’t wind up having to drink any hemlock, although some in the crowd might have relished the thought, had it occurred to them. In any event, by the end of the evening, two things were clear. Woods had resoundingly lost the debate, and no one had touched his argument. But why not? What was the argument, and why did it fall (so resoundingly?) on deaf ears? The beginning of an answer to these questions is contained, I think, in the article that I want to draw attention to here. As I said earlier, the article contains the core of Woods’ argument against Morgentaler. This argument has two important parts. Its first part consists of an interesting and important taxonomy of social debates in modern liberal democracies. Many of our most significant social debates are, according to Woods, standoffs, an idea for which he proposes a concise and multi-layered definition. Standoffs of what he calls force 1 meet two conditions:
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(1) people disagree over the point at issue in the standoff; (2) there is no agreement on a process that might settle the issue. Standoffs of force 2 add the condition (3) there is no agreement to disagree. Standoffs of force 3 add the condition (4) there is no agreement on referring the issue to third-party arbitration. And finally, standoffs of force 4 add the condition (5) neither side of the disagreement is prepared to lose at the legislative or judicial level of social decision making. Force 4 standoffs are of a sort to threaten civil society; should state institutions act in one way or the other – and typically, in standoffs of force 4, not to act is thereby to act, de facto, in one way or the other – one or the other of the two sides of the standoff will countenance acts of civil disobedience. And this is in fact what had happened in Canada regarding abortion: in ongoing acts of civil disobedience, Morgentaler set up abortion clinics outside what the law allowed, and pro-choice advocates assisted women in finding their way to these clinics, often running hostile gauntlets of pro-life protesters stationed outside the clinic doors. On the other side, the building housing one of Morgentaler’s clinics, in Toronto, was destroyed by fire, and Morgentaler himself was attacked by a man wielding a pair of garden shears.2 Standoffs of force 4 are thus serious threats to civil society; they involve serious harms on each side’s understanding of the underlying moral issue, and a serious level of social uncertainty, since neither side is able to press its claims, reasonably and successfully, against the other. As such, standoffs of force 4 might be thought to provide an appropriate context for the ‘maximin’ rule of social decision making: examine the losses that might accrue in acting one way or another, and maximize the minimum outcome that might result, however the issue might be decided. Woods calls his version of this general principle RR, and defines it as follows:
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RR: Settle the issue in such a way as minimizes the realization of the greater possible cost.3 Applied to the standoff of the abortion debate, RR tells us, according to Woods, not to allow abortions until we can attain a sufficient level of social certainty to the effect that they do not involve the murder of innocent human beings, where such human beings might be fetuses, embryos, or zygotes, from the moment of conception onward. RR is so clearly reasonable, according to Woods, and the abortion debate so clearly a standoff of force 4, that it is utterly remarkable that a public appeal to RR does not immediately resolve the issue. Thus did Woods expect to win the debate against Morgentaler hands down, and thus does he continue on, in the article here under scrutiny, to puzzle out why this argument did not bring down the house, either then or since. Before turning to Woods’ response to this problem, which Woods sees, in his usual fashion, as at base a logical conundrum, we need to examine in some detail his general appeal to the maximin rule of social choice. The maximin rule makes its most widely heralded entrance into the social and political philosophy of the twentieth century in 1971, in John Rawls’ simply entitled book A Theory of Justice.4 At the heart of this book is Rawls’ effort to resolve a general standoff of at least force 3, the conflict between liberty and equality that is at the heart of modern democracies. In their 1908 book, the even more simply entitled Ethics, John Dewey and James Tufts devoted the final third of their discussion of modern ethics to the salient practical problem of their own day, that of finding a middle ground position between the rapacious individualism championed by the robber barons of early twentieth-century economic might in the United States, and the calls for greater social equality coming, in response, from socialists and the early trade union movement.5 The acuity of this early twentieth-century problem was historically blunted by progressive politics, the relative success of the trade union movement, and, finally, the burst of economic growth that followed the Second World War. But the problem itself has hardly gone away, either in the United States or, as we have now come to understand more of the dimensions of the problem, globally. Rawls published his book in 1971, before global concerns had come to dominate the international agenda, and so his focus is naturally on resolving the conflict between liberty and equality within the confines
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of a single nation-state. Rawls thinks this conflict needs to be faced at the very foundation of social justification, at the level of the fundamental principles that will themselves structure a constitutional convention, the latter and more particular principles of which will structure later stages in the justification of particular social policies. It is only at the most fundamental level of social choice that Rawls thinks the maximin rule of social choice is appropriate, the level at which we are choosing fundamental principles of justice that will affect the general shape of society for all generations to come. Rawls is well aware of the general limitations of the maximin principle, but he thinks that because of what is at stake in the original position, and how uncertain those stakes are for any family line, a principle as conservative as maximin is fully appropriate. This appropriateness has not gone unchallenged in subsequent Rawls scholarship, but the basic lines of Rawls’ defence are clear: if you are playing for everything, for yourself and all your future offspring, with a future that is completely uncertain to you, you had better pay attention to the worst outcome and avoid it. Even here, however, the limitations of the maximin principle are clear: in directing our attention solely to the worst possible outcome, it has us completely ignore other possibly very bad outcomes, and worse, it does not allow us to weight these outcomes with even the equiprobability of chance. Maybe Rawls wins the debate at the level of the fundamental principles of society, maybe he does not. My point here is that with his own appeal to RR, Woods is on far shakier ground than Rawls. Woods wants to use the maximin principle far downstream from any process that would determine the basic institutional structure of society, to resolve, directly, particular problems of social policy in a way that disallows us to look at just how bad outcomes beyond the worst outcome might possibly be, and in a way that completely ignores any appeal to the probabilities of these outcomes being as bad as they might be. RR is hardly by itself an argumentative trump; at best, it is the start of an argument that will be hard going. That this argument will be particularly hard going in the case of ending fetal life through abortion is my own argument in the remainder of this paper. RR, applied to the abortion debate, would tell us that in the face of a standoff of force 4, the worst policy outcome that could occur in this standoff is the death of large numbers of fetuses, and that we can safely ignore any other harms because in force 4 standoffs we are in a situation of such great social uncertainty that all talk about probabilities must likewise be ignored.
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Against these claims, I think there are significant reasons why RR is inappropriate in the context of the abortion debate, reasons the significance of which goes far beyond the confines of this one particular debate regarding under what conditions a human life may be legitimately ended. To get at these reasons, I want to examine Woods’ own diagnosis of why RR fails to resolve the abortion debate, and then turn to a more fruitful diagnosis of the problem, one that Woods himself moots in his earlier book on engineered death.6 In the article we began with, to understand the apparent illogic of social change when it comes to abortion, Woods turns to the logic of social change involving taboos. At its briefest, Woods’ argument is that while there has been a long-standing taboo against killing human beings, this taboo, as such, was never justified by social reasons widely understood and accepted by all. As the taboo has been called into question by those demanding more liberal policies on abortion, and now, even more recently, by those demanding more liberal policies on euthanasia and genetic engineering, the lack of justificatory reasons behind the taboo has become all too publicly apparent, and so the taboo has simply collapsed. Such is the fate of even our deepest taboos in an age of reason now reaching its years of full maturity. Something like this has been going on, I think. But the process is much more complicated than Woods suggests either in his article or in his earlier book. The book picks up several important strands of the process, but not all the strands, and it does not investigate the ways in which these strands are intertwined with one another. The process of social change that has culminated in, among other things, more liberal policies on abortion, is both inherently more messy, and more illogical, than Woods would have us believe.7 One important strand of the process of social change that Woods discusses in both his article and his book is the fact that by the twentieth century we no longer have an adequate public understanding of the harmfulness of death, or even of death itself.8 At the beginning of the twentieth century, Bertrand Russell was quite confident about what would happen to him after he died: ‘when I die I shall rot.’ This follows from the fact that ‘man is part of nature,’ and as such, our ‘thoughts and ... bodily movements [must] follow the same laws that describe the motions of stars and atoms.’9 A similar evocation of mechanism, if not materialism, is to be found, across the ocean in North America, in George Santayana’s claim:
Engineered Death and the (Il)logic of Social Change 459 TABLE 24.1 Search results for 1990s Category Brain death The meaning of life/death Existentialism and death Hospitalized death Euthanasia Suicide Immortality Aging and death Total
Number of entries
Per cent
44 68 12 17 70 11 7 6
19 29 5 7 30 5 3 2
235
100
I believe that nothing is immortal ... No doubt the spirit and the energy of the world is what is acting in us, as the sea is what rises in every little wave; but it passes through us; and cry out as we may, it will move on. Our privilege is to have perceived it, as it moved.
