copyrighted material-taylor & francis · 2015-06-04 · 1.1 the first sorites 8 1.2 chrysippan...

Copyrighted Material-Taylor amp Francis

Systems

Vagueness

If you keep removing single grains of sand from a heap when is it no longera heap This question and many others like it soon lead us to the problemof vagueness

Timothy Williamson traces the history of the problem from discussionsof the heap paradox in ancient Greece to modern formal approaches suchas fuzzy logic He discusses the view that classical logic and formalsemantics do not apply to vague languages and shows that none of thealternative approaches can give a satisfying account of vagueness withoutfalling back on classical logic

Against this historical and critical background Williamson thendevelops his own epistemicist position Vagueness he argues is anepistemic phenomenon a kind of ignorance there really is a specific grainof sand whose removal turns the heap into a non-heap but we cannot knowwhich one it is

Williamsonrsquos argument has ramifications far beyond the study ofvagueness It reasserts the validity of classical logic and semantics moregenerally it makes the thoroughly realist point that even the truth about theboundaries of our concepts can be beyond our capacity to know it

The approach throughout keeps technicalities to a minimum this ispartly to counter the illusion encouraged by the emphasis on formalsystems that vagueness can be studied in a precise metalanguage For thetechnically minded an appendix shows how the epistemic view can beformalised within the framework of epistemic logic

Timothy Williamson is Professor of Logic and Metaphysics at theUniversity of Edinburgh He is the author of Identity and Discrimination


The Problems of Philosophy

Founding editor Ted HonderichEditors Tim Crane and Jonathan Wolff University College London

This series addresses the central problems of philosophy Each book gives a freshaccount of a particular philosophical theme by offering two perspectives on thesubject the historical context and the authorrsquos own distinctive and originalcontribution

The books are written to be accessible to students of philosophy and relateddisciplines while taking the debate to a new level

DEMOCRACYRoss Harrison

THE EXISTENCE OF THEWORLDReinhardt Grossman

NAMING AND REFERENCER J Nelson

EXPLAINING EXPLANATIONDavid-Hillel Ruben

IF P THEN QDavid H Sanford

SCEPTICISMChristopher Hookway

HUMAN CONSCIOUSNESSAlastair Hannay

THE IMPLICATIONS OFDETERMINISMRoy Weatherford

THE INFINITEA W Moore

KNOWLEDGE AND BELIEFFrederic F Schmitt

KNOWLEDGE OF THEEXTERNAL WORLDBruce Aune

MORAL KNOWLEDGEAlan Goldman

MINDndashBODY IDENTITYTHEORIESCynthia Macdonald

THE NATURE OF ARTA L Cothey

PERSONAL IDENTITYHarold W Noonan

POLITICAL FREEDOMGeorge G Brenkert

THE RATIONALFOUNDATIONS OF ETHICST L S Sprigge

PRACTICAL REASONINGRobert Audi

RATIONALITYHarold I Brown

THOUGHT AND LANGUAGEJ M Moravcsik

THE WEAKNESS OF THE WILLJustine Gosling

THE MIND AND ITS WORLDGregory McCulloch

PERCEPTIONHoward Robinson

THE NATURE OF GODGerard Hughes

Also available in paperback


Vagueness

Timothy Williamson

London and New York


First published 1994by Routledge11 New Fetter Lane London EC4P 4E29 West 35th Street New York NY 10001

This edition published in the Taylor amp Francis e-Library 2001

Paperback edition 1996

copy 1994 Timothy Williamson

All rights reserved No part of this book may be reprinted or reproduced or utilizedin any form or by any electronic mechanical or by any other means now known orhereafter invented including photocopying and recording or in any informationstorage or retrieval system without permission in writing from the publishers

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library

Library of Congress Cataloging-in-Publication DataA catalogue record for this book is available from the Library of Congress

ISBN 0ndash415ndash03331ndash4 (hbk)ISBN 0ndash415ndash13980ndash5 (pbk)ISBN 0-203-01426-X Master e-book ISBNISBN 0-203-17453-4 (Glassbook Format)


For Nathan Isaacs and Jeff Williamson



Contents

Preface xiIntroduction 1

1 The early history of sorites paradoxes 811 The first sorites 812 Chrysippan silence 1213 Sorites arguments and Stoic logic 2214 The sorites in later antiquity 2715 The sorites after antiquity 31

2 The ideal of precision 3621 The emergence of vagueness 3622 Frege 3723 Peirce 4624 Russell 52

3 The rehabilitation of vagueness 7031 Vagueness and ordinary language 7032 The BlackndashHempel debate 7333 Family resemblances 8434 Open texture 89

4 Many-valued logic and degrees of truth 9641 Overview 9642 Truth-functionality 9743 Three-valued logic beginnings 10244 Three-valued logic Halldeacuten 10345 Three-valued logic Koumlrner 10846 Three-valued logic second-order vagueness 11147 Continuum-valued logic a rationale 11348 Continuum-valued logic truth-tables 114


viii Contents

49 Fuzzy sets and fuzzy logic 120410 Degree-theoretic treatments of sorites paradoxes 123411 Comparatives and modifiers 124412 Vague degrees of truth 127413 Non-numerical degrees of truth 131414 Degree-functionality 135415 Appendix axiomatizations of continuum-valued

logic 138

5 Supervaluations 14251 Incomplete meanings 14252 Origins 14353 Logic and semantics 14654 The elusiveness of supertruth 15355 Supervaluational degrees of truth 15456 Supervaluations and higher-order vagueness 15657 Truth and supertruth 162

6 Nihilism 16561 Despair 16562 Global nihilism 16663 Local nihilism appearances 17164 Local nihilism colours 180

7 Vagueness as ignorance 18571 Bivalence and ignorance 18572 Bivalence and truth 18773 Omniscient speakers 19874 The supervenience of vagueness 20175 Meaning and use 20576 Understanding 20977 Decidable cases 212

8 Inexact knowledge 21681 The explanatory task 21682 The crowd 21783 Margins for error 22684 Conceptual sources of inexactness 23085 Recognition of vague concepts 23486 Indiscriminable differences 23787 Inexact beliefs 244


Contents ix

9 Vagueness in the world 24891 Supervenience and vague facts 24892 Determinacy in the world 24993 Unclarity de re 257

Appendix The logic of clarity 270

Notes 276References 307Index 320



Preface

This book originated in my attempts to refute itsmain thesis that vaguenessconsists in our ignorance of the sharp boundaries of our concepts andtherefore requires no revision of standard logic For years I took thisepistemic view of vagueness to be obviously false asmost philosophers doIn 1988 Simon Blackburn then editor of the journal Mind asked me toreview Roy Sorensenrsquos intriguing book Blindspots which includes adefence of the epistemic view It did not persuade me I could not see whatmakes us ignorant and Sorensen offered no specific explanation Analternative treatment of vagueness supervaluationism looked more or lessadequate ndash unlike other popular alternatives such as three-valued andfuzzy logic which on technical grounds have always looked like blindalleys However I continued to think about the epistemic view for thestandard objections to it did not seem quite decisive It was not clear thatthey did not assume a suspect connection between what is true and what wecan verify It then struck me that the notion of a margin for error could beused to give a specific explanation of ignorance of the sharp boundaries ofour concepts and the epistemic view began to look more plausible Alimited version of it was tentatively proposed in my book Identity andDiscrimination (Oxford Blackwell 1990) The more closely theobjections to it were analysed the weaker they seemed The next step wasto focus on the fact that the meaning of vague expressions can be stated onlyin a language into which those expressions can be translated it is a mistaketo treat the language in which one theorizes about vagueness as though itwere precise Mark Sainsburyrsquos inaugural lecture at Kingrsquos CollegeLondon lsquoConcepts without Boundariesrsquo helped to bring the significance


xii Preface

of this point home to me although we used it in quite different ways Itpermits the formulation of arguments against a wide range of non-epistemic views including the supervaluationism that had previouslylooked adequate (my objection to it however is not the one made inSainsburyrsquos lecture) The balance of arguments seemed to have movedfirmly onto the side of the epistemic view A book-length treatment wasclearly needed This is the result

Some of the research for this book was carried out in late 1990 whilst Iwas a Visiting Fellow at the Research School of Social Sciences of theAustralian National University in Canberra Many people helped to makethe visit a success Philip and Eileen Pettit stand out Gratitude is also dueto University College Oxford and Oxford University for allowing me extraleave of absence in that academic year

Ted Honderich kindly permitted me to substitute a volume on vaguenessin this series for one planned on another subject One result of working onthe past of the problem of vagueness for which I am particularly grateful isa better sense of the richness of Stoic logic In this connection I thank DavidSedley for permission to quote translations from the first volume of a workhe edited with AA Long The Hellenistic Philosophers (CambridgeCambridge University Press 1987)

Parts of Chapters 7 and 8 are drawn from two previously publishedarticles of mine lsquoVagueness and ignorancersquo Aristotelian Society suppl 66(1992) 145ndash62 and lsquoInexact knowledgersquo Mind 101 (1992) 217ndash42 I amgrateful to the Aristotelian Society and The Mind Association forpermission to use this material

For written comments on predecessors of parts of this book many thanksgo to Michael Bacharach Justin Broackes Myles Burnyeat PeterCarruthers Bill Child Jack Copeland Dorothy Edgington TimothyEndicott Graeme Forbes Brian Garrett Bill Hart Dominic Hyde FrankJackson Rosanna Keefe Peter Lipton Andrei Marmor GregoryMcCulloch Karina and Angus McIntosh David Over Peter Pagin PhilipPercival Philip Pettit Mark Sainsbury David Sedley Jonathan SuttonCharles Travis and David Wiggins Peter Simons replied to lsquoVagueness andignorancersquo in an enjoyable symposium at Reading chaired by MarkSainsbury More people than I can name helped with critical questions aftertalks on the epistemic view of vagueness inexact knowledge and related


