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Preface and Acknowledgements
I started grappling with the philosophical challenges presented by vagueness in the early 1970s. At that time, I think it fair to say, almost nothing of real significance had been written on the topic since the contributions of Eubulides of Megara.1 In the modern era, in particular, philosophers of language from Frege on had been for the most part content to theorize in ways that marginalized vagueness, or to focus on idealized languages in which there was none. No one writing before 1970 seemed fully to have taken the measure of the awkwardness of the Sorites paradox,2 or the depth of its roots, as usually formulated, in our intuitive thinking about what kind of ability mastery of a language is.
My curiosity about the topic was originally piqued by conversations with my friend the mathematician Aidan Sudbury and with Michael Dummett, then my colleague at All Souls, who around that time was working on the stunning lecture that he later published as ‘Wang’s Paradox’ (Dummett 1975). My own interest initially stemmed from concerns in the philosophy of mathematics: I was drawn to the thought that the apparent open-endedness of the extension of a vague predicate might provide a fruitful model for the manner in which a finitist should think about the putatively infinite extension of natural number, and that a correct logic of vagueness might accordingly be appropriate for a finitist number theory. My subsequent paper ‘Strict Finitism’ (Wright 1982) was the upshot of my reflections in that direction. But while thinking about finitism I became preoccupied with the Sorites paradox itself. Dummett’s paper argued, inter alia, that vague expressions do indeed affect natural language with inconsistency—that is, that the paradox shows that our use of vague expressions is governed by rules that are actually inconsistent. That struck me then as an incredible conclusion,3 but one that was nevertheless forced by a certain conception of the significance of the kind of theory of meaning for a natural language to which philosophers of the time aspired, at least in Oxford in the mid-1970s, in the throes of the then reverberating
1 An interesting study of the early history of the Sorites is Moline (1969).
2 An exception is Black (1937).
3 Others, of course, have endorsed Dummett’s response, notably Matti Eklund. A useful conspectus of his views is Eklund (2019).
‘Davidsonic boom’. This conception is what I punningly dubbed the ‘governing view’—crudely, that understanding a natural language is, through and through, a rule-governed competence. The idea, crudely, was that we are able to parse a novel sentence by (in some sense) working out the conjoint implications of the rules of syntax relevant to its mode of construction and the semantical rules governing its occurrent primitive expressions. My first two chapters4 elaborate and critique that thought, and indeed my efforts to refine it resurface in several places in this volume. But at that time I attempted no specific resolution of the paradox other than, in this way, to try to undercut one kind of motivation for (one form of) its major premise.
It was more than a decade before I felt that I had anything further to say on the issues. By then Hilary Putnam (1983) had suggested that a resort to intuitionist rather than classical logic might contribute to a solution. After some skirmishing (Wright (1987a, this volume, Chapter 4), I became convinced that there might be something to this. The other development in my thinking at that time was the realization that we need to distinguish a variety of Sorites paradoxes, differing in the form taken by their major premises, the various lines of motivation for those premises, and even—in recognition of the so-called Forced March Sorites—in whether they involve explicit inference from premises at all (Wright 1987a, this volume, Chapter 4). I was, however, still thinking of vagueness as essentially a phenomenon of semantics—as some kind of deficiency, or partiality, of content, or lack of instruction from semantic rules—and it was only after trying to come to terms with Timothy Williamson’s brilliant book (1994, critiqued in this volume, Chapter 6) in the mid-1990s that a different way of thinking about the matter began to dawn on me. In essentials, Williamson’s ‘Epistemicism’ grafts together two thoughts: a classical, bivalent metaphysics of indicative content coupled with a view of the vagueness of a predicate as essentially an epiphenomenon of our difficulty in judging its application in the area close to the sharp ‘cut-off’ required by the first thought. It now occurred to me that dispensing with the first thought while developing the second (shorn, therefore, of the presupposition of sharp cut-offs) might provide the motivation for a thoroughgoing intuitionistic treatment of the
4 ‘On the Coherence of Vague Predicates’ (this volume, Chapter 1) was eventually published in the same volume of Synthese as ‘Wang’s Paradox’. The volume also included Kit Fine’s ‘Vagueness, Truth and Logic’ (Fine 1975), and proved to be key in launching the intense discussion of vagueness and the Sorites, now into its fifth decade, that has followed since. ‘Language-Mastery and the Sorites Paradox’ (1976, this volume, Chapter 2) was published in Gareth Evans’s and John McDowell’s influential edited anthology Truth and Meaning, exploring the issues raised by Davidson’s proposal for meaning-theory.
topic—a treatment broadly modelled on the Mathematical Intuitionists’ treatment of classical number theory and analysis—providing both for a satisfying deconstruction of the plausibility of the major premises in Sorites paradoxes, and for a well-motivated framework in which those premises can be denied without the essentially and regrettably superstitious resort to sharp cut-offs. That remains my view.
But it takes a bit of working out. My first explicit foray in the intuitionistic direction was ‘On Being in a Quandary’ (Wright 2001b, this volume, Chapter 7; also published in Philosopher’s Annual, 24). The paper was initially rejected by Mind with a comment by the referee that it ‘contained no discernible line of argument’. I requested the editor at the time, Mark Sainsbury, to determine whether that should be the final view of the journal. Mark solicited other opinions and, gratifyingly, subsequently saw fit to publish. The intuitionistic project underwent further motivation and development in a paper I wrote for the memorable Liars and Heaps conference organized by Jc Beall and Michael