Commenting on this passage, Will Durant adds that ‘[m]echanism is universal, and prevails even in the inmost recesses of the soul.’10 From continental Europe there arises an even deeper pessimism about natural mechanisms and the inmost recesses of the soul. God, according to Freudian psychology, is a fantasy about the ultimate father, naturally produced by the inmost mechanisms of the human mind, a mind that is unable to comprehend its own mortality. According to Freud, ‘[w]henever we make the attempt to imagine our death, [we] perceive that we really survive as spectators.’11 This view about the unimaginability of our own death is echoed by other early twentieth-century psychologists and philosophers, perhaps reaching its apogee with Martin Heidegger.12 Where had things gotten to by the end of the century? In 1998 a research assistant and I searched The Philosopher’s Index, using the term ‘death,’ for the 1990s, the 1980s, and the 1970s, the decade in which Woods’ Engineered Death was published. For the 1990s, the results arranged themselves as shown in table 24.1. The biggest combined category is euthanasia, suicide, and brain death (54 per cent), with most articles devoted to the clinical management of end-of-life health care; the articles on hospitalized death might
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also be added to this larger category, bringing it to 61 per cent of the total entries. The second-largest combined category is the meaning of life/death and existentialism and death (34 per cent), but most of the existentialist articles struggle over how to interpret Heidegger’s obscure notions of the ‘dasein’ and its ‘thrown-ness’ into the world, and most of the analytic articles struggle with the Epicurean view that death is nothing to be feared because it is, in fact, nothing. The 1970s and 1980s showed equally unimpressive results as far as yielding any evidence of a general understanding of death that might be of direct concern to twentieth-century human beings in the English-speaking world. One self-conscious response to these results comes from Joseph Amato in an anomalous entry from 1993: Majorities present, like majorities past, prefer their amphitheatres to philosophers’ books. They wish simple explanations, impossible hopes, and good stories, however contradictory and imprecise they are, to great abstraction and arcane analysis. Death causes people to tell stories.13
By the end of the century, then, as philosophy completed its movement toward abstract analysis, art and literature were a far better place to look for a genuine, human understanding of death than was philosophy, however imprecise and contradictory such an understanding might be.14 The great pleasures and insights of art and literature to one side, I think that as philosophers we need a clearer understanding of how twentieth-century philosophy, science, and technology have all played integrated roles in our contemporary inability to understand death; the abstractness of twentieth-century philosophy itself is hardly the main problem here. In Engineered Death, Woods mentions, but does not explore, several important strands of the problem: Our underlying philosophy is liberal Utilitarianism, the view that man is by nature free and his most profound expression of freedom lies in his use of the world in pursuit of his own happiness and in accordance with his own values. The underlying epistemology is empiricism, the view that genuine knowledge of the world is impossible without strong perceptual confirmation. The underlying scientific methodology is positivism – operationism in
Engineered Death and the (Il)logic of Social Change 461 physics, behaviourism in the social sciences, formalism in mathematics and emotivism in metaethics.15 [Commenting on doctors’ reluctance to terminate life-saving treatments:] it would be the sheerest nonsense to suggest that a medical interest in the prolongation of life is as such misplaced. But it is very much misplaced if is to be perceived as an end in itself. So perceived, it implies an understanding of medicine that can only diminish its humanity, by allowing for the replacement of concrete human goals with the abstract objectives of a runaway technology.16
These are some of the strands of explanation for our contemporary inability to understand death, but there are others, and again, they are intertwined. In an effort to begin gathering together these strands and making sense of them as an integrated whole, let me take up where Woods’ last point has left us, with what we might call, following Woods, the abstract importance of modern technology, particularly as it pertains to health care. At the turn of the twentieth century, hospitals were a place you went to die, if you were not well off enough to die at home, with female family members attending you, perhaps with the assistance of a private nurse. Early twentieth-century medical technology, however, allowed for the beginnings of the medical control of death. In particular, new diagnostic techniques were developed, especially for the early detection and prevention of cancer and cardiac disease, the two kinds of diseases with the highest mortality rates in the twentieth century. Chief among these techniques were electrocardiography and X-rays, and in the case of the treatment of cancer, surgery and radiation. All of these techniques were available mainly in hospitals, a centralized location that also allowed for the standardization of professional training for both physicians and nurses. Nursing, in particular, became a profession open to women, and as hospitals developed, nurses preferred them to private homes because they offered more certain employment opportunities and working conditions. The later discoveries of antibiotics and vaccines gave hospitals an increased importance in the medical control of death, and with increasing urbanization, hospitals became a central location in the community for the delivery of health care, particularly health care involving life-threatening medical conditions. In Canada, the steady and upward trend toward hospitalized death began early in the century, long before the Canada Health Act of
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1984 ensconced hospitals, doctors, and nurses at the centre of the Canadian health system. The removal of death from the home meant at least two things, abstractly and concretely: death was removed from the common experience of the private home, and it was something to be technologically managed rather than merely accepted and accommodated. Especially important for the later abortion debate, I think, was the decreased incidence of childhood death and the decreased risk of death during childbirth. The early technologies of modern medicine, like X-rays and radiation treatments for cancer, were closely tied to the rise of modern physics. Physics brought with it a metaphysics of materialism, the materialism at the heart of Russell’s comments about what would certainly happen to him after he died. In the essay from which the quote was taken, and in related essays, Russell articulated a philosophy appropriate to the rise and power of modern physics and its concomitant technologies.17 According to Russell, the meaning of human life, and the values that might guide it, could only be found in human nature itself. Human beings, for Russell, have a tendency to create social arrangements that result in a great deal of misery; our only saving grace is to be found in the fact that we are naturally attracted to both knowledge and sympathy, capacities that enable us to change our social arrangements for the better, if we can get the better of the fearfulness that leads us to mistreat our fellows in the first place. Again, on the other side of the Atlantic, in North America, John Dewey was articulating similar views about the betterment of human society.18 Though humans were naturally selfcentred, we were also sympathetic, and through continuing social experimentation we could achieve institutional structures that led to a greater and greater harmonization of our individual interests. What mattered in human life was the elimination of misery and the greater satisfaction of individual interests through greater social harmony. A similarly materialistic approach to the human mind and human motivation spilled over into the new discipline of psychology. Of more lasting influence than Freud was the behaviourism being developed in North America and Russia.19 According to behaviourism, the mind, including beliefs and intentions, was of no scientific interest. What explained human action was, on the one hand, observable stimuli and, on the other, observable responses. What linked them was locked in the physiology of the organism, in mechanisms then invisible to empirical observation, chiefly the brain and central nervous system. While behavourism as such collapsed under subsequent criticism, its legacy
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continues among animal behavourists, neuroscientists, and evolutionary psychologists. Behaviourism was a close ally of logical positivism, which was, as Woods reminds us, closely connected to emotivism in ethics. The early emotivists declared that because moral claims could not be empirically tested, they were literally meaningless and were, at best, expressions of an emotional attraction toward or repulsion from their subject, a verbal expression that might be emotionally contagious to those around the speaker, much like the cry of one baby in a maternity ward quickly spreading into the cries of others.20 As the century progressed, positivism and emotivism, like behaviourism, gradually sank under their own weight; but like behaviourism, both doctrines have had a marked influence on the remainder of the century. The dominant view in metaethics is an anti-realism about moral values, combined with various efforts to account for the fact that moral language and moral arguments appear to be about moral truth, even if there is no such thing, at least apart from our thinking so.21 All these factors have deeply affected twentieth-century thinking about death and its harmfulness: emotivism in ethics, behaviourism in psychology, the materialism inherent in the new physics, and, most closely related to concerns directly related to human death, the impressive advances in medical technology made possible by physics. But the second science behind the advances in medical technology, the science of biology, has been perhaps even more important in terms of shaping twentieth-century attitudes regarding death and its meaning for human beings. The coming of age of the theory of evolution in the waning days of the nineteenth century and in the beginning years of the twentieth helped bind together the strands of social change already mentioned, with each other and with three additional and equally important strands of social change: liberal individualism, libertarianism, and the anti-war movement of the 1960s and 1970s. Before moving on to these final three strands, we need to consider the central importance of Darwin’s revolution in biology for twentieth-century thinking about the fact of human death and what it might mean to us as individuals and social groups. Materialism, the search for moral values in human nature, and the naturalness of death were all reinforced by Darwin’s theory of evolution. While earlier scientific revolutions could easily be squared with a belief in God, if not a literal interpretation of the Bible, the theory of evolution could be squared with neither. Even if the Copernican revo-