Preface xiii

topics at the universities of Bradford Bristol Cambridge (the MoralSciences Club) Dundee Edinburgh Heidelberg Leeds Lisbon London(University College) New England (Armidale) Nottingham OsloOxford Queensland Stirling Stockholm and Uppsala the AustralianNational University and Monash University and to a meeting of the LisbonPhilosophical Society in May 1991 an AnglondashPolish Symposium on thePhilosophy of Logic and Language at Oriel College Oxford in September1991 the Second Workshop on Knowledge Belief and StrategicInteraction at Castiglioncello in June 1992 and the Joint Session of theAristotelian Society and the Mind Association at Reading University inJuly 1992 Invidiously I pick out George Bealer and Peter Menziesbecause there is a particularly direct causal link between their questions andsections of the book Early versions of several chapters were used in classesat Oxford and were considerably improved as a result Ron ChrisleyMichael Martin and Roger Teichmann were particularly persistentquestioners I have also been helped by conversations with MariaBaghramian Joatildeo Branquinho John Campbell David Charles Kit FineOlav Gjelsvik and Peter Strawson (not to mention anyone previouslymentioned) Juliane Kerkhecker guided me through Lorenzo Vallarsquos Latin

Those who know Elisabetta Perosino Williamson will guess how shehelped in the writing of this book and how much It is dedicated to a great-uncle and an uncle whose open-minded rationality (amongst other things)I rightly tried to imitate with only mixed success



Introduction

Logicians are often accused of treating language as though it were preciseand ignoring its vagueness Their standards of valid and invalid reasoningare held to be good enough for artificial precise languages but to breakdown when applied to the natural vague languages in which we actuallyreason about the world that we experience A perfectly precise language forsuch reasoning is an idealization never to be realized Although we canmake our language less vague we cannot make it perfectly precise If wetry to do so by stipulating what our words are to mean our stipulations willthemselves be made in less than perfectly precise terms and the reformedlanguage will inherit some of that vagueness

The problem is not confined to logic Attempts to describe the semanticsof natural languages in formal terms are also frequently supposed to ignorevagueness and therefore to misdescribe the meanings of ordinaryexpressions Of course a theory might ignore vagueness and remain ausefulapproximation for some purposesbut it is also legitimate to ask whatchanges of theory are needed to take vagueness into account

At the core of classical (ie standard) logic and semantics is the principleof bivalence according to which every statement is either true or false Thisis the principle most obviously threatened by vagueness When forexample did Rembrandt become old For each second of his life one canconsider the statement that he was old then Some of those statements arefalse others are true If all of them are true or false then there was a lastsecond at which it was false to say that Rembrandt was old immediatelyfollowed by a first second at which it was true to say that he was old Whichsecond was that We have no way of knowing Indeed it is widely felt to be


2 Vagueness

just silly to suppose that there was such a second Our use of the word lsquooldrsquois conceived as too vague to single one out On such grounds the principleof bivalence has been rejected for vague languages To reject bivalence isto reject classical logic or semantics

At some times it was unclear whether Rembrandt was old He wasneither clearly old nor clearly not old The unclarity resulted fromvagueness in the statement that Rembrandt was old We can even use suchexamples to define the notion of vagueness An expression or concept isvague if and only if it can result in unclarity of the kind just exemplifiedSuch a definition does not pretend to display the underlying nature of thephenomenon In particular it does not specify whether the unclarity resultsfrom the failure of the statement to be true or false or simply from ourinability to find out which The definition is neutral on such points oftheory Just as we might agree to define the term lsquolightrsquo or lsquopoetryrsquo byexamples in order not to talk past each other when disagreeing about thenature of light or poetry so we can agree to define the term lsquovaguenessrsquo byexamples in order not to talk past each other when disagreeing about thenature of vagueness

The phenomenon of vagueness is broad Most challenges to classicallogic or semantics depend on special features of a subject matter the futurethe infinite the quantum mechanical For all such a challenge impliesclassical logic and semantics apply to statements about other subjectmatters Vagueness in contrast presents a ubiquitous challenge It is hardto make a perfectly precise statement about anything If classical logic andsemantics apply only to perfectly precise languages then they apply to nolanguage that we can speak

The phenomenon of vagueness is deep as well as broad It would beshallow if it could be adequately described in precise terms That is notgenerally possible The difficulties presented by the question lsquoWhen didRembrandt become oldrsquo are also presented by the question lsquoWhen didRembrandt become clearly oldrsquo At some times it was unclear whether itwas unclear whether Rembrandt was old The limits of vagueness arethemselves vague The same difficulties are presented by the questionlsquoWhen did Rembrandt become clearly clearly oldrsquo the point reiterates adinfinitum This is the phenomenon of higher-order vagueness It means that


Introduction 3

the meta-language in which we describe the vagueness of a vague languagewill itself be vague

The use of non-classical systems of logic or semantics has beenadvocated for vague languages New and increasingly complex systemscontinue to be invented What none has so far given is a satisfying accountof higher-order vagueness In more or less subtle ways the meta-languageis treated as though it were precise For example classical logic is said tobe invalid for vague languages and is then used in the meta-language Suchproposals underestimate the depth of the problem

The problem is not solved by the pessimistic idea that no system of logicor semantics classical or non-classical is adequate for a vague languageThat idea still permits one to ask for perspicuous descriptions of vaguenessin particular cases No one has given a satisfying and perspicuousdescription of higher-order vagueness without use of classical logic Ofcourse the nature of vagueness might be to defy perspicuous descriptionbut that counsel of despair should prevail only if there is good evidence thatit does not overestimate the depth of the problem

The thesis of this book is that vagueness is an epistemic phenomenon Assuch it constitutes no objection to classical logic or semantics In cases ofunclarity statements remain true or false but speakers of the language haveno way of knowing which Higher-order vagueness consists in ignoranceabout ignorance

At first sight the epistemic view of vagueness is incredible We maythink that we cannot conceive how a vague statement could be true or falsein an unclear case For when we conceive that something is so we tend toimagine finding out that it is so We are uneasy with a fact on which wecannot attain such a first-personal perspective We have no idea how weever could have found out that the vague statement is true or that it is falsein an unclear case we are consequently unable to imagine finding out thatit is true or that it is false we fallaciously conclude that it is inconceivablethat it is true and inconceivable that it is false Such fallacies of theimagination must be laid aside before the epistemic view can be adequatelyassessed

Most work on the problem of vagueness assumes that the epistemicview is false without seriously arguing the point If the epistemic view istrue that work is fundamentally mistaken Even if the epistemic view is


4 Vagueness

false that work is ungrounded until cogent arguments against the viewhave been found The assessment of the epistemic view is therefore oneof the main tasks facing the study of vagueness This book contributes tothat task

The assessment of the epistemic view has ramifications far beyond the

study of vagueness As already noted classical logic and semantics are atstake More generally the epistemic view implies a form of realism that

even the truth about the boundaries of our concepts can outrun our capacity

to know it To deny the epistemic view of vagueness is therefore to imposelimits on realism to assert it is to endorse realism in a thoroughgoing form

The first part of the book is historical It traces the slow and intermittentrecognition of vagueness as a distinct and problematic phenomenon up

to the origins of the theories of vagueness that have been popular over thelast two decades This part is also critical It argues that none of the extant

non-epistemic theories of vagueness is adequate Not only do theyabandon classical logic or semantics for alternatives of doubtful

coherence those sacrifices are not rewarded by adequate insight into thenature of vagueness The second part of the book is constructive It

develops and applies an epistemic view of vagueness finds the standardobjections to it fallacious and concludes that the epistemic view provides

the best explanation of the phenomenon of vaguenessThe Greeks introduced the problem of vagueness into philosophy in

the guise of the original sorites paradox if the removal of one grain froma heap always leaves a heap then the successive removal of every grain

still leaves a heap Chapter 1 sketches the history of this paradox and itsvariants from their invention to the nineteenth century Stoic logicians are

interpreted as taking an epistemic view of sorites paradoxesWhat makes a sorites paradox paradoxical is the vagueness of its

central term eg lsquoheaprsquo Historically however such paradoxes wereidentified by their form Vagueness as such became a topic of

philosophical discussion only at the start of the twentieth century whenit presented an obstacle to the ideal of a logically perfect language

associated with the development of modern logic Only with difficultywas the phenomenon of unclear boundaries separated from other

phenomena such as lack of specificity to which the term lsquovaguenessrsquo is


Introduction 5

also applied in everyday usage Chapter 2 discusses three stages in the

emergence of the philosophical concept of vagueness in the work ofFrege Peirce and Russell

As philosophical attention turned to ordinary language vaguenessacquired a more positive image It was seen no longer as a deviation from

an ideal norm but as the real norm itself Assuch it was described by BlackWittgenstein and Waismann Their work is discussed in Chapter 3

Formal treatments of vagueness have become common only in the last

few decades One main approach relies on many-valued logic whichreplaces the dichotomy of truth and falsity by a manyfold classification

Chapter 4 follows the development of its application to the problem ofvagueness from the use of three-valued logic to the growth of lsquofuzzy logicrsquo

and other logics based on infinitely many values and then of moresophisticated accounts appealing to a qualitative conception of degrees of

truth These views are criticized on several grounds None has adequatelytreated higher-order vagueness degrees of truth are not connected with

vagueness in the requisite way the generalization from two-valued logic tomany-valued logic has highly counter-intuitive consequences when

applied to natural languagesA technically subtler approach to vagueness is supervaluationism with

which Chapter 5 is concerned It preserves almost all of classical logic atthe expense of classical semantics by giving a non-standard account oftruth It also treats higher-order vagueness in a promising way However itis argued that the treatment of higher-order vagueness undermines the non-standard account of truth making supervaluationism as a wholeunmotivated

On a more pessimistic view vagueness is a form of incoherence If thisview is taken globally Chapter 6 suggests all rational discourse issubverted for vagueness is ubiquitous However local forms of nihilismmight be coherent They have been defended in the special case of conceptsused to describe perceptual appearances on the grounds that such conceptscannot differentiate between perceptually indiscriminable items yetperceptually discriminable items can be linked by a sorites series of whicheach member is perceptually indiscriminable from its neighboursHowever careful attention to the structure of the relevant concepts shows


6 Vagueness

that the paradoxical arguments are unsound In particular they falselyassume that appearances are just what they appear to be