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lution ended the music of the spheres, the natural laws it trailed in its wake could still be used to highlight the beauty and integrity of the natural mechanisms created by the designer of the universe. Through a more accurate understanding of the creator’s work, one could find greater empirical evidence for his sublime intelligence and goodness. Not so with the aspect of nature exposed by Darwin’s theory of evolution. Mutations were sloppy and haphazard, and natural selection was ruthless in its elimination of poor experiments, experiments tied, in many cases, to sentient creatures who perished in miserable ways. The theory of evolution, and the growing evidence that supported it, gave real force to Hume’s prescient objections to the argument from design: Look round this Universe. What an immense Profusion of Beings, animated and organiz’d, sensible and active! You admire this prodigious Variety and Fecundity. But inspect a little more narrowly these living Existences, the only Beings worth regarding. How hostile and destructive to each other! How insufficient all of them for their own Happiness! How contemptible or odious to the Spectator! The whole presents nothing but the idea of a blind Nature, impregnated by a great vivifying Principle, and pouring forth from her Lap, without Discernment or parental Care, her maim’d and abortive Children.22
All Hume lacked was his ‘great vivifying Principle’; Darwin supplied it, at least partially, with his own principle of natural selection. The world created by Darwinian evolution was an ugly one, both from an engineering perspective and a moral perspective. Life was cheap, and there was nothing particularly special about human life. Our species was subject to the chance ebb and flow of mutation and natural selection just like any other. From early on Darwin himself agonized about this aspect of his theory and its wider social implications, in a way that his early champions in the arena of public discourse, such as Thomas Henry Huxley and Herbert Spencer, did not. Both Huxley and Spencer, in their dramatically different ways, thought evolutionary ideas freed men (and to some extent women) from historically imposed social hierarchies, and that just as the evolution of species led from lower forms of organization to higher forms, so too might social progress undo the less efficient and less beautiful forms of human social organization that had hitherto marked human history.23 In this way thinkers like Huxley were natural allies of liberal thinkers like John Stuart Mill and John Dewey, ‘liberal Utilitarians,’ to use
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Woods’ somewhat misleading phrase. With Huxley and Spencer, however, the strands of social change become twisted. On the one hand, Huxley and Mill lead quite nicely to the socially progressive liberal views of Dewey and Russell, both extremely important public philosophers in the first part of the twentieth century.24 Spencer, however, begets, in the early twentieth century, William Graham Sumner, Andrew Carnegie, and Social Darwinism. At the beginning of the century, the ideals of this extremely individualistic view of human value exerted a powerful hold on the North American imagination of the United States. Like behaviourism and emotivism, Social Darwinism gradually receded in social importance, but not under its own weight; Dewey, progressive politics and the trade union movement finally got the better of it. But just the same as behaviourism and emotivism, Social Darwinism lived on through the end of the century in a more attenuated and powerful form, in this case, libertarianism. Woods wrestles briefly and unsuccessfully with libertarianism in Engineered Death. Perhaps the single most influential article in the North American debate over abortion is Judith Jarvis Thomson’s ‘A Defense of Abortion.’25 Woods takes it up at some length on pages 77 to 82 of his book, arguing that while it is one thing to unplug oneself from Thomson’s violinist, it is quite another to ‘unplug’ oneself from a fetus. Woods’ claim is that the analogy fails, and that the very things that make us think it is morally permissible for us to do something that will cause the death of the violinist are the very things that are absent from the abortion case. Being attached to the violinist creates, says Woods, a social monstrosity: our privacy is completely invaded, our mobility severely limited, and with it the ability to lead anything like a normal human life. Not so in a normal pregnancy, or even in imaginary cases that are closer to the situation of a normal pregnancy. What Woods misses here are two of the most important aspects of Thomson’s example. The first is that human life is not always a trumping value. If the inconvenience is great enough, a human life is not as important as the amount of inconvenience this life may impose on the lived experiences of another individual. Underneath this first point is a second: in a libertarian world, none of us owes anyone else, as a matter of individual rights, anything more than non-interference. If the only thing that would save my life is the cool hand of Henry Fonda on my brow, to cite another of Thomson’s examples, I have no right against Henry Fonda for that single, life-saving gesture. But if Woods is innocent of the reactionary concerns of late-century
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libertarians to any social policy that would curtail individual freedom, he is equally innocent of any feminist concerns that might be geared toward greater social equality. In an argument that for me is the most troubling of the book, to be found in the same chapter as his response to Thomson, Woods compares a woman’s duty to sacrifice her own life for the life of her fetus to a man’s duty to sacrifice his life for his country in war.26 A woman’s duty to protect the life of her fetus is equal to her duty to protect her own life, says Woods, and a husband’s duty to protect his wife’s life is equal to his duty to protect the life of a fetus his wife might be carrying.27 Whichever way one chooses, husband or wife, the moral tragedy is equal; the tragedy is compounded, one supposes, if the wife chooses for her own life against the opposite choice of her husband. Still, Woods has put his finger on an important connection here, the connection between war and pregnancy; what he has missed, however, is the more particular connection of the feminist movement of the 1970s to the anti-war movement of the 1960s, which was itself, in part, a response to the twentieth-century technology of warfare. We began our discussion of the social factors leading to the abortion debate with a discussion of how science and technology led to the twentieth-century trend toward hospitalized death, a general social situation in which most people die in hospital, with the timing and nature of their deaths carefully and professionally managed by doctors and nurses. The other application of life-and-death technology was to the waging of war, with utterly disastrous results. The First World War brought with it machine guns, tanks, and poisonous gases. The casualties that resulted were unprecedented, and they affected at least three generations of parents, sons, and daughters. In the Second World War casualties were moved, in a new and impressive way, from the battlefields to the cities. Having conquered the air more completely than in the First World War, both sides heavily bombed densely populated urban areas. The century’s greatest triumph of science and technology, the atomic bomb, was used to level two Japanese cities, Hiroshima and Nagasaki. In the cold war that followed, the populations of major cities were held hostage against one another in an era of mutually assured destruction. Again, generations were affected, and affected deeply. Near the end of the cold war the United States became involved in Vietnam. Unsure of the moral legitimacy of the war, significant numbers of young males protested. Many avoided the war, some of them by fleeing to Canada. Unless the war was clearly shown to be morally
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legitimate, they were not going to risk their lives to fight it. Out of the anti-war movement, and the civil rights movement, grew what has come to be called the second wave of the feminist movement. Like young men, young women would make up their own minds about what uses their bodies would be put to. Thomson’s libertarian argument about the inviolability of individual freedom is not to be confused with the apparently similar feminist argument for women’s control of their own bodies, an argument that is tied to social equality, and for some feminists, to pacifism.28 For feminists, an acceptance of the traditional duties of women was the fundamental roadblock to greater social equality, to a world where all lives mattered in a way they seemed not to matter in the world defined by male-centred values. In his 1971 satirical novel Our Gang, Philip Roth neatly put his finger on the connection between the war and the abortion debate. In the opening passage of the book, a troubled citizen confronts the president of the United States, Trick E. Dixon, with a worry about the Mai Lai massacre: citizen: ... inasmuch as I feel as you do about the unborn, I am seriously troubled by the possibility that Lieutenant Calley may have committed an abortion. I hate to say this, Mr President, but I am seriously troubled when I think that one of those twenty-two Vietnamese civilians Lieutenant Calley killed may have been a pregnant woman. tricky: Now just one minute. We have a tradition in the courts of this land that a man is innocent until he is proven guilty. There were babies in that ditch at Mai Lai, and we know there were women of all ages but I have not seen a single document that suggests the ditch at Mai Lai contained a pregnant woman. citizen: But what if, sir – what if one of the twenty-two was a pregnant woman? Suppose that were to come to light in your judicial review of the lieutenant’s conviction. In that you personally believe in the sanctity of human life, including the life of the yet unborn, couldn’t such a fact seriously prejudice you against Lieutenant Calley’s appeal? I have to admit that as an opponent of abortion, it would have a profound effect upon me.29
What we see here is the taboo against killing other human beings clearly under threat. By the time of the Vietnam war, the taboo, if such it is, was certainly facing challenges – challenges of realpolitik, not logic – but insofar as the satire of the passage above succeeds, the gen-
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eral wrongness of killing human beings seems to be standing up under fire. More to the point, what we see here, in the face of all the changes of the twentieth century detailed above, is the incredibility of the principle RR as applied to the abortion debate. The bald, unadorned claim that the death of the fetus is the worst outcome imaginable is just not believable. The application of RR to the abortion debate thus fails all on its own accord, without a collapsing, or collapsed, taboo against killing human beings as the cause of that failure. The failure of this particular application of RR is not a failure of logic at any level, but the result of the combined social changes that we have seen drawn together here.30 But perhaps with this point we might still give Woods the last word on how to understand the abortion debate. In Engineered Death, Woods appeals to an early version of the principle he comes in his later work to call RR. This early version of RR comes in the form of what Woods calls a ‘burden of proof argument’ against those who would countenance abortion.31 If abortion is murder, says Woods, in countenancing abortion we are countenancing murder. But if abortion is not murder, all we are doing in forbidding abortion is forbidding some actions that are not, as it turns out, murderous. The bigger of the two mistakes is to fail to count actions that are in fact murders as such, and so, says Woods, the burden of proof is squarely on the side of those who would permit abortions. This burden of proof argument, however, depends on a background moral tradition that takes seriously the possibility that killing fetuses might be murder. For example, a Hindu might urge, using similar reasoning, that Albertans refrain from eating beef until they have discharged a burden of proof to the effect that killing cattle is not murder. Without the right sort of shared moral tradition to back it up, the burden of proof argument falls flat, as Woods himself is quick to point out. But this is just what we can see happening when we pull together the strands of social change described above: a significant change in social tradition, rather than the loss of a powerful taboo. The loss of a taboo suggests a sudden hole in the social fabric of public moral discourse. Changes in social tradition are slower, and not without a certain amount of social struggle. But once such a change is underway, burden of proof arguments like the ones Woods suggests in Engineered Death and ‘Privatizing Death’ will have lost their purchase on our moral thinking. Thus does Woods’ argument against Morgentaler fail to convince, both in the event and in the paper discussed here. But the
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reasons for this failure are, I think, extremely interesting, not only regarding engineered death, but engineered life.32 As the revolution in genetic engineering in the late twentieth century continues on into the next, our thinking about the meaning of human death, and with it, the meaning of human life, will continue to change. The ethical dilemmas of engineered death will be ramified by the ethical dilemmas of engineered life, and vice versa, in ways that we are now only beginning to imagine. notes 1 In Life and Death: Metaphysics and Ethics, Midwest Studies in Philosophy Volume 24, Peter A. French and Howard K. Wettstein, eds. (Oxford: Blackwell, 2000), 199–218. In what follows I offer a close reading of the argument of this article, perhaps too close for comfort in the context of a Festschrift. Woods, however, has been a superlative model for all of us in the University of Lethbridge, Department of Philosophy, returning to his desk with all the more eagerness and energy in the face of hard criticism of his arguments. So this was the nicest retirement present I could think of giving him. 2 Kathleen McDonnell, Not an Easy Choice: A Feminist Re-examines Abortion (Toronto: Women’s Press, 1984), 81–2. 3 Woods, ‘Privatizing Death,’ 206. In this formulation, the general principle involved here is more appropriately referred to under the title of ‘minimax,’ but the underlying point is exactly the same. ‘RR’ is an abbreviation for ‘resolution rule.’ 4 John Rawls, A Theory of Justice (Cambridge: Harvard University Press, 1971). For Rawls’ use of the maximin principle see 150–61. For some early and influential criticisms of Rawls’ use of the principle see David Lyons, ‘Rawls versus Utilitarianism,’ Journal of Philosophy 69, no. 18 (1972): 535–45; Kenneth J. Arrow, ‘Some Ordinalist-Utilitarian Notes on Rawls’ Theory of Justice,’ Journal of Philosophy 70, no. 79 (1973): 245–63, and John C. Harsanyi, ‘Can the Maximin Principle Serve as a Basis for Morality? A Critique of John Rawls’ Theory,’ American Political Science Review 69 no. 2 (1975): 594– 606. 5 John Dewey and James Tufts, Ethics (New York: Henry Holt and Company, 1908). 6 John Woods, Engineered Death: Abortion, Suicide, Euthanasia and Senecide (Ottawa: University of Ottawa Press, 1978). 7 In what follows I draw heavily on the work of a joint research project funded by the NHRDP, ‘Social and Health Care Trends Influencing Pallia-
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8 9
10
11
12
tive Care and the Location of Death in Twentieth-Century Canada.’ The project’s principal investigator was Donna Wilson, a nurse interested in public policy issues, who headed a research team comprising Marjorie Anderson, a nurse interested in the history of nursing; Robin Fainsinger, a palliative care specialist; Herbert Northcott, a sociologist; Susan Smith, a historian; and myself, a philosopher. My student research assistant for the project was Ardis Anderson. The research group submitted a report of over 500 pages to the NHRDP in November 1998, subsequent to which several related articles have appeared: Donna M. Wilson, Marjorie C. Anderson, Robin L. Fainsinger, Herbert C. Northcott, Susan L. Smith, and Michael J. Stingl, ‘Twentieth Century Social and Health Care Influences on Location of Death in Canada,’ Canadian Journal of Nursing Research 34, no. 3 (2002): 141–62; Donna M. Wilson, Herbert C. Northcott, Corrine D. Truman, Susan L. Smith, Marjorie C. Anderson, Robin L. Fainsinger, and Michael J. Stingl, ‘Location of Death in Canada,’ Evaluation and the Health Professions, 24 no. 4 (2001): 385–403; Susan L. Smith and Dawn D. Nickel, ‘From Home to Hospital: Parallels in Birthing and Dying in Twentieth-Century Canada,’ Canadian Bulletin of Medical History 16 (1999): 49–64. Subsequent work relevant to the current argument can be found in Herbert C. Northcott and Donna M. Wilson, Dying and Death in Canada (Aurora, ON: Garamond Press, 2001). ‘Privatizing Death,’ 199–201; Engineered Death, 4–6, 136–42, and 145–60. Bertrand Russell, ‘What I Believe,’ in Why I Am Not a Christian (London: Unwin, 1925), 43–69 (1975 edition). To retrieve the original pithiness of this line, one needs to remember the Victorian obsession with immortality and its intellectual backdrop of philosophical idealism; see Peter Gay, Schnitzler’s Century: The Making of Middle-Class Culture, 1815–1914 (New York: Norton, 2001). American pragmatism was a similar response to idealism; see Louis Menand, The Metaphysical Club: A Story of Ideas in America (New York: Farrar, Straus and Giroux, 2001). The Santayana passage, quoted and commented on by Will Durant, comes from Durant’s article on Santayana in the fourteenth edition (1929) of the Encyclopedia Britannica. Sigmund Freud, ‘Thoughts for the Time on War and Death,’ Collected Papers (New York: Basic Books, 1959), 4: 288–317. Originally published in 1915. For a discussion of Heidegger on this point, see Peter Koestenbaum, ‘The Vitality of Death,’ The Journal of Existentialism 5 (1964): 139–66. The point, of course, is logically fallacious, as Paul Edwards points out in ‘My Death,’ in The Encyclopedia of Philosophy, vol. 5, Paul Edwards, ed. (New York: Macmillan, 1967), 416–19. I can imagine myself going to the symphony tomor-
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13 14
15 16 17
18
19
20
21
22
row or not; that I must be thinking such a thought in the present entails nothing at all about my imagined future presence or absence from the symphony performance in question. (For a further interesting note on the Victorians and immortality, see J.B. Schneewind’s short entry on Frederic W.H. Myers, immediately after the entry ‘My Death,’ 419.) Joseph A. Amato, ‘Death, and the Stories We Don’t Have,’ The Monist 76, no. 2 (1993): 252–67. See also Konstantin Kolenda, ‘Facing Death: Four Literary Accounts,’ Philosophic Exchange 15–16 (1984–5): 29–43, and Heinrich Mohr, ‘Death and Nothingness in Literature,’ in The Ancients and the Moderns, Reginald Lilly, ed. (Bloomington: Indiana University Press, 1996), 295–310. This passage and the two immediately preceding it are from Engineered Death, 141. Ibid., 109. Other important essays are ‘A Free Man’s Worship,’ in Mysticism and Logic (London: Unwin, 1917); ‘Do We Survive Death,’ ‘The Existence of God,’ and the title essay of Why I Am Not a Christian; ‘The Value of Scepticism,’ ‘Machines and Emotions,’ ‘Behaviorism and Values,’ and ‘Freedom in Society,’ in Sceptical Essays (New York: Norton, 1928). In addition to the book with Tufts, Ethics, other representative works by Dewey are Human Nature and Conduct: An Introduction to Social Psychology (New York: Modern Library, 1922), and The Quest for Certainty: A Study in the Relation of Knowledge and Action (New York: G.P. Putnam’s Sons, 1929). See, for example, John B. Watson, ‘Psychology as a Behaviorist Views It,’ Psychological Review 20 (1913): 158–77, Psychology from the Standpoint of a Behaviorist (Philadelphia: J.B. Lippincott, 1919), and Behaviorism (New York: Norton, 1925). Michael Stingl, ‘Ethics I (1900–1945),’ in Philosophy of Meaning, Knowledge and Value in the Twentieth Century, vol. 10 of the Routledge History of Philosophy, John V. Canfield, ed. (London: Routledge, 1996), 134–62. Robert L. Arrington, ‘Ethics II (1945 to the present),’ in Philosophy of Meaning, Knowledge and Value, 163–96. For two relatively recent and influential views that moral truth might exist only to the extent that we might think that things are so, see Gilbert Harman, The Nature of Morality (New York: Oxford University Press, 1977), and John L. Mackie, Ethics: Inventing Right and Wrong (New York: Penguin, 1977). David Hume, The Natural History of Religion and Dialogues Concerning Natural Religion, A. Wayne Colver and John Valdimir Price, eds. (Oxford: Oxford University Press, 1976), 241. By the turn of our own century, intelligent design theorists are reduced to fighting a rearguard action; see Freder-
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23
24
25
26
27
28
29 30
31 32
ick Crews, ‘Saving Us from Darwin,’ New York Review of Books 48, no. 15 (2001): 24–7, and ‘Saving Us from Darwin, Part II,’ New York Review of Books 48, no. 16 (2001): 51–5. See Adrian Desmond, Huxley: From Devil’s Disciple to Evolution’s High Priest (Reading, MA: Perseus Books, 1994). See esp. chapters 12–15 and 19. For an interesting corrective on Spencer’s views see Paul Thompson, ed., Issues in Evolutionary Ethics (Albany: SUNY Press, 1995), 8–17, and Robert J. Richards, Darwin and the Emergence of Evolutionary Theories of Mind and Behavior (Chicago: University of Chicago Press, 1987), chaps. 6 and 7. For rich and elegant discussions of both their ideas and their public relevance, see Alan Ryan’s Bertrand Russell: A Political Life (New York: Oxford University Press, 1988), and John Dewey and the High Tide of American Liberalism (New York: Norton, 1995). Judith Jarvis Thomson, ‘A Defense of Abortion,’ Philosophy and Public Affairs 1, no. 1 (1971): 47–66; the same year, we might note, as John Rawls’ A Theory of Justice. To be sure, by the time Woods comes to this particular point he has spent two earlier chapters arguing at great length that if fetuses are not persons, they are the moral equivalent of persons. Depending (in part) on how one feels about those earlier arguments, Woods is either bravely or foolishly following them out to what he sees as their logical conclusion. What we have here may be another instance of what Woods calls ‘Philosophy’s Most Difficult Problem’ in ‘Privatizing Death,’ 205. There are significant legal issues here that Woods fails to anticipate; see Bernard M. Dickens, ‘Comparative Judicial Embryology: Judges’ Approaches to Unborn Human Life,’ Canadian Journal of Family Law 9, no. 1 (1990): 180–92. On feminism, abortion, and social equality, see Susan Sherwin, No Longer Patient: Feminist Ethics and Health Care (Philadelphia: Temple University Press, 1992). That Sherwin is no libertarian comes out clearly in her denial of a woman’s right to avail herself of the new reproductive technologies. For feminism and pacifism see Sara Ruddick, Maternal Thinking: Toward a Politics of Peace (New York: Ballantine Books, 1989). Philip Roth, Our Gang (New York: Random House, 1971), 5. There’s that year again. I leave open the more general question of whether a principle like RR is ever applicable at any social level less fundamental than the selection of the basic institutions of society, if indeed it is even applicable there. Engineered Death, 41–2. In the area of genetics we also encounter burden of proof arguments,
Engineered Death and the (Il)logic of Social Change 473 e.g., that because the dangers of genetic engineering are unprecedented, proving safety should always be a burden fully and completely discharged by those wishing to proceed with programs of genetic research. Needless to say, such arguments are seldom permanent bulwarks against such proceedings.