Chapter 7 defends the epistemic view of vagueness First it argues that itis incoherent to deny the principle of bivalence for vague statements inunclear cases It then questions our ability to think through coherently theconsequences of a non-epistemic view of vagueness Obvious objectionsto the epistemic view are analysed and shown to be fallacious A picture oflinguistic understanding is sketched on which we can know that a word hasa given meaning without knowing what the boundaries of that meaning arein conceptual space

Chapter 8 develops the epistemological background to the epistemicview It gives independent justification for principles about knowledge onwhich the ignorance postulated by the view was only to be expected as aspecial case of a much wider phenomenon inexact knowledgeNevertheless the case is special for the source of the inexactness isdistinctive in being conceptual Higher-order vagueness has a central placein this account for a central feature of inexact knowledge is that one canknow something without being in a position to know that one knows itwhen the inexactness takes the form of vagueness this becomes unclarityabout unclarity The epistemology of inexact knowledge is then used toanalyse in greater depth the phenomena of indiscriminability to which thenihilist appeals

It is controversial whether in any sense the world itself as opposed to ourrepresentations of it can be vague Chapter 9 examines the issue It arguesthat the epistemic view permits objects to be vague in a modest sense forthe impossibility of knowing their boundaries may be independent of theway in which the objects are represented

The Appendix identifies the formal system uniquely appropriate for thelogic of clarity and unclarity on the epistemic view of vagueness

Most of the book keeps technicalities to a minimum The gain inintelligibility will it is hoped outweigh the loss in rigour There is also aphilosophical reason for minimizing technicality The emphasis on formalsystems has encouraged the illusion that vagueness can be studied in aprecise meta-language It has therefore caused the significance of higher-order vagueness to be underestimated Indeed to use a supposedly precisemeta-language in studying vague terms is to use a language into which by


Introduction 7

hypothesis they cannot be translated Since vague terms are meaningfulthis is an expressive limitation on the meta-language It is not an innocentone The argument in Chapter 7 for the incoherence of denials of bivalencein unclear cases can be stated only in a language into which the relevantvague terms can be translated To deny bivalence is in the end to treat vagueutterances as though they said nothing Vagueness can be understood onlyfrom within


Chapter 1

The early history ofsorites paradoxes

11 THE FIRST SORITES1

The logician Eubulides of Miletus a contemporary of Aristotle wasfamous for seven puzzles One was the Liar if a man says that he is lyingis he telling the truth Another was the Hooded Man how can you knowyour brother when you do not know that hooded man who is in fact yourbrother The Electra turned on the delusion of Orestes in his madness whotook his sister Electra for a Fury The Elusive Man will appear later Therewas also the Horned Man since you still have what you have not lost andyou have not lost horns you still have them (hence the horns of a dilemma)The remaining puzzles were the Bald Man and (accompanying five menand one woman) the Heap In antiquity they were usually formulated asseries of questions2

Does one grain of wheat make a heap Do two grains of wheat make aheap Do three grains of wheat make a heap Do ten thousand grains ofwheat make a heap It is to be understood that the grains are properly piledup and that a heap must contain reasonably many grains If you admit thatone grain does not make a heap and are unwilling to make a fuss about theaddition of any single grain you are eventually forced to admit that tenthousand grains do not make a heap

Is a man with one hair on his head bald Is a man with two hairs on hishead bald Is a man with three hairs on his head bald Is a man with tenthousand hairs on his head bald It is to be understood that the hairs areproperly distributed and that a man with reasonably few hairs is bald If youadmit that a man with one hair is bald and are unwilling to make a fussabout the addition of any single hair you are eventually forced to admit thata man with ten thousand hairs is bald


The early history of sorites paradoxes 9

The standard ancient terms for the Heap and the Bald Man were lsquosoritesrsquo

and lsquophalakrosrsquo respectively The Greek adjective lsquosoritesrsquo comes from thenoun lsquosorosrsquo for lsquoheaprsquo and means literally heaper as in lsquothe heaperparadoxrsquo Perhaps Eubulides coined the word himself The primaryreference was to the content of the puzzle although there may also havebeen a secondary allusion to the heaping up of questions in its form Theterm was extended to similar puzzles such as the Bald Man They were also

known as little-by-little arguments3

So far as is known Eubulides invented the Heap and the Bald Manhimself The Heap may have been inspired by a different puzzle the MilletSeed propounded by Zeno of Elea a century earlier If the fall of a seed tothe ground were completely silent so would be the fall of a bushel of seedwhich it is not thus each seed must make a noise when it falls to the groundThat puzzle can certainly be adapted to sorites form Does one seed make anoise when it falls to the ground Do two seeds Three However Zenoseems to have based his puzzle on something much more specific aprinciple that the noise made when some grains fall to the ground is

proportional to their weight4Eubulides may well have been the first tofocus on the general difficulty that sorites questioning presents he isunlikely to have overlooked the common form of the Heap and the BaldMan however much their difference in content caused them to be listed asdistinct puzzles

It is not known what Eubulides used sorites puzzles for ndash funtroublemaking or some graver purpose Many philosophical doctrineshavebeen suggested as the target he intended them to destroy the coherence ofempirical concepts (such as lsquobaldrsquo and lsquoheaprsquo) the law of non-contradiction (lsquoNot both P and not Prsquo) the law of excluded middle (lsquoEitherP or not Prsquo) pluralism (the existence of more than one thing) Aristotlersquostheory of infinity (as potential rather than actual) Aristotlersquos theory of themean (as the place of virtue between vicious extremes) The evidence giveslittle support to any of these suggestions Eubulides is indeed said to haveattacked Aristotle but in slanderous terms the sources do not connect thedispute with any of the puzzles Aristotle betrays no clear awareness ofsorites reasoning in any of his extant works Some later commentators didconsider its use against Aristotlersquos theory of the mean but withoutsuggesting that either Eubulides or Aristotle had done so Eubulidesrsquo


10 Vagueness

interestswere described aspurely logical if he had a specific target in mind

it is likely to have been a logical one5 But if he had no specific target inmind it does not follow that his interest in sorites puzzles was intellectuallyfrivolous for one can treat any absurdity apparently licensed by acceptedstandards of argument as a serious challenge to those standards A puzzlethreatens the whole system of thought within which it is formulated and forthat very reason refutes no element of it in particular until a positivediagnosis isgiven fixing the source of the trouble There is no evidence thatEubulides had such a diagnosis

The sorites puzzles can be traced conjecturally forward from Eubulidesin the mid fourth century BCE through a chain of teacherndashpupil links Onepupil of Eubulides was Apollonius Cronus one of his pupils was DiodorusCronus David Sedley argues that it was Diodorus who gave Eubulidesrsquopuzzles wide circulation and intellectual status According to SedleyDiodorus inherited from Eubulides lsquohis sophistical leanings hisflamboyancy and his love of showmanshiprsquo he could present a puzzle sothat it not only caught the imagination but emerged as a serious challenge

to philosophical theory6 Diodorus taught dialectic lsquothe science ofdiscoursing correctly on arguments in question and answer formrsquo thestandard medium of philosophical argument and that in which the soritespuzzles were formulated

No explicit mention of the sorites by Diodorus survives However healmost certainly discussed it his intellectual ancestors and descendantsdid and it is just the kind of puzzle he liked Moreover he propounded anargument about motion with very strong affinities to the Heap Its details asthey survive are so unconvincing that one must hope them to have beengarbled It begins with the promising general principle that a predicatewhich can apply lsquoabsolutelyrsquo can also apply lsquoby predominancersquo forexample a man can be grey-headed because all the hairs on his head aregrey but he can also be grey-headed because a majority of them are (theBald Man involves a different form of aging) The idea is then to supposethat a body of three particles is moving by predominance and to addstationary particles until a body of ten thousand particles would also bemoving by predominance when only two particles are Heaps are indeed

mentioned in the course of the argument7 Although a sorites puzzle couldbe constructed out of such material this is not it the words lsquoa majorityrsquo



would need to be replaced by something like lsquomostrsquo in the definition oflsquopredominancersquo The puzzle would then turn on lsquomostrsquo rather thanlsquomovingrsquo It is tempting to connect the sorites with some of Diodorusrsquo otherinterests too For example he argued that the present truth of a past tensestatement such as lsquoIt has movedrsquo or lsquoHelen had three husbandsrsquo does notrequire the past truth of a present tense statement such as lsquoIt is movingrsquo or

lsquoHelen has three husbandsrsquo8Might the question lsquoWhenrsquo be similarly outof place with respect to the creation or destruction of a heap AgainDiodorus studied the minima of perceptibility somelater sorites paradoxesdepend on the fact that imperceptible differencescan add up to a perceptibleone Unfortunately there is no evidence to verify these speculativeconnections

Diodorus is unreliably reported to have died of shame when he failedto solve a puzzle set him by another philosopher in front of Ptolemy SoterKing of Egypt After him sorites puzzles became standard weapons indisputes between two rival schools of philosophy the Stoa and theAcademy He influenced both In about 300 his ex-pupil Zeno of Citiumbegan to teach in a public hall in Athens the Stoa Poikile from which hispupils were known as Stoics In about 273 Arcesilaus became head ofPlatorsquos Academy and achieved a remarkable coup by converting it to

scepticism Arcesilaus was influenced by Diodorusrsquo logic9The scepticsof the Academy then attacked the Stoic lsquodogmatistsrsquo Their chief weaponsincluded sorites arguments against which they took the Stoics to have nonon-sceptical defence A sceptic does not feel obliged to answer any ofthe sorites questions he can simply plead ignorance If a Stoic is obligedto answer each question lsquoYesrsquo or lsquoNorsquo he will find himself in anembarrassing position

The Stoic theory of knowledge was an obvious focus for scepticalattack and sorites reasoning a particularly apt means The central conceptof Stoic epistemology was that of a cognitive impression something likea Cartesian clear and distinct idea Cognitive impressions have apropositional content and are essentially such as to derive from andrepresent real objects with complete accuracy They are the foundation ofthe Stoic account of the possibility of knowledge The sceptics proceededto construct sorites series from cognitive to non-cognitive impressions