25 Incorrect English MICHAEL WREEN
Moderate views are attractive to many people, and sometimes that’s at least partly because they’re so-called. Abortion, I tend to think, is a case in point. Few philosophers consider themselves extreme conservatives, extreme liberals, or even conservatives, though a substantial number think of themselves as liberals, with ‘liberal’ probably meaning, at least in their minds, much the same thing as ‘moderate.’ In any case, one of the best-known defences of the so-called moderate position on abortion is Jane English’s ‘Abortion and the Concept of a Person.’ First published in 1975, English’s article is, by my reckoning, one of the four most frequently reprinted articles of the last thirty years, the others being Judith Jarvis Thomson’s ‘A Defense of Abortion,’1 James Rachels’ ‘Active and Passive Euthanasia,’2 and Mary Ann Warren’s ‘On the Moral and Legal Status of Abortion.’3 But while the other articles have attracted a great deal of attention in the literature, there’s practically nothing on English’s piece. This, as far as I know, is the first article devoted exclusively to it. English’s main thesis is that (M) Our concept of a person cannot and need not bear the weight that the abortion controversy has thrust upon it.4 To establish it, she argues for three independent subtheses: (1) a conclusive answer to the question of whether the fetus is a person is unattainable;5 (2) even if the fetus is a person, abortion is still justifiable in many cases;6 and (3) even if the fetus is not a person, killing it is still wrong in many cases.7 Taken together, (1), (2), and (3) provide strong support for (M), she thinks, and thus for skepticism about any argu-
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ment on the morality of abortion that depends heavily on the concept of a person, as arguments for the conservative, extreme conservative, and extreme liberal positions typically do. This paper is a critical analysis of English’s arguments for (2). Space limitations prevent examining her arguments for (1), (3), and (M). There’s plenty to do in relation to (2), however, and (2) is interesting and important in its own right, as well as being held by virtually every moderate. 1 Two different argumentative strategies, not clearly distinguished by English, are used to argue for (2). Both are based on self-defence. The first is to cite cases about which we have clear intuitions that harming another in self-defence is morally permissible, and then to extract a principle from them. The principle is then applied to show that abortion is justifiable in at least some cases. ‘How severe an injury may you inflict in self defense?’ English asks. Her answer is that In part this depends upon the severity of the injury to be avoided: you may not shoot someone merely to avoid having your clothes torn. This might lead one to the mistaken conclusion that the defense may only equal the threatened injury in severity; that to avoid death you may kill, but to avoid a black eye you may only inflict a black eye or the equivalent. Rather, our laws and customs seem to say that you may create an injury somewhat, but not enormously, greater than the injury to be avoided. To fend off an attack whose outcome would be as serious as rape, a severe beating or the loss of a finger, you may shoot; to avoid having your clothes torn, you may blacken an eye ... Aside from this, the injury you may inflict should only be the minimum necessary to deter or incapacitate the attacker. Even if you know he intends to kill you, you are not justified in shooting him if you could equally well save yourself by the simple expedient of running away. Self defense is for the purpose of avoiding harms rather than equalizing harms.8
The principle of self-defence here is (SD) The severity of an injury a person may inflict to avoid a threatened harm is the minimum necessary, and may, at its maximum, somewhat
476 Michael Wreen exceed the severity of the harm threatened; however, it may not greatly exceed it.
(SD) implies that if a pregnancy threatens a woman’s major interests, and if terminating the pregnancy is the only way to avert the threat, then abortion is permissible. Abortion in cases of life-threatening pregnancy is thus permissible, but it’s also permissible if a woman’s health is threatened in a major way – say, she’s likely to suffer kidney failure, or sink into a prolonged, severe depression if the pregnancy continues – or if other of her major interests are threatened – say, the welfare of her other children would be seriously compromised if a new baby were added to the family, or her career would be seriously and irreparably damaged by the birth of a child. (SD) doesn’t underwrite abortion on demand, however, since not every pregnancy brought to term threatens a woman’s life, health, major interests, or life prospects. On the assumption that the fetus is a person, not every abortion is permissible, but many more than a staunch conservative would think are. 2 The key to English’s argument here is her principle of self-defence. (SD) sounds good but is actually very problematic. It’s both too strong and too weak. It’s too strong because it would allow me to kill a potential colleague. Suppose I’ve spent years pursuing a PhD in philosophy, only to find myself facing an extremely tight job market at the time of graduation. Luckily, though, I find myself one of two finalists for a position in my area of specialization. The other candidate for the job is at least as well qualified as I am, and that worries me. As I remember English’s principle, however, my spirits soar. The other candidate’s existence poses a serious threat to my life prospects – the direction of my life will change markedly and irreparably for the worse if he gets the job – so it’s permissible to kill him if that’s the only way to secure the post. English’s principle thus shows just how wrong it is to say that philosophy bakes no bread. At least in this case, it enables a philosopher to make a little dough. The problem with (SD) is that it allows a person to harm or even do away with another person if his continued existence means that the first person’s life will take a major turn for the worst, at least if there’s no other way to neutralize the threat that he poses. Lots of people,
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though, put a major damper on our lives and the pursuit of our projects, and killing them, or harming them in a major way, may be the only way to secure whatever good for ourselves is in question. This is true not only of career-related objectives, but of others that are important to us and that help us to define ourselves. Athletic achievement, for example, is a major objective of many people, and the only way to win some event or secure a coveted place on a team may be to hobble one’s opponents. On English’s principle, Tanya Harding, the infamous ice-skater who apparently arranged to have her chief rival kneecapped, did no wrong. At the root of many such counter-examples is what Bentham called the sweetest of all pleasures: self-approbation. Honour and thinking well of oneself may be the basis of many, though I doubt all, of the projects and values that we define ourselves in terms of. A philosopher who undermines a hated critic’s career, who makes sure that, even though the latter is tenured, he doesn’t get published, isn’t invited to conferences, or can’t secure a better position, is protecting a major interest, his ego, or positive self-image. That image would suffer greatly if the arguments of his opponent were given greater airplay, or if the dais he presented them from were raised a little higher. ‘The combat of wits is the fiercest,’ Hobbes said with some justice, and ‘vain glory,’ to use another of his terms, may well generate such situations, even in the august field of philosophy. English’s principle tells us that undermining our critics in underhanded ways is permissible if our egos are big enough, or if we suffer from pride, in the biblical sense of the term. Actually, the counter-intuitive consequences of (SD) don’t end there, for it also allows criminals to resist arrest, and even to kill or seriously injure the police if the jail sentence they face is long enough. And Little Johnny, he’d be allowed to lie and say that Little Jimmy actually broke the window, if English were correct, this being especially true if punishment or payment for the broken window were a consideration in addition to reputation as a boy who didn’t obey his parents’ strict orders. Another untoward implication of English’s principle concerns duties that are thrust upon us, and duties to our relatives. Caring for an aged and sickly relative, or even just raising a child, can throw a person into a prolonged depression, curtail his education, hamper the advancement of his career, prevent him from starting his own business, deprive him of the time and energy needed to pursue athletics or a highly valued hobby, or even damage his health. Are we allowed to kill our sickly relatives or our children?
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On this last point, though, English has an answer ready. Contrasting the situation of the fetus with that of a born child, she writes: What if, after birth, the presence of an infant or the need to support it posed a grave threat to the woman’s sanity or life prospects? She could escape this threat by the simple expedient of running away. So a solution that does not entail the death of the infant is available because of the biological dependence of the fetus on the woman. Birth is the crucial point not because of any characteristics the fetus gains, but because after birth the woman can defend herself by a means less drastic than killing the infant. Hence self-defense can be used to justify abortion without necessarily thereby justifying infanticide.9
This, however, is merely to say that abandonment looks good when compared with murder. Moreover, even that contrast fades when it’s realized that English’s principle doesn’t require, nor does it mention, making provisions for the child – or sick relative – abandoned. Death resulting from abandonment may not be killing, but morally speaking, it’s little different from killing. 3 For very different reasons, English’s principle is also too weak. Suppose that fifty angry students, armed to the teeth, were to attack me simultaneously. Each is determined to see me die, whether by his hand alone or by acting in concert with others. The only way that I can escape death is by tossing a hand grenade into their midst. Doing so isn’t permissible on English’s principle, though: pulling the pin would inflict harm enormously greater – forty-nine deaths greater – than the harm I’m trying to avoid, my own single death. Nor can this conclusion be avoided by countering that in such circumstances what a person does is first kill one attacker, then a second, then a third, and so on, thus simply repeatedly applying (SD) rather than violating it. That’s not so. The lethal act is unique, the deaths of the attackers are simultaneous, and the intent of each attacker, the threat that each attacker poses, is to contribute anything from all to only one-fiftieth of the effort actually needed to kill me. I may even intend to kill them all at once, knowing their frame of mind; and, regardless of that, I do kill them all at once, in a single act. An implication of this, and one that English would probably find especially disturbing, concerns abortion. If a woman’s pregnancy is lifethreatening, abortion may still not be morally permissible. If the woman
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is pregnant with twins or triplets, and the babies – persons, on her own assumption – can be delivered live, then the harm inflicted by abortion would greatly outweigh the harm she herself is threatened with. Some would say that (SD) is too weak for another reason. Imagine a person who, once every few weeks or so, is robbed of $300 by a second person. This goes on year after year, decade after decade. For whatever reasons – say, because the robbery takes place in a corrupt totalitarian country – legal recourse isn’t possible. The person being robbed, a typical member of the middle class, is undoubtedly harmed by the incessant theft, but not enormously, not as much as he would be if his career objectives or his life prospects were drastically altered, or his arm cut off. The thief himself is psychologically much odder. Although he has no desire to assault his victim, he’s so determined to extract $300 from him every third Thursday that he would fight to the death – his death – to get the money. Given the determination of the thief, the victim knows that the only way to prevent his slow but endless financial bleeding is to warn the thief that the next time he tries to pick his pocket, he’ll fight him to the death to defend his money. Let’s suppose that that’s exactly what he does. Some people, and even some academic philosophers, think that doing so is permissible. Basically, their idea is that no one has to remain hostage to endless wrongdoing just because the wrongdoer is determined unto death to continue his wrongdoing. This position is controversial, however, and needn’t be pressed into service as a counterexample. (SD) has problems enough without a possibly unfit draftee fighting against it. Most of those problems, including those concerning its being too strong as well as its being too weak, stem from the fact that (SD) is simply a restricted and impure principle of utility. It’s restricted, because it doesn’t concern every action and speaks only of permissibility, and it’s impure, because even in its limited domain of application, it allows for acts that don’t maximize utility and permits a person to favour himself. But it’s still a principle of utility, because the only thing it considers relevant to justifying ‘self-defence’ is utility. Counter-examples showing that it’s too strong or too weak are possible because it neglects the crucial factors present in paradigmatic cases of justified self-defence, such as action that is threatening – as opposed to mere threats to one’s interests – initial wrongdoing or harmful action or attempts at such, intention, mens rea, rights, and voluntariness. Neglecting such factors, it’s prey to counter-examples, even those of a utilitarian sort, such as the one mentioned above involving fifty angry students attacking a lovable, sweet, intelligent, knowledgeable, and good-looking philosophy professor.
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4 English’s second strategy is to use a pure ‘case law’ approach. That is, in order to establish that even if the fetus is a person, abortion is still permissible in some cases, she eschews principles and simply argues by analogy from other cases. That’s exactly Judith Jarvis Thomson’s strategy for precisely the same conclusion in her famous article (though in Thomson’s case, that was all she was trying to prove). English’s first analogy has you suppose that a mad scientist ... [has] hypnotized innocent people to jump out of the bushes and attack innocent passers-by with knives. If you are so attacked, we agree that you have a right to kill the attacker in self defense, if killing him is the only way to protect your life or to save yourself from a serious injury. It does not seem to matter here that the attacker is not malicious but himself an innocent pawn, for your killing of him is not done in a spirit of retribution but only in self defense.10
Later, she varies the case slightly by having the person attacked hire a bodyguard to protect him.11 This complication, however, one that concerns the fact that it’s usually a doctor, and not the pregnant woman herself, who performs an abortion, will be ignored here. It’s not a detail that would help English in any case, as the role-related duties of a doctor and a bodyguard are substantially different, and it’s eminently arguable that no doctor ever has a role-related duty to kill anyone against his will, or without consulting him or those representing his best interests. 5 In her analogy, English assumes that we have a strong intuition that it’s morally permissible to kill such an innocent attacker; the argument for abortion being permissible in cases of pregnancy that pose a serious threat to a woman’s life or health can’t get off the ground without a strong intuition to that effect securely in place. Confidence in it may be shaken somewhat, however, when it’s pointed out that the description of the case is prejudicially skewed by English. She identifies ‘you’ with the innocent person being attacked, not with the equally innocent person hypnotized and put into motion by the mad scientist. That inevitably introduces a strong element of selfinterest and a feeling of unfair victimhood into assessment of the case,
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and invites a judgment of permissibility. As a balance, to make sure that our confidence that killing the hypnotic is permissible is well placed, we should also imagine that we’re the innocent hypnotic, and re-ask the question. Better still would have been to present the case from a thirdperson point of view in the first place, in order to avoid any tainting whatsoever. Also imperative is to separate the two cases English considers: (i) that in which the harm threatened is the loss of life, and (ii) that in which the harm threatened is bodily injury. Intuitions about the two cases may differ, or at least differ in strength, and discussing the two as though they are one, as English does, may well hobble our judgment about the latter by conflating it with the former. Last, we should make explicit two details of the cases that English tacitly assumes: first, if the hypnotic did kill or seriously maim the passerby, he would then snap out of it and lead the rest of his life in a normal, moderately happy way; and second, the hypnotic looks and acts normally – say, like Tom Cruise greeting his fans. 6 Even with these corrections and qualifications in place, most people’s considered intuition would probably still be that it’s permissible to kill the hypnotized attacker if he threatens life or limb. Some of these people, however, might be less sure, and it’s a strong, sure intuition that English needs, since her overarching argument is analogical, and substantial confidence in the conclusions of the hypnotic cases is thus needed in order to transfer them with any assurance to more questionable cases involving pregnancy. Worse still, there might be those who give up their intuition in one or both of the cases. Consider first someone who thinks that it’s not permissible to kill if only bodily injury is threatened. A person who thinks that it’s permissible for John Simms to kill innocent Tom Cruise only if threatened with the loss of his life might argue that since it’s one life lost in any case, it’s permissible for Simms to choose which life is lost and to favour himself. Morality may require us to be impartial, other things being equal, but since other things – namely, consequences, intent, mens rea, and so on – are equal in this case, morality doesn’t require Simms to sacrifice himself. For just that reason, though, English’s second case is very different. Everything except consequences are equal when only major bodily injury is threatened, and since morality does require
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impartiality, it’s not permissible for Simms to kill Cruise if that’s all he’s threatened with. Loss of life is a much greater harm than major bodily injury, and, as just said, everything else is equal. Or not quite. In addition, Simms could be compensated for his loss, whereas Cruise couldn’t if he were killed. That, too, is morally important. Even more strongly, it could be argued that it would be wrong for Simms to kill Cruise even if Cruise threatened his life. Simms and Cruise are both simply victims of bad luck, the argument goes, the bad luck of being in the wrong place at the wrong time. A coin should be flipped to determine who lives and who dies – or rather, a coin has already been flipped, for both Simms and Cruise simply find themselves in the situation they’re in, victims of circumstances beyond their control. Chance has already flipped the coin. For Simms to favour himself or to flip another coin to decide who lives would be to decide that he didn’t like the way that chance had first decided the outcome, and to insist on one more favourable to himself, or at least to see if luck would smile on him if the coin were flipped again. Compare the following situation, the argument would conclude. Standing in a tightly enclosed space with a stranger, Simms sees a meteor heading right for the spot where he’s standing. His only chance for survival is to exchange places with the stranger, Tom Cruise, standing next to him. He does this by manipulating Cruise: he leans closer and closer to Cruise until, feeling uncomfortable, Cruise inches into his position and he into Cruise’s. The fact that, in the original analogy, there are only two things, Simms and Cruise, while in this there are three, Simms, Cruise, and the meteor, doesn’t vitiate the analogy, it would be argued, for though in the original Cruise is, so to speak, both a person and a meteor, the fact that he’s a deadly agent is separable from his identity as a person. He was hypnotized, so in a very important sense, it’s not him who’s the deadly agent. Cruise’s deadly force is supplied by an external agency, just as the meteor’s is. That’s why we might consider him simply a Cruise missile. Finally, it would be pointed out that the above argument applies just as surely if only bodily injury were in question. 7 While I have a great deal of sympathy for the arguments of the previous section, I don’t think that they’re quite right. First of all, other things aren’t really equal as far as the situations of Cruise and Simms
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are concerned. Cruise, even though he’s hypnotized, performs the initial threatening action. This is one of the factors that, in paradigmatic cases of permissible self-defence, justifies harming another. Admittedly, the action in question isn’t Cruise’s in one sense, since it doesn’t flow from him or represent him. Still, the action is extensionally equivalent to such an action, and its resemblance to it enables us to call it Cruise’s action in a secondary, extended sense of the term, much as, as Aristotle says, a hand detached from a body isn’t a hand, but can be considered such in a secondary, derivative sense of the term. In morality, and especially in the sort of extreme and bizarre situations under consideration, such secondary senses have to be given at least some weight as far as the protection of important interests is concerned. Something similar holds in respect to the murderous intention that Cruise has but Simms lacks, another asymmetry in the case. Cruise may have such an intention, but the intention isn’t his – or at least that’s a natural way to describe the situation. Still and all, the close resemblance to a paradigmatically ‘owned’ intention is enough to make it count, morally speaking; and much the same goes for differences in mens rea in general between the two. A second major problem with the arguments concerns impartiality. Although morality does require impartiality, and one of its main functions is to curb the unbridled pursuit of self-interest, it also allows for the pursuit of self-interest, even in some circumstances in which others would be better off if it did not. For most people in affluent countries, the good that comes from pursuing projects and ideals which are largely self-centred wouldn’t come close to the good that could be done by sacrificing self-interest to serve others in much poorer circumstances. If, as most people think, pursuing such projects and ideals is permissible even so – which isn’t to say that people don’t have a duty to be charitable – then morality does permit at least some partiality in one’s own favour, even if others would suffer. And if any situation is one in which such partiality is permitted, it is the kind of circumstances English describes, at least if it’s death that a person is faced with and not some lesser evil. Connected with this is the fact that being partial to oneself to some extent probably has to be part of morality. As (I think) Frankena put it, morality is made for man, not man for morality. I take this to mean that the requirements, prohibitions, and permissions of morality can’t be at odds with fundamental human psychology – or, as earlier philosophers would say, human nature – to any very appreciable extent. A morality