12 Vagueness

replacing each impression by a virtually indistinguishable one and took

themselves to have undermined Stoic claims to knowledge10

The Stoics needed a defence against sceptical sorites attacks But theymust have discussed sorites arguments prior to the institution ofscepticism in the Academy for Zeno emphasized the development oftechniques for dealing with just such puzzles as part of a training indialectical techniques He was succeeded as head of the Stoa by Cleanthesin 262 and Cleanthes by Chrysippus (c280ndashc206) in about 232 It wasChrysippus who was responsible for the systematic construction of Stoiclogical theory In antiquity he was reputed the greatest logician of all(Aristotle being the greatest scientist and Plato the greatest

philosopher)11 He also devised strategies to handle attacks by thesceptics and in particular their use of sorites arguments Unfortunatelythe two volumes (ie scrolls of papyrus) he is known to have written Onthe Little-by-Little Argument do not survive nor do his three volumes On

Sorites Arguments Against Words12There are only fragments andreports What follows is a speculative reconstruction

12 CHRYSIPPAN SILENCE

The Stoics firmly accepted the principle of bivalence every propositionis either true or false Unlike Aristotle they did not reject it for futurecontingencies it is true or false that there will be a sea-fight tomorrowAccording to Cicero Chrysippus lsquostrains every nerve to persuade us that

every axioma [proposition] is either true or falsersquo13 Thus for everyproposition P there is one right answer to the question lsquoPrsquo it is lsquoYesrsquo ifP is true and lsquoNorsquo if P is false Consequently for every sequence ofpropositions P1 Pn there is one sequence of right answers to the

questions lsquoP1rsquo lsquoPnrsquo each member of which is either lsquoYesrsquo or lsquoNorsquo

In a sorites sequence the right answers to the first and last questions areobvious and opposite The Stoics used lsquoAre i fewrsquo as the schematic form

of the ith question it will be convenient to follow their example14 Thusthe right answers to the first and last questions are lsquoYesrsquo and lsquoNorsquorespectively This fits the Bald Man rather than the Heap but thedifference is not significant Now the usual sorites sequences aremonotonic in the sense that a question rightly answerable lsquoNorsquo never



comes between two questions rightly answerable lsquoYesrsquo nor vice versa Ifi are few then a fortiori fewer than i are few if i are not few then afortiori more than i are not few Thus the sequence of right answers to thequestions lsquoIs one fewrsquo lsquoAre ten thousand fewrsquo consists of lsquoYesrsquo acertain number of times followed by lsquoNorsquo a certain number of times Inparticular there is a last question lsquoAre i fewrsquo rightly answerable lsquoYesrsquoimmediately followed by a first question lsquoAre i + 1 fewrsquo rightlyanswerable lsquoNorsquo i are few and i + 1 are not few i is a sharp cut-off pointfor fewness

The argument from bivalence to the existence of a sharp cut-off pointassumes that the sentences lsquoOne is fewrsquo lsquoTen thousand are fewrsquo doexpress propositions The Stoics themselves distinguished the propositionasserted from the sentence by means of which it is asserted Howeversomeone who utters lsquoi are fewrsquo with the sense lsquoA man with i hairs on hishead is baldrsquo does assert something which on the Stoic view requires thesentence to express a proposition The assumption gave no escape from theargument to a sharp cut-off point

There is independent evidence that the Stoics accepted sharp cut-off

points15 First and most tenuous in Cicerorsquos account Chrysippus compareshimself to a clever charioteer who pulls up his horses before he comes to aprecipice what is the sharp drop if not fromtruth to falsity Second in othercases which look susceptible to sorites reasoning the Stoics insisted onsharp cut-off points For example they denied that there are degrees ofvirtue holding that one is either vicious or perfectly virtuous An analogywas drawn with a drowning man as he rises to the surface he is comingcloser to not drowning but he is drowning to no less a degree until he breaksthe surface when he is suddenly not drowning at all Third in rebutting thesorites argument against cognitive impressions Chrysippus dealtexplicitly with the case lsquowhen the last cognitive impression lies next to thefirst non-cognitive onersquo cognitiveness has a sharp cut-off point The Stoicswere prepared to apply bivalence to sorites reasoning and swallow theconsequences

For the Stoics there are sharp cut-off points The difficulty in answeringthesoritesquestionsmust come not fromthenon-existence of right answersbut from our ignorance of what they are The sorites is a puzzle inepistemology This book is a defence of that Stoic view The immediateneed however is not to defend the view but to explore its consequences


14 Vagueness

One might answer the questions lsquoIs one fewrsquo lsquoAre i fewrsquo lsquoYesrsquoand the questions lsquoAre i + 1 fewrsquo lsquoAre ten thousand fewrsquo lsquoNorsquo butone would be guessing No one has such knowlege of cut-off points noone knows both that i are few and that i + 1 are not few Such a pattern ofanswers is forbidden by the principle that one should give an answer onlyif one knows it to be correct If they were to respect that principle theStoics needed an alternative pattern of answers

The problem concerns what one should believe not just what oneshould say If one believes that i are few and that i + 1 are not few oneviolates the principle that one should believe only what one knows to becorrect The Stoics were committed to this principle in a sense but theelucidation of that sense requires a little more background in Stoic

epistemology16

The Stoics made a threefold distinction between mere opinion (doxa)cognition (katalepsis) and scientific knowledge (episteme) The ordinaryman cognizes by assenting to cognitive impressions He believes theirpropositional content as in normal perception Cognitive impressionstend to cause us to assent to them We are strongly but not irresistiblyinclined to take what is clearly the case to be the case However theordinary man tends to assent indiscriminately to all impressions Whenhe does not realize that conditions are abnormal he may land himself witha mere opinion by assenting to a non-cognitive impression The wise manthe Stoic ideal does not err in that way (or any other) He has trainedhimself to assent only to cognitive impressions This is possible becausethere is always some qualitative difference however slight between acognitive and a non-cognitive impression given the Stoic principle that

no two entities are absolutely indiscernible17 In case of doubt the wiseman suspends judgement Where certainty is not to be had he may act onplausible assumptions but without believing them what he believes is

just that they are plausible18 Since cognitive impressions cannot be falsethe wise man has no false beliefs He is not omniscient but he is

infallible19 His beliefs and only his are sure to withstand attemptedrefutations For example he will not be tricked by the sorites questionerinto denying the proposition that ten thousand grains make a heap towhich he had previously assented on the basis of a cognitive impressionOnly he has scientific knowledge



Understandably the Stoics were not sure that there had been any wisemen Perhaps Socrates was one but the number was very small and theStoics did not themselves claim to be wise Thus they could not claim tosay or believe only what they scientifically knew However they didaspire to wisdom In particular they sought to train themselves to assentonly to cognitive impressions to what is clear The term lsquoknowledgersquounqualified by lsquoscientificrsquo will be used for assent to what is clear ie totruths guaranteed as such by cognitive impressions it covers bothcognition and scientific knowledge In circumstances for which they hadtrained ordinary Stoics might indeed say and believe only what theyknew in this looser sense Moreover they could not hope to dealsuccessfully with the sorites interrogation if they assented to non-cognitive impressions in the course of it Thus any good strategy forresponding to the interrogation would involve one in saying and believingonly what one knows This principle will help to explain the Stoictreatment of the puzzle

Chrysippus recommended that at some point in the sorites

interrogation one should fall silent and withhold assent20 The wise manwould and the ordinary Stoic should suspend judgement making nostatement and forming no belief either way thereby avoiding error in boththought and speech

If the source of the puzzle is just that one does not know whether lsquoYesrsquoor lsquoNorsquo is the right answer to some of the questions it turns out to be on alevel with other matters of which one is simply ignorant For example theStoics readily admitted that they did not know the right answer to thequestion lsquoIs the number of stars evenrsquo If no more is involved the Stoiccould confidently face a sorites interrogation armed only with the threepossible answers lsquoYesrsquo lsquoNorsquo and lsquoI donrsquot knowrsquo If he knew i to be few hewould answer lsquoYesrsquo if he knew i not to be few he would answer lsquoNorsquo inevery other case he would answer lsquoI donrsquot knowrsquo Why should such anhonest admission of ignorance not completely dissolve the puzzle

The Stoics did not classify the sorites interrogation merely as a list ofquestions some of whose answers were unknown they classified it as asophism The question about the stars makes it very easy to say only whatone knows to be true The sorites makes it very hard not to say what oneknows to be false At first sight the epistemic approach seems to lose the


16 Vagueness

difficulty of the puzzle What follows is an attempt to think through theepistemology of sorites series more carefully on Stoic lines The resultsare subtler than one might expect and help to explain some otherwisepuzzling features of the Stoic strategy

Recall that Chrysippus did not say that one should admit ignorance hesaid that one should fall silent Under interrogation saying lsquoI donrsquot knowrsquois quite a different policy from saying nothing In the former case but notthe latter one denies knowledge The Stoic is supposed not to make astatement unless he knows it to be correct Now to say lsquoI donrsquot knowrsquo inanswer to the question lsquoAre i fewrsquo is in effect to make the statement lsquoIneither know that i are few nor know that i are not fewrsquo just as lsquoYesrsquo istantamount to the statement lsquoi are fewrsquo and lsquoNorsquo to the statement lsquoi are notfewrsquo Thus the Stoic is supposed to answer lsquoI donrsquot knowrsquo only if he knowsthat he neither knows that i are few nor knows that i are not few The lsquoYesrsquolsquoNorsquolsquoDonrsquot knowrsquo strategy requires the Stoic to answer lsquoI donrsquot knowrsquowhenever he does not know The strategy is therefore available on Stoicterms only if for each i

(1) If one neither knows that i are few nor knows that i are not few thenone knows that one neither knows that i are few nor knows that i arenot few

Knowledge was defined above as assent to what is clear Thus what isknown is clear Conversely since cognitive impressions strongly incline usto assent what is clear tends to be known The lsquoYesrsquolsquoNorsquolsquoDonrsquot knowrsquostrategy is therefore very close to the following if i are clearly few saylsquoYesrsquo if i are clearly not few say lsquoNorsquo otherwise say lsquoUnclearrsquo It will besimplest to begin with the latter strategy Once its flaws have emerged theformer can be reconsidered

The lsquoYesrsquolsquoNorsquolsquoUnclearrsquo strategy is available on Stoic terms only if thethird answer is clearly correct when neither the first nor the second is (1)corresponds to

(2) If i are neither clearly few nor clearly not few then i are clearly neitherclearly few nor clearly not few

This is equivalent to the conjunction of two simpler principles21



(3a) If i are not clearly few then i are clearly few(3b) If i are not clearly not few then i are clearly not clearly not few

(3a) and (3b) are two instances of what is called the S5 principle forclarity if it is not clear that P then it is clear that it is not clear that P Thusthe lsquoYesrsquolsquoNorsquolsquoUnclearrsquo strategy is available only if clarity satisfies theS5 principle

On a Stoic view clarity cannot in general satisfy the S5 principle Forexample not knowing that the light is abnormal I assent to the non-cognitive impression that x is yellow when in fact x is white Thus x is not

clearly yellow but it is not clearly not clearly yellow22 If it were I shouldhave a cognitive impression to the effect that x is not clearly yellow suchan impression would strongly incline me to judge that x is not clearlyyellow But I have no such inclination at all it does not occur to me thatthe situation is other than a normal case of seeing something to be yellowMy problems come from glib assent to a non-cognitive impression notfrom resistance to a cognitive one Of course the Stoic view is that if Ihad been more attentive I could have avoided the mistake Myattentiveness might have enabled me to have a cognitive impression to theeffect that my perceptual impression was non-cognitive but it does notfollow that I have such a second-order cognitive impression in my actualstate I have no such impression

The S5 principle seems equally vulnerable in the case of (3a) and (3b)Just as fewness is sorites-susceptible so is clear fewness One is clearlyfew ten thousand are not clearly few (for they are not few) By Stoic logicthere is a sharp cut-off point for clear fewness for some number i i minus 1are clearly few and i are not clearly few One is in no better a position tosay what that number is than to locate the cut-off point for fewness itselfThe cognitive impression that i minus 1 are few is too relevantly similar to thenon-cognitive impression that i are few for one to judge reliably that thelatter is indeed a non-cognitive impression One cannot discriminate sofinely Although i are not clearly few they are not clearly not clearly fewSome instance of (3a) is false By a parallel argument so is some instanceof (3b) Thus (2) which entails them is false too The lsquoYesrsquolsquoNorsquolsquoUnclearrsquo strategy fails on Stoic terms for at some points in the seriesnone of those three answers is clearly correct so none is known to becorrect One cannot recognize the last of the clear cases


18 Vagueness

Jonathan Barnes has argued to the contrary that lsquoit seems relatively easyto show that if there is a last clear case then we can recognise it as suchrsquoTransposed to present notation his reasoning runs

Consider any relevant case are i clearly few or not One may answerlsquoNorsquo lsquoYesrsquo or lsquoI donrsquot knowrsquo but if one doesnrsquot know whether i areclearly few then i are not clearly clearly few and hence not clearly fewHence one can always answer lsquoNorsquo or lsquoYesrsquo to the question lsquoAre i

clearly fewrsquo hence the last clear case if it exists is recognizable23

The reasoning is perfectly general If it were valid it would show thatclarity satisfies the S5 principle everywhere In particular it would applyto my mistake about the colour of x But then the fallacy is clear If in myglib state I am asked whether x is clearly yellow I shall answer lsquoYesrsquo but

I shall be wrong24

One might try to revive Barnesrsquos argument by adding an extra premiseIt was noted above that the Stoic can expect to survive the soritesinterrogation intact only if he has trained himself not to assent to anyrelevant non-cognitive impression One might therefore add the premisethat the Stoicrsquos answers are clearly correct This excludescounterexamples of the foregoing kind If one answers lsquoYesrsquo to thequestion lsquoAre i clearly fewrsquo then i are clearly clearly few if one answerslsquoNorsquo they are clearly not clearly few However it still needs to be shownthat if neither of these answers is available then one clearly doesnrsquot knowwhether i are clearly few for otherwise the answer lsquoI donrsquot knowrsquo isunavailable too

One might hope to make the answer lsquoI donrsquot knowrsquo available just bysuspending judgement as to whether i are clearly few whenever one didnot say lsquoYesrsquo or lsquoNorsquo For knowledge requires assent if one assentsneither to the proposition that i are clearly few nor to the proposition thatthey are not clearly few one knows neither proposition Moreover onecan suspend judgement clearly to make onersquos ignorance clear One thenclearly fails to know that i are clearly few However what needs to beshown at this point in the argument is that i are not clearly clearly few Butthis does not follow for onersquos failure to know is guaranteed only by onersquosfailure to assent If one can refuse assent to what is clear one might refuse



assent to the proposition that i are clearly few even when i are clearlyclearly few Thus the argument seems to require clarity to be sufficient aswell as necessary for the assent of a well-trained Stoic This is tantamountto the assumption that such a person can discriminate with perfectaccuracy between cognitive and non-cognitive impressions But that is ineffect what the argument was supposed to show

One cannot reliably answer the question lsquoAre i clearly fewrsquo just byfollowing the advice if you hesitate to say lsquoYesrsquo say lsquoNorsquo If that policyworked one would unhesitatingly judge that i were clearly few if and onlyif i were clearly few whatever one thought was right would be right Butclarity is an independently desirable epistemic feature about whoseapplication one can be wrong as well as right Unless one is reasonablycautious one will often unhesitatingly judge that i are clearly few whenthey are not in fact clearly few (or even few) On the other hand if one isreasonably cautious one will often hesitate over what turns out to begenuinely clear one suspects a hidden catch and finds there is none Thestory is told of a mathematician lecturing who began a sentence lsquoIt is clearthat rsquo was seized by sudden doubt spent several minutes in agonizedthought and then resumed lsquoIt is clear that rsquo The point of the story is thathe may have been right

There is no universal algorithm for detecting clarity Chrysippus seemsto have held that not even the wise man can discriminate with perfectaccuracy between cognitive and non-cognitive impressions SextusEmpiricus writes

For since in the Sorites the last cognitive impression is adjacent to thefirst non-cognitive impression and virtually indistinguishable from itthe school of Chrysippus say that in the case of impressions which differso slightly the wise man will stop and become quiescent while in thecases where a more substantial difference strikes him he will assent to

one of the impressions as true25

Not even the wise man can locate the last clear case with perfect accuracyPractice has improved his powers of discrimination but not made themperfect In order to avoid the risk of assent to a non-cognitive impressionhe refrains from assenting to impressions that he cannot discriminate from


20 Vagueness

non-cognitive ones He will therefore sometimes refrain from assenting towhat is in fact a cognitive impression When he refrains from assenting toan impression he does not always classify it as unclear for if he did hewould sometimes be mistaken He does not say lsquoUnclearrsquo he falls silentThe ordinary Stoic who takes him as a model should do the same

If one answers the simple sorites question lsquoAre i fewrsquo with lsquoYesrsquowhenever that answer is clearly correct on Stoic assumptions one stopsanswering lsquoYesrsquo either at the point when it ceases to be clearly correct or atsome later stage In the former case one has located the cut-off point forclarity with perfect accuracy in the latter one has violated the constraintthat all onersquos answers should be clearly correct Since one cannot reliablylocate the cut-off point for clarity with perfect accuracy one will reliablysatisfy the constraint only if one stops answering lsquoYesrsquo before it has ceasedto be the clearly correct answer One must undershoot in order to avoid therisk of overshooting

The Stoic will fall silent before the end of the clear cases Chrysippus is

reported as advising just that26 Barnesrsquos claim that the end of the clearcases can easily be shown to be recognizable as such makes him suggestthat Chrysippus may have been misreported and merely suggested that oneshould stop at the end of the clear cases What has emerged is that there isno reason to reject the report for the advice it attributes to Chrysippus isgood advice on Stoic terms

There remains a trivial sense in which the lsquoYesrsquolsquoNorsquolsquoDonrsquot knowrsquostrategy is feasible unlike the lsquoYesrsquolsquoNorsquolsquoUnclearrsquo strategy For if onersquosrefusal of assent is very clear one can recognize it and thereby come toknow that one does not know A trained Stoic may satisfy (1) throughout asorites series if he assents to no relevant non-cognitive impressionHowever lsquoI donrsquot knowrsquo now in effect reports his refusal to assent apsychological episode not the state of the evidence available to him whathe does not know may nevertheless be clear Such a report would be of littleinterest to the questioner Silence remains intelligible

The moral that one should stop before the end of the clear cases can begeneralized One would like to obey two injunctions

(4a) If lsquoYesrsquo is a good answer say lsquoYesrsquo(4b) If lsquoYesrsquo is not a good answer do not say lsquoYesrsquo



The goodness of an answer is some truth-related property of it and doesnot simply consist in its being given There is play between theantecedents and consequents of (4a) and (4b) in an imperfect world theywill sometimes come apart In such a case one either fails to say lsquoYesrsquowhen lsquoYesrsquo is a good answer violating (4a) or says lsquoYesrsquo when lsquoYesrsquo isnot a good answer violating (4b) If one regards violations of (4a) and(4b) as equally serious one may simply aim to say lsquoYesrsquo when and onlywhen it is a good answer Other things being equal onersquos misses are aslikely to fall on one side of the target as on the other and no matter Butone might regard a violation of (4b) as worse than a violation of (4a)given the choice one would rather err by omission not saying lsquoYesrsquo whenit is a good answer than by commission saying lsquoYesrsquo when it is not a goodanswer For example one may prefer failing to make true or warrantedstatements to making false or unwarranted ones In that case one willfollow a policy of saying nothing when in doubt One decreases the riskof more serious violations by increasing the risk of less serious ones Atthe limit the price of never violating (4b) is sometimes violating (4a)That is the choice the Stoic made in falling silent before the end of theclear cases here clarity is goodness It was worse to say lsquoYesrsquo in anunclear case than not to say it in a clear one Those who take the oppositeview should fall silent after the end of the clear cases