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that is well nigh impossible to live with is no morality at all, and to require an individual to let himself be killed in the circumstances English envisions is, for very deep psychological reasons, beyond the pale for most human beings. Doing so may be admirable, may well be an act of supererogation, but it can’t be obligatory. That’s probably just as true if major bodily injury and not death is the harm threatened. All things considered, then, I agree that it would be permissible for Simms to kill Cruise if threatened with the loss of his life and, somewhat more reluctantly, that it would also be permissible to do so if only major bodily injury were threatened. But the cases aren’t simple or obvious, as English thinks they are, and their proper location is close to the periphery of permissibility, and not at its centre. That means that even if English’s analogy is perfect, the permissibility of abortion in cases of pregnancy that are life threatening, or that threaten major bodily harm, is also peripheral. 8 In theory, the hypnotic and pregnancy cases could differ not at all, or could differ in ways that strengthen, weaken, or, as a mixed bag, strengthen in some respects and weaken in others, the inference to English’s conclusion that abortion is permissible in the two cases of problematic pregnancy that she considers. I see several differences that weaken the inference and one that, at least prima facie, seems to strengthen it. One big difference between the cases is that with pregnancy, selfdefence isn’t in question, not even in the extended sense of the term applicable in the hypnotic cases. The fetus doesn’t attack the woman, not in any sense at all, nor does it have lethal intentions or mens rea even in the atrophied senses of those terms applicable in the hypnotic cases. Rather, it’s the continued existence or growth of the fetus that poses a threat to the woman’s life or health. In fact, with pregnancy, even the term ‘threat’ changes its conceptual locale: unlike any attacker, even an innocent hypnotized one, the fetus poses no threat, though its existence or growth does. If the distinction I have in mind here isn’t clear, compare it with the case of a man who tries for years to get his paralyzed wife to smile. When he finally succeeds, the joy of seeing her smile causes him to have a fatal heart attack. His wife didn’t kill him, nor did she even cause his death. Her smile, or her smiling, did, though. To return: if abortion is justified in the cases English is
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interested in, that won’t be because the factors that make self-defence justifiable also make such abortions justifiable. Not quite killing in self-defence but in the near vicinity is killing in a desperate, life- or health-threatening situation. The classic example is two men hiding from nearby Nazis. The men will escape capture if both remain quiet. Unfortunately, one has to sneeze, and a sneeze will give away their location. The second, though, quickly, painlessly, and quietly kills the first before he can say ‘ah’ much less ‘choo.’ He was in a desperate, life-threatening situation, and killed another in order to save himself. Many people feel that such killing is justifiable. By parity of reasoning, then, abortion is also justifiable in cases of life-threatening pregnancy. Abortion when the pregnancy is only health threatening may be more of a stretch, and may not be justifiable, but abortion when the pregnancy is life threatening certainly is. But there are two problems with this. The first is that the permissibility of killing the would-be sneezer under the conditions described is, at very best, highly controversial. Any feeling that it would be okay to kill him is probably motivated by the belief that if he sneezes, both men will die at the hands of the Nazis. The justification then would be that it’s permissible to kill an innocent person if not doing so would result in his death anyway, plus an additional death. Even that’s somewhat controversial, but it needn’t be lingered over, for it’s simply irrelevant. The correct analogy is a scenario in which, if the person sneezes, he will be spared – say, he’s really the son of a high-ranking official – while the person he’s hiding with will die. In that case, I take it that the intuition that it would be okay to kill the innocent would-be sneezer largely evaporates. Killing him would be little different from grabbing an innocent person to use as a human shield when being shot at by a gunman. Actually, it would be worse than that, because in the case of the sneezer, the other party hiding from the Nazis actually does the killing, not a third party. My own view, which won’t be explored here, is that the transition from killing in self-defence, in either the primary or secondary sense of the term, to killing in a desperate, life-threatening situation also marks a transition from justification to mitigation. The cry that goes up in cases of the latter sort is ‘But what was I to do?’ and ‘I had no other choice,’ and such remarks sound more like reminders of mitigating circumstances than justifications. If the sort of basic human psychology noted near the end of the previous section is taken into consideration,
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and it’s remembered that as fundamental a disvalue as loss of life is in question, then perhaps such remarks should carry at least some weight as mitigation, even if they shouldn’t in other cases. We needn’t pay attention to a person who says ‘But there was nothing else I could do!’ after embezzling millions of dollars to keep his business afloat and his family living in the style to which they’d become accustomed. 9 Bad as all of that is, the situation is actually worse as far as abortion is concerned. A second important difference between the hypnotic and pregnancy cases is that, with pregnancy, the person whose life or health is threatened is the very person responsible for the existence and location of the (by assumption) person whose continued existence or growth now threatens her life or health. The woman, in other words, is actually in the position of a mad scientist, a Dr Frankenstein, who creates a Tom Cruise whose existence or growth then threatens her life or health. Granted, she creates that situation inadvertently. Still, she also does so knowing of the possibility of danger to herself. If this is too abstract a way to put the point, it can be made concrete by imagining that Simms, a mad but not wicked Dr Frankenstein, whips up a sleeping Tom Cruise on his lab table, knowing that there’s a slight but not nonexistent chance that Cruise’s body chemistry, when Cruise awakens, will be temporarily, but only temporarily, out of whack, resulting in the emission of radiation fatal to whoever is closest to him at the time, or at least detrimental to that person’s health. Cruise himself would be unaffected by the radiation, however. (As the previous sections have made evident, that’s far from a completely accurate analogy, but it will serve to illustrate the point at hand.) The so-called attacker isn’t just there, waiting for Simms to happen by, as in English’s original analogy, but there only because of Simms’s previous acts, acts he knew had some chance of eventuating in an innocent ‘attack’ on himself. The same is true of the woman in the case of life- or health-threatening pregnancy. For Simms to kill Cruise would be to play an especially nasty trick on him, since it’s Simms himself who’s at least partly responsible for the attack. Certainly no one else is. And yet Simms wants Cruise to pick up the entire tab for his actions by paying with his life. Morally speaking, that’s an unwarranted form of Cruise control. A third difference between the two cases that works against English is that a woman is related by blood, and, short of cloning, in the most intimate way possible, to the (by assumption) person who is her fetus,
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while there’s no blood relation between Simms and Cruise, not even if the case is modified à la Dr Frankenstein. While many philosophers currently pooh-pooh the moral import of blood relations as such, and chalk up beliefs to the contrary to lingering speciesism or one of its kin – that is, to mere biology mistakenly being allowed to have moral import – the man in the street feels otherwise, and accepts blood relations as morally basic. I myself side with the general public and think that blood is thicker than water. In fact, I think that’s true even if, as we all know is too often the case, an individual hates his relatives and would like to see a little of their blood shed. At any rate, if English’s analogy is to be accurate, Simms would have to be Cruise’s mother or father, and yet still kill him. A final difference between the cases is that a fetus is inside the woman’s body, whereas Cruise isn’t inside Simms’. For many people, that’s a crucial fact about abortion in general, and a difference between the hypnotic and pregnancy cases that, in contrast to the other differences mentioned above, strengthens English’s argument. ‘But it’s my body!’ ‘Keep your laws off my body!’ and so on are rallying cries for many pro-choice advocates. On the assumption that the fetus is a person, however, I doubt that they carry much weight. Even though it’s the woman’s body, another person is lodged inside it – and is there, as stressed earlier, only because of the woman’s previous actions. Rights to one’s own body carry little weight if someone else is unjustly harmed, or someone else’s rights are violated, in exercising those supposed rights, a fact illustrated by rules against smoking on airplanes, among other things. But to use yet another analogy to illustrate the point as far as abortion is concerned: imagine that Walter L. Weber feeds his daughter Waltina part of the mushroom that made Alice, of Wonderland fame, shrink so small, and then draws her up in a syringe and injects her into his arm. The girl is able to breathe the oxygen in her father’s arteries, let’s say, and extract nutrients from his blood. Could Walter then say, ‘But it’s my body!’ and proceed to pound on the vein in his forearm where Waltina is located, knowing fully well that doing so would kill her? ‘But it’s my body!’ is a powerful argument if a body part is in question; it’s decidedly less so if a person in a body is. All analogies are imperfect to some extent, but the analogy that seems to me closest to the case of life-threatening pregnancy on the assumption that the fetus is a person is this: Fred meets Shaggy, a much younger and smaller man, and invites him on a tour of his vast estate, or at least implicitly makes no objection
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to Shaggy going along with him. As they walk around the grounds, they come upon an abandoned bank that has fallen into a state of disrepair. In the bank is a vault. The two enter the vault, but while they are there the wind blows the door shut, locking them in. There’s a timer on the lock and the door can’t be opened again for another nine hours. Unfortunately, there’s not enough oxygen in the room for both of them for nine hours. Being younger, smaller, and more resilient, Shaggy can survive, even with Fred there, though he’d be temporarily a little worse for wear after the ordeal. For Fred it’s a very different story. If Shaggy weren’t there, or if he weren’t breathing, he could survive until the door swung open nine hours later and Scooby, his good friend, greeted him. But if Shaggy does continue to breathe for nine hours ... Fred faces a desperate, life-threatening situation due to Shaggy’s continued existence or breathing, a desperate situation that he could remedy if he killed Shaggy. Shaggy, incidentally, is actually Fred’s longlost son, a fact discovered just before their walk around the grounds. The scenario here could be modified to cover health-threatening pregnancy by having Fred not die but suffer kidney failure or become seriously diabetic if he spends nine hours in the vault with Shaggy. The analogy isn’t perfect – among other things, a fetus chooses nothing, not even to be in its mother’s womb (a difference, incidentally, that works against the permissibility of abortion) – but it at least brings out many of the salient features of a typical case of life-threatening pregnancy. 10 There’s one other point I’d like to make about English’s argument for (2). After presenting the analogies that I’ve been examining at length, she modifies the case once more: Suppose you are a highly trained surgeon when you are kidnapped by the hypnotic attacker. He says he does not intend to harm you but to take you back to the mad scientist who, it turns out, plans to hypnotize you to have a permanent block against all your knowledge of medicine. This would automatically destroy your career which would in turn have a serious adverse impact on your family, your personal relationships and your happiness. It seems to me that if the only way to avoid this outcome is to shoot the innocent attacker, you are justified in so doing. You are defending yourself from a drastic injury to your life prospects. I think it is