The Chrysippan strategy results from two levels of precaution At thefirst level goodness in (4a) and (4b) is simply truth The Stoics were notalone in holding it to be worse to give a false answer than to fail to give atrue one For truth (4a) rather than (4b) is to be violated This preferencemotivates the constraint that one should give an answer only if it is clearBut then clarity takes on a life of its own as a cognitive end and again theStoic takes the cautious option (4a) rather than (4b) is to be violated forclarity too

The Chrysippan strategy is incomplete if it gives no clue as to whereamongst the clear cases one should fall silent If the advice is to fall silenta little before the end of the clear cases it is very vague and alsopresupposes that the respondent can predict the future course of thequestioning ndash an impossible task in the case of non-numerical soritessuch as those about cognitive impressions and others considered belowA sceptic might suspend judgement until the questioning was over but


22 Vagueness

that was not what Chrysippus recommended Perhaps one is supposed tobe silent for all the cases one cannot discriminate from unclear cases Forreasons of a kind already given there is no algorithm for discriminabilityfrom unclear cases one will sometimes be wrong about it However ifgoodness is taken to be discriminability from unclear cases one mightregard violation of (4b) as no worse than violation of (4a) at this thirdlevel That would bring the regress to an end

13 SORITES ARGUMENTS AND STOIC LOGIC

A paradox may be defined as an apparently valid argument withapparently true premises and an apparently false conclusion One oftenspeaks of a sorites paradox and there was mention above of soritesarguments Yet the Heap and the Bald Man have been presented as theyusually were in antiquity as series of questions not as arguments withpremises and conclusions According to Barnes lsquowe can ndash and the

ancients did ndash see a logical structure behind that dialectical faccediladersquo27

Consider the following argument with premises above the line andconclusion below

1 is fewIf 1 is few then 2 are fewIf 2 are few then 3 are fewIf 9999 are few then 10000 are few

10000 are few

The argument appears to be valid if its premises are true its conclusionwill be true too The relevant rule of inference is modus ponens whichallows one to argue from lsquoPrsquo and lsquoIf P then Qrsquo to lsquoQrsquo it appears impossiblefor its premises to be true and its conclusion false By modus ponens lsquo1is fewrsquo and lsquoIf 1 is few then 2 are fewrsquo entail lsquo2 are fewrsquo In the same waylsquo2 are fewrsquo and lsquoIf 2 are few then 3 are fewrsquo entail lsquo3 are fewrsquo After 9999applications of modus ponens one finally reaches the conclusion lsquo10000are fewrsquo The premise lsquo1 is fewrsquo is apparently true and the conclusion



lsquo10000 are fewrsquo apparently false The gradualness of the sorites seriesmakes each of the conditional premises appear true Thus the apparentlyvalid argument has apparently true premises and an apparently falseconclusion At least one of these appearances is misleading for theconclusion cannot be both true and false

The argument is valid by the standards of orthodox modern logic itcannot have true premises and a false conclusion It is also valid by the

standards of Stoic logic28 Two logical principles are at stake One is modus

ponens it was the first indemonstrable (basic) form of argument in Stoiclogic lsquoIf the first then the second but the first therefore the secondrsquo Theother is the principle sometimes known as Cut that valid arguments can bechained together for example the valid argument from lsquo1 is fewrsquo and lsquoIf 1is few then 2 are fewrsquo to lsquo2 are fewrsquo can be chained together with the validargument from lsquo2 are few and lsquoIf 2 are few then 3 are fewrsquo to lsquo3 are fewrsquogiving a valid argument from lsquo1 is fewrsquo lsquoIf 1 is few then 2 are fewrsquo and lsquoIf2 are few then 3 are fewrsquo to lsquo3 are fewrsquo The third Stoic ground-rule for theanalysis of complex arguments is the relevant form of Cut when from twopropositions a third is deduced and extra propositions are found fromwhich one of those two can be deduced then the same conclusion can bededuced from the other of the two plus those extra propositions Supposethat one can deduce lsquon are fewrsquo from lsquo1 is fewrsquo lsquoIf 1 is few then 2 are fewrsquo lsquoIf n minus 1 are few then n are fewrsquo By modus ponens from the twopropositions lsquon are fewrsquo and lsquoIf n are few then n + 1 are fewrsquo one deducesthe conclusion lsquon + 1 are fewrsquo The ground-rule then permits one to deducethe conclusion lsquon + 1 are fewrsquo from the premises lsquo1 is fewrsquo lsquoIf 1 is few then2 are fewrsquo lsquoIf n are few then n + 1 are fewrsquo By continuing in this wayone eventually reaches a complete Stoic analysis of the sorites argument

above into basic inferences29

On Stoic terms the argument is valid its first premise is true and itsconclusion false Thusnot all the conditional premises are true By the Stoicprinciple of bivalence at least one of them is false Yet the gradualness ofthe sorites makes them all appear true How can one of the conditionals lsquoIfi are few then i + 1 are fewrsquo be false

At this point there is a complication The truth-conditions ofconditionals were the subject of a fierce controversy that went back toDiodorus and his contemporary Philo and was taken up by the Stoics In


24 Vagueness

Alexandria the poet Callimachus wrote lsquoEven the crows on the roof topsare cawing about the question which conditionals are truersquo Philo treatedthe truth-value of the conditional as a function of the truth-values of itscomponents lsquoIf P then Qrsquo is true in three cases lsquoPrsquo and lsquoQrsquo are true lsquoPrsquois false and lsquoQrsquo is true lsquoPrsquo and lsquoQrsquo are false It is false in case lsquoPrsquo is trueand lsquoQrsquo is false The Philonian conditional is the weakest construction toobey modus ponens and therefore the weakest kind of conditional it istrue if any conditional with antecedent lsquoPrsquo and consequent lsquoQrsquo is true ThePhilonian lsquoIf P then Qrsquo is equivalent to lsquoNot P and not Qrsquo In contrastDiodorus held lsquoIf P then Qrsquo to be at least as strong as lsquoNot ever P and notQrsquo Chrysippus went still further for him a conditional is true if and onlyif its antecedent is incompatible with the negation of its consequent ThuslsquoIf P then Qrsquo becomes equivalent to lsquoNot possible P and not Qrsquo In modernterms Philorsquos conditional is material implication and Chrysippusrsquo is strictimplication Later Stoics tended to follow Chrysippus

In the sorites argument some conditional premise lsquoIf i are few then i +1 are fewrsquo is supposed to be false If the conditional is Chrysippan it isfalse if and only if lsquoi are fewrsquo is compatible with lsquoi + 1 are not fewrsquoHowever this conclusion looks banal who ever thought themincompatible Chrysippus might cheerfully allow that all the conditionalpremises so taken are false To know the falsity of such a conditional isnot to identify a cut-off point it is merely to know that a certain point isnot logically debarred from being the cut-off Some modern philosopherswould disagree holding that sorites puzzles arise because vague conceptsare subject to tolerance principles which do rule out the possibility of cut-off points For them lsquoi are fewrsquo does threaten to be incompatible with lsquoi +1 are not fewrsquo making the Chrysippan conditional lsquoIf i are few then i + 1are fewrsquo true But the Stoics did not take that view and may not haveregarded the argument with Chrysippan conditional premises as

genuinely challenging30

The most challenging form of the sorites argument uses Philonianconditionals Its premises claim the least and are therefore the hardest todeny Since the Philonian reading of the conditional was not the standardone the premises had to be formulated explicitly as negatedconjunctions Just that was done in standard Stoic accounts of theargument



Not 2 are few but not 3 as well Not the latter but not 4 as well And so

on up to 10 But 2 are few Therefore 10 are few as well31

The manoeuvre does nothing to solve the paradox For example it cannotremove any semantic or conceptual pressure against cut-off points If theargument with Chrysippan conditionals were in any respect moreparadoxical than the argument with negated conjunctions it would be mereevasion to replace the former by the latter The point is the reverse toconfront the paradox in its most telling form As already noted theargument with Chrysippan conditionals may not have been regarded asparadoxical at all In any case the Chrysippan conditionals entail thenegated conjunctions so anything that generates the argument withChrysippan conditionals will also generate the argument with Philonianconditionals

Once the explicit conditional has been eliminated modus ponens can nolonger be used to drive the argument forward But Stoic logic still obligesIts third indemonstrable principle is lsquoNot both the first and the second butthe first therefore not the secondrsquo Thus from lsquoi are fewrsquo and lsquoNot i are fewand i + 1 are not fewrsquo one can derive lsquoNot i + 1 are not fewrsquo and Stoic

principles allowed the double negation to be eliminated32 Such argumentscan be chained together by the third ground-rule just as before Thus anyargument of this form is valid on Stoic terms

P1

Not P1 and not P2

Not P2 and not P3

Not Pnminus1 and not Pn

Pn

Indeed Chrysippus sometimes supported his own doctrines by argumentswith exactly the same form as the sorites33Another Stoic example is

It is not the case that while fate is of this kind destiny does not exist northat while destiny exists apportionment does not nor that while


26 Vagueness

apportionment exists retribution does not nor that while retributionexists law does not nor that while law exists there does not exist rightreason enjoining what must be done and prohibiting what must not bedone But it is wrong actions that are prohibited and right actions thatare enjoined Therefore it is not the case that while fate is of this kind

wrong and right actions do not exist34

The sorites argument with negated conjunctions is valid its firstpremise is true and its conclusion false Thus some premise of the formlsquoNot i are few and i + 1 are not fewrsquo is false Hence i are few and i + 1 arenot few Thus i is a sharp cut-off point for fewness Since one cannotidentify such a point one is in no position to deny any of the premisesOne can only suspend judgement The challenge lsquoWhich premise isfalsersquo is unfair for one may be unable to find out even though one knowsthat at least one premise is false

What has been gained by presenting the sorites as an argument withpremises and conclusion Its logical structure was not the heart of theproblem for the argument is formally valid according to those whom itthreatens the Stoics They used arguments with that structurethemselves As for the sceptics they could suspend judgement on itslogical status it was enough for their purposes that their opponents tooksuch arguments to be valid The logical structure provides a convenientway of laying out the problem but so far nothing more

It is tempting to argue for a dialectical structure behind the logicalfaccedilade First the use of conditionals in the sorites argument is adistraction for the sorites interrogation shows that one can set the puzzlegoing in a language whose only resources are lsquoYesrsquo lsquoNorsquo and simplesentences (without logical connectives such as lsquoifrsquo lsquoandrsquo and lsquonotrsquo) in theinterrogative mood Second the argument has been persuasive so far notbecause its premises commanded assent but because they forbad dissentThe problem was not that one could say lsquoNot i are few and i + 1 are notfewrsquo but that one could not say lsquoi are few and i + 1 are not fewrsquo One is notpresumed to believe the premises of the sorites argument The point of thequestions is to force one to take up attitudes for or against the individualpropositions for any pattern of such attitudes leads one into troublehence the power of the interrogative form Since the premises of the



sorites argument seem compelling only when one is interrogated on theircomponents the question form takes primacy

The situation is transformed if the premises of the sorites argument canbe given positive support If they can the argument form takes primacythe question form leaves too much unsaid Moreover Chrysippan silenceis no longer an adequate response for it does not undermine the positivesupport for the premises One strand in later developments was theattempt to provide that support

14 THE SORITES IN LATER ANTIQUITY

Chrysippusrsquo strategy may have satisfied later Stoics There is no sign ofattempts to take the investigation further The puzzles became a standard

and perhaps stale item in the Stoic logical curriculum35

The sceptics were not satisfied with Chrysippusrsquo silence It was mostnotably attacked half a century after his death by Carneades whorenewed the scepticism of the Academy lsquoFor all I care you can snore notjust become quiescent But whatrsquos the point In time therersquoll be someoneto wake you up and question you in the same fashionrsquo Chrysippus wasdialectically no better off than he would have been had he fallen asleeplsquoWhy should your pursuer care whether he traps you silent or

speakingrsquo36Carneadesrsquo attitude was that of a chess-player with what hetakes to be a winning strategy whose opponent simply refuses to make amove (in a game without time-limits)

Suspension of judgement was the sceptical attitude and Carneadesfastened on the extent to which Chrysippusrsquo strategy allowed it to spreadIf Chrysippus suspended judgement in clear cases on what basis did heobject to the scepticrsquos suspension of judgement The question does notreduce the strategy to immediate incoherence for some sort of reply isopen to Chrysippus do not suspend judgement when the case isdiscriminable from the unclear cases Nevertheless the Stoic is in a verydelicate position He stops at clear cases avoiding the risk of givinganswers that are not clearly correct only at the cost of failing to giveanswers that are clearly correct (in the Stoic sense) The Stoicsrsquoepistemological caution enlarged the concessions to scepticism that theirbivalent semantics forced them to make under sorites questioning The


28 Vagueness

concessions did not amount to surrender for cases remained in whichthey could still claim knowledge but these cases were marked off by adisputed no manrsquos land rather than a compelling principle Perhaps that isjust what knowledge is like

Carneades made particular use of sorites reasoning against the Stoictheory of the gods

If Zeus is a god Posidon too being his brother will be a god But ifPosidon [the sea] is a god the [river] Achelous too will be a god And ifthe Achelous is so is the Nile If the Nile is so are all rivers If all riversare streams too would be gods If streams were torrents would be Butstreams are not Therefore Zeus is not a god either But if there were

gods Zeus too would be a god Therefore there are no gods37

Unlike standard sorites series this one has varying relations betweensuccessive members Posidon is the brother of Zeus but a stream is asmaller version of a river Nevertheless it is clearly a little-by-littleargument The moral is not intended to be that there are no gods rather itis that Stoic rational theology fails because its attempts to demarcatedivinity by non-arbitrary general principles provide it with no way to resistany one step Carneades supports each premise by separate considerationswhose cogency the Stoic is supposed to grant Mere silence would not be

an adequate defence against this argument38

If the screw is to be tightened in the usual sorites arguments theirpremises need support Awareness of this is shown by Galen (CE c 129ndashc199)

If you do not say with respect to any of the numbers as in the case of the100 grains of wheat for example that it now constitutes a heap butafterwards when a grain is added to it you say that a heap has now beenformed consequently this quantity of corn becomes a heap by theaddition of the single grain of wheat and if the grain is taken away theheap is eliminated And I know of nothing worse and more absurd than

that the being and non-being of a heap is determined by a grain of corn39

Chrysippus could not suspend judgement on the general claim that onegrain does not make the difference between a heap and a non-heap He must



deny it for it contradicts the existence even of an unknown cut-off pointFor him the addition of one grain can turn a non-heap into a heap

Galenrsquos interest in sorites puzzles was connected with a long-runningdispute between Empirical and Dogmatic (one might say Rationalist)Doctors The Empirical Doctors based their medical knowledge oninductive inferences holding it to be reliable only if derived fromsufficiently many observations their opponents applied soritesreasoning against the notion lsquosufficiently manyrsquo The Empirical Doctorsreplied that the argument proved too much if it destroyed the notionlsquosufficiently manyrsquo it would by parity of reasoning destroy much of thecommon sense on which all must rely They gave the examples of amountain strong love a row a strong wind a city a wave the open seaa flock of sheep and a herd of cattle the nation and the crowd boyhoodand adolescence the seasons none would exist if sorites arguments weresound The Empirical Doctors could reasonably claim that soritesarguments were unsound without being able to say exactly where theflaw lay Not even Chrysippus could say which premise in negated

conjunction form was false40

There are also signs of a rather different Empiricist point The soritesquestioner is compared to someone who asks a shoemaker what last willshoe everyone the question has no answer for different feet requiredifferent lasts The idea may be that the required number of observationsdepends on the circumstances of the particular case There is no general

answer to the question lsquoAre fifty observations enoughrsquo41 The point hasbeen repeated by modern philosophers and is correct as far as it goes butthat is not very far For the questions can be asked about a particular caseand the Empiricist still cannot plausibly claim to know all the answersSimilarly fifty grains may make a heap in one arrangement and not inanother but in any particular process of heaping up grains one by onethere will come a point at which the right answer to the question lsquoIs this aheaprsquo is unknown

As already noted the Empiricists knew that susceptibility to soritespuzzles is widespread They had no interest in logic and did not to attemptto demarcate the phenomenon in logical terms The logicians themselvesregarded sorites arguments as instantiating a logically valid form thetrouble lying in the premises For both sorites susceptibility depends on


30 Vagueness

content but how The Empiricists attributed it to everything which has lsquoameasure or multitudersquo Burnyeat complains that this characterization istoo narrow since it excludes puzzles based on qualitative rather thanquantitative variation such as a series of shades of colour from red to

orange42 However the Empirical Doctorsrsquo own example of strong lovesuggests that they had a broader notion in mind for the strength of love isintensive rather than extensive Perhaps they meant that anything whichcomes in degrees can give rise to sorites puzzles

It was known that for every sorites series which proceeded by adding(as Eubulidesrsquo original series seem to have done) a reverse soritesseries proceeded by subtracting Thus examples tend to come in pairs ofopposites rich and poor famous and obscure many and few great and

small long and short broad and narrow43 The awareness ofreversibility no doubt helped to check the tendency to think of a soritespuzzle as showing its conclusion to be strange but true for theconclusion of one sorites argument contradicts the first premise of thereverse argument

Some modern writers have argued for a special connection betweensorites puzzles and concepts applied on the basis of observation Thepreceding examples show no sign of this Some followers of Aristotledid make a connection but not quite the modern one In commentarieson Aristotlersquos Nicomachean Ethics both Aspasius (CE c 100ndash150) andan anonymous writer took the puzzles to show that empirical conceptsmust be applied case by case General rules are no substitute forobservation and good judgement Where the modern writers takeobservation to pose the puzzles the Aristotelian commentators took itto solve them But can even a man of good judgement always tell by

looking whether something is a heap as the grains pile up one by one44

The Dogmatic Doctors presumably hoped that their use of soritesarguments against rivals would carry conviction with potential clientswho might use the debate to choose their doctor It would not have beenprofitable to use arguments that everyone knew were unsoundNevertheless sorites paradoxes were well known outside philosophyHorace playfully used a sorites series for lsquooldrsquo against the belief that oldpoets are best and Persius one for lsquorichrsquo against the desire to become

rich45The general reader was expected to catch allusions to heaps The



notoriety of slippery slope arguments no more prevented their frequentemployment than it does today

One late development may be mentioned The fourth-century rhetoricianMarius Victorinus applied the term lsquosorites syllogismrsquo to the iterated

syllogisms already discussed by Aristotle46

All F1s are F2sAll F2s are F3sAll Fnminus1 are Fns

All F1s are Fns

This should be compared with the Stoic forms of argument discussedabove They are not quite the same The Stoics iterated rules of inferenceinvolving complex premises builtout of simpler propositions in lsquoIf the firstthen the second but the first therefore the secondrsquo lsquothe firstrsquo and lsquothesecondrsquo are variables for propositions What Victorinus iterated was theAristotelian syllogistic form with premises lsquoAll Fs are Gsrsquo and lsquoAll Gs areHsrsquo and conclusion lsquoAll Fs are Hsrsquo where the variables stand in for sub-propositional terms such as lsquohorsersquo and lsquotreersquo In modern terms the Stoicswere doing propositional logic the Aristotelians predicate logic The pointof analogy is the chaining together of arguments not the nature of thearguments so chained together

His combination of Stoic and Aristotelian terminology aside Victorinuswas not an innovator in logic However his writings had some influence inthe Christian world of later antiquity for he was a Christian convert andtaught St Jerome That made him an acceptable channel for thetransmission of logical doctrine and lsquosorites syllogismrsquo would laterbecome standard terminology47 But it did so simply as a name forcompound syllogisms conceived as unproblematic The connection withthe sorites as a puzzle had faded out

15 THE SORITES AFTER ANTIQUITY

A thousand years may be passed over in a few words Sorites puzzlesformed no part of the medieval logic curriculum no doubt because they