Incorrect English 489 no exaggeration to claim that unwanted pregnancies (most obviously among teenagers) often have such adverse life-long consequences as the surgeon’s loss of livelihood.12
Besides inheriting all the problems of her stronger analogies, those concerning threats to a person’s life or health, this one has problems of its own. If I’m allowed to kill the innocent hypnotic if he threatens ‘a drastic injury to [my] life prospects,’ I’m also allowed to kill my job rival, mentioned in section II above. Even waiving that point, pace English, it is an exaggeration, and a wild one, to claim that ‘unwanted pregnancies ... often have such adverse life-long consequences’ as she describes. I seriously doubt that giving birth and raising a child for twenty years ‘often’ has such consequences, but that’s not the point. The point is that pregnancy never does, and it’s pregnancy that’s in question, not raising a child. The issue is whether a woman must carry to term, or is permitted to abort before that time. The issue isn’t whether a woman is permitted to abort before nine months are up, or must – apparently single-handedly – give birth and raise a child for twenty years. Like many people, English wrongly conceptualizes the issue. Adoption isn’t even a blip on the radar screen. 11 Not every argument for (2) has been examined here. A number have been, though, and the ones that have are, in many respects, typical of arguments for the claim that abortion is permissible if a woman’s major interests are at stake, even if the fetus is a person or has a right to life.13 Given that that’s so, it’s probably pretty difficult to justify abortion under those circumstances, and it may not be possible to do so at all.14 That conclusion won’t sit well with many people, and for more than a few, I’m sure that it’s very inconvenient and upsetting. The business of philosophy, though, is to follow the argument wherever it may lead, not lead the argument where you want it to follow. As Socrates well knew, the conclusions reached are sometimes very inconvenient and upsetting. notes My thanks to Walter L. Weber for his biting criticism of an earlier draft of this paper.
490 Michael Wreen 1 Judith Jarvis Thomson, ‘A Defense of Abortion,’ Philosophy and Public Affairs 1, no. 1 (1971): 47–66. 2 James Rachels, ‘Active and Passive Euthanasia,’ New England Journal of Medicine 292, no. 2 (1975): 78–80. 3 Mary Ann Warren, ‘On the Moral and Legal Status of Abortion,’ The Monist 57, no. 1 (1973): 43–61. 4 Jane English, ‘Abortion and the Concept of a Person,’ originally published in Canadian Journal of Philosophy 5, no. 2 (1975). I’ll be quoting from the article as reprinted in The Problem of Abortion, 2nd ed., Joel Feinberg, ed. (Belmont, CA: Wadsworth Publishing, 1984), 151–60. 5 Ibid., 153. 6 Ibid., 151. 7 Ibid. 8 Ibid., 155. 9 Ibid., 157. 10 Ibid., 154. 11 Ibid., 156. 12 Ibid. 13 Although personhood is focused on above, it’s really the right to life that’s doing all the work. In this context, no harm is done by conflating the two concepts, despite the fact that they’re logically independent, because common ground in the abortion debate is that all persons have a right to life. 14 Other proposed cases of permissible abortion, even on the assumption that the fetus is a person, include (1) pregnancy due to incest or rape, (2) cases of severe fetal abnormality, and (3) some life- or health-threatening pregnancies not considered above, such as those in which both the woman and the fetus will die if an abortion is not performed, and those in which the pregnancy is due to rape.
26 Ameliorating Computational Exhaustion in Artificial Prudence PAUL VIMINITZ
1. A Brief Pre-History of the Project Those of us long-since committed to the view that the languages of war, politics, economics, and morality are reducible without remainder to game theory, are wont to dismiss the worries of our skeptics, precisely because we’re so immersed in our Weltanschauung that these worries no longer resonate in us. To the claim that ‘The human condition is simply too complex to be captured by such a sparse set of axioms and primitives!’ we’re wont to reply, ‘Just watch us!’ To the insistence that ‘There’s more to understanding than prediction and control!’ we’re wont to wonder, ‘What?!’ And to the charge that, ‘Powerful as the game theoretic take on the human condition might be, it’s offensive!’ we’re wont to answer that, Every paradigm shift in our self-understanding offended against the ancien one! The world didn’t come to an end when it ceased to be the centre of the cosmos. The Origin of Species didn’t undermine human dignity. Having been told that it’s the brain rather than the heart wherein the emotions reside, no one fell out of love. Neither, then, need we fear the end of civilization as we know it should it turn out that our most heartfelt moral intuitions and cherished political institutions are just intra- and extramental responses respectively to otherwise dilemmatic mixed-motive games. But there are other challenges to the foundations to our enterprise we dismiss only at the peril of the enterprise itself. For example: Are tuistic preferences primitive or derivative? Can we distinguish between a parametric choice situation and a strategic one without adverting to mental states? If not, need we be realists about mental states, or just Dennettian instrumentalists about them? If our notion of rationality is
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parasitic on our notion of a preference, and if a preference for A over B is just a disposition to choose the former over the latter were the two offered to us, how could one ever demonstrate her own irrationality? And if we can never be irrational, what work remains for a theory of rational choice, social or otherwise, to do? Is an algorithm defined as an invariant? If so, no sense could be made of an input changing an algorithm. So shall we say that the real algorithm is the meta-one – that is, the one allowing the input which directs the first-order algorithm to change? But if we say that, and allow it to apply recursively, is our reduction not in danger of devolving into analyticity? And there are end-of-pipe worries as well. We claim – in fact we have to claim – that our models will give our clients prediction and control. But who are our clients? Certainly not the widow whose mite is funding our research. We’re wont to say that the models we’re building can be used by feminists to understand how patriarchal strategies divide and conquer, as readily as they can to fine-tune those strategies themselves. But likewise could unleashing the power of the atom have done away with coal slags rather than turn Hiroshima into one. The question to be asked is not ‘How could our reductions be used?’ but ‘How will they?’ In short, I suspect that much of the resistance to our reduction is driven not by skepticism but by its very opposite – fear! And that fear is not entirely unwarranted. Much of what we do is sullied by the lucre of the right-wing think-tanks that financed its inception.1 David Gauthier may be remembered not for The Logic of Leviathan2 or Morals by Agreement3 but for his advocacy of retaliationism in the nuclear deterrence debate.4 That he since recanted does little to correct the perception that game theoreticians are nasty people, bent on an even nastier moral and political agenda. That it can be shown that our reduction doesn’t necessarily redound to that agenda won’t do. What needs to be shown – if, that is, these fears are to be laid to rest – is that it redounds to anything but that agenda. In any event, it was with that in mind that I set myself to task designing Artificial Prudence. What I wanted to show is that contractarianism is not ‘a moral and political theory for the last generation on earth.’5 Rather – or so I think I’ve shown – it can both account for and approve of our concern for future generations.6 What I wanted to show – and what I think I have shown – is that there’s no more gap between what the game-theoretic model dictates and the phenomenology of the moral point of view, than there is between Deep Blue and Gary Kasparov. That Deep Blue can beat Kasparov quite regularly takes nothing
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away from the fact that Kasparov is a damn good chess player. But that Kasparov can sometimes beat Deep Blue means only that Deep Blue stands in need of some fine-tuning. So perhaps the fact that people do things that contradict the cruder versions of the game-theoretic model means only that either the model or people or both stand in need of like fine-tuning. But, as we’re about to see, models sophisticated enough to close the gap between rationality and morality are expensive. This paper is about those expenses and how to minimize them. 2. Artificial Prudence Artificial Prudence (APr) is almost entirely parasitic on Peter Danielson’s Artificial Morality (AM).7 But since, as we’re about to see, AM turns out to be just a ‘special case’ of APr, I’ll forego any rehearsal of AM per se and proceed directly to APr. APr, not unlike AM, confines itself to four dispositions defined by their behaviour in a Prisoners’ Dilemma.8 That is, UC unconditionally, CC conditionally, and RC reciprocally, cooperates, and UD unconditionally defects. UC and UD are self-explanatory. CC cooperates if doing so is sufficient to elicit its co-players’ cooperation. RC cooperates if doing so is necessary. So the difference between CC and RC is that, whereas CC generously cooperates with UC, RC ruthlessly exploits her. Let u, c, r, and s be the number of UCs, CCs, RCs, and UDs, respectively, in a population. And let $4, $3, $2, and $1 be a set of payoffs such that $4 > $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
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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-
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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
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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
19
20
21
22
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-
508 John Woods
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
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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