32 Vagueness

were absent from the works of Aristotle and his followers48 Their revivalhad to wait for what was in other respects the corruption and decline of logicin the Renaissance

Lorenzo Valla (1407ndash57) was one of the chief instigators of a shift inthe curriculum from the formal rigour of scholastic logic to the moreliterary pursuit of humanist rhetoric An ordinary language philosopherof the Renaissance he insisted that argument should be conducted inidiomatic Latin The achievements of medieval logic were dismissed assolecisms of grammar for they depended on the artificial regimentationof Latin as a semi-formal language Valla taught the art of arguing but asan essentially informal and practical skill In this context he discussed thesorites of which he knew from Cicerorsquos account The aim was toinculcate the ability to distinguish between sound and sophisticalarguments in particular cases

Valla uses the term lsquocoacervatiorsquo for the heaping ndash the iteration ndash ofsyllogisms As such he treats it as a valid form of argument He gives theexample lsquoWhatever I wish my mother wishes whatever my motherwishes Themistocles wishes whatever Themistocles wishes theAthenians wish therefore whatever I wish the Athenians wishrsquo This isa clearly valid sorites syllogism if its premises are true so is itsconclusion Vallarsquos main concern is to bring out the difference betweenthis kind of reasoning and invalid kinds easily mistaken for it One of hisexamples of the latter is lsquoRome is the most beautiful of all cities thisdistrict is the most beautiful of all districts in Rome this house is the mostbeautiful of all houses in this district therefore this house is the mostbeautiful of all houses in all citiesrsquo This argument is clearly fallaciousthe beauty of the whole is not determined only by the beauty of its mostbeautiful part The individual links in the chain are non-syllogistic Theproblem lies not in the heaping but in what is heaped Valla suggests thatsuch arguments may be invalid when they concern wealth nobilitystrength beauty and knowledge but will be valid when they concernwarmth cold height and distance perhaps the intended contrast isbetween many-dimensional and one-dimensional phenomena Vallarsquossuggestion is at best a fallible rule of thumb the warmest house may notbe found in the warmest city Such a rule is of little use to the theory ofreasoning but may be a useful contribution to its practice Valla also



considers standard sorites puzzles mainly in question and answer formHe insists that each grain and each hair makes some relevant differencehowever small and ends by quoting Cicerorsquos remark that in such cases wecan never know exactly where to draw the line This is quite consistentwith the claim that we can learn to know unsound slippery slope

reasoning when we see it49

After Valla the history of the sorites syllogism diverged from the historyof the sorites sophism For textbooks of logic a sorites was a multiple

syllogism in the orthodox sense of lsquosyllogismrsquo50 A favourite examplecame from Romans 8 2930 lsquoFor whom he did foreknow he also didpredestinate to be conformed to the image of his Son Moreover whomhe did predestinate them he also called and whom he called them he alsojustified and whom he justified them he also glorifiedrsquo This sense of theterm lsquosoritesrsquo survived as long as syllogistic logic and is still in occasionaluse The liveliness of this tradition may be judged from its majorinnovation Rudolf Goclenius (1547ndash1628) Professor of Logic andMetaphysics at Marburg listed the premises of sorites syllogisms inreverse order in his Isagoge in Organum Aristotelis Sorites syllogismswere thenceforth known as Goclenian when so presented and as

Aristotelian when in the original order51 The distinction betweenAristotelian and Goclenian forms became the main item of information inmany a textbookrsquos chapter on lsquoSoritesrsquo The etymology of the word wassometimes explained as a heaping up of premises without reference to theparadox

Sorites paradoxes appeared far less often in logic textbooks and thenusually under alternative terminology such as lsquosophisma polyzeteseosrsquo thefallacy of continual questioning They were preserved by their appearancein sceptical works by Cicero and Sextus Empiricus which became well

known in the seventeenth century52 One philosopher of sceptical cast gavethem attention ndash Pierre Gassendi in his history of logic De Origine etVarietate Logicae ndash but little use was made of the sorites for scepticalpurposes in theperiod The sceptics of the Academy had exploited it againstStoic logic only with the rise of modern logic at the end of the nineteenthcentury would it again be perceived as a serious threat

Leibniz mentions both the Heap and the Bald Man in New Essays onHuman Understanding his reply to Lockersquos Essay Concerning HumanUnderstanding Locke had argued that the boundaries of sorts and species


34 Vagueness

are fixed by the mind not by the nature of the things classified supportinghis claim by appeal to the existence of monsters on the boundaries betweenspecies Leibniz replies that such borderline cases could not show that allclassification is purely stipulative The line between man and beast orbetween stabbing and slashing is fixed by the nature of what it dividesHowever he allows that in cases of insensible variation some stipulation isrequired just as we must fix our units of measurement (nature does notrequire us to measure in inches) and associates such cases with sorites

paradoxes Bald men do not form a natural kind nor do heaps53 Aborderline case is then a matter of opinion different opinions may beequally good Leibniz gives the outer limits of colours as an example Healso mentions Lockersquos remark that the same horse will be regarded as largeby a Welshman and as small by a Fleming because they have differentcomparison classes in mind ndash although here one might wish to distinguishbetween vagueness and context dependence Leibniz concludes thatalthough different species could be linked by sorites series of insensibletransitions in practice they usually are not so that few arbitrary stipulations

are required in the classification of what actually exists54

Unlike the Stoics Leibniz holds that some of the questions in a soritesinterrogation do not have right and wrong answers The indeterminacywould remain even if we were perfectly acquainted with the inner naturesof the creatures in question Our inability to answer is not simply a productof ignorance A similar moral would later be drawn by Alexander Bain inhis Logic Under the canons of definition he discussed borderline casesconnecting them with the Heap and concluded lsquoA certain margin must beallowed as indetermined and as open to difference of opinion and such amargin of ambiguity is not to be held as invalidating the radical contrast of

qualities on either sidersquo55

Sorites paradoxes play a quite different role in Hegelrsquos logic He usesthem to illustrate not indeterminacy but the way in which a quantitativedifference can issue in a qualitative one The temperature of water rises orfalls until it turns to steam or ice The peasant adds ounce after ounce to hisassrsquos load until its spine snaps Grain after grain is added and a heap startsto exist The assimilation of these cases is surprising for it seems to requirea sharp cut-off point between non-heap and heap so that quality can be adiscontinuous function of quantity Hegel had something broader in mind



however as his next examples show A small increase or decrease inexpenditure may not matter but more will amount to prodigality or avariceThe constitution of a state need not be changed every time a citizen is bornor dies or an acre is added to or subtracted from its territory butnevertheless the constitution appropriate to a small city state isinappropriate to a vast empire Differences of degree however slightcannot be considered without implication for differences of kind InunHegelian terms sorites paradoxes refute generalizations of the form if x

and y differ by less than quantity q and x has quality Q then y has quality

Q56

According to the principle of excluded middle everything is either aheap or not a heap Given Hegelrsquos low opinion of this principle lsquothe maximof the definite understanding which would fain avoid contradiction but indoing so falls into itrsquo one may find it surprising that he did not use the

sorites as a stick with which to beat it57 Only when philosophers accordedmore respect to the principle did the paradox emerge as one of the mostsignificant obstacles to its application


Vagueness

If you keep removing single grains of sand from a heap when is it no longera heap This question and many others like it soon lead us to the problemof vagueness

Timothy Williamson traces the history of the problem from discussionsof the heap paradox in ancient Greece to modern formal approaches suchas fuzzy logic He discusses the view that classical logic and formalsemantics do not apply to vague languages and shows that none of thealternative approaches can give a satisfying account of vagueness withoutfalling back on classical logic

Against this historical and critical background Williamson thendevelops his own epistemicist position Vagueness he argues is anepistemic phenomenon a kind of ignorance there really is a specific grainof sand whose removal turns the heap into a non-heap but we cannot knowwhich one it is

Williamsonrsquos argument has ramifications far beyond the study ofvagueness It reasserts the validity of classical logic and semantics moregenerally it makes the thoroughly realist point that even the truth about theboundaries of our concepts can be beyond our capacity to know it

The approach throughout keeps technicalities to a minimum this ispartly to counter the illusion encouraged by the emphasis on formalsystems that vagueness can be studied in a precise metalanguage For thetechnically minded an appendix shows how the epistemic view can beformalised within the framework of epistemic logic

Timothy Williamson is Professor of Logic and Metaphysics at theUniversity of Edinburgh He is the author of Identity and Discrimination
































Vagueness

Timothy Williamson

London and New York














Contents







viii Contents


logic 138






Contents ix






Preface



xii Preface







Preface xiii





Introduction





2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56


































Vagueness

Timothy Williamson

London and New York














Contents







viii Contents


logic 138






Contents ix






Preface



xii Preface







Preface xiii





Introduction





2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56
















Contents







viii Contents


logic 138






Contents ix






Preface



xii Preface







Preface xiii





Introduction





2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




viii Contents


logic 138






Contents ix






Preface



xii Preface







Preface xiii





Introduction





2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




Contents ix






Preface



xii Preface







Preface xiii





Introduction





2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56





Preface



xii Preface







Preface xiii





Introduction





2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




xii Preface







Preface xiii





Introduction





2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




Preface xiii





Introduction





2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56





Introduction





2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




2 Vagueness






Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




Introduction 3








4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




4 Vagueness




















Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




Introduction 5















6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




6 Vagueness








Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




Introduction 7



Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




Chapter 1















10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56












10 Vagueness















12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56











12 Vagueness




















14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56











14 Vagueness



epistemology16













16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56











16 Vagueness
















18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56











18 Vagueness





I shall be wrong24












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56












20 Vagueness














22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56









22 Vagueness







10000 are few












24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56













24 Vagueness












P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56










P1

Not P1 and not P2

Not P2 and not P3


Pn




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56




26 Vagueness

















28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56














28 Vagueness




















30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56












30 Vagueness















All F1s are Fns






32 Vagueness















34 Vagueness











Q56









All F1s are Fns






32 Vagueness















34 Vagueness











Q56




32 Vagueness















34 Vagueness











Q56














34 Vagueness











Q56







Q56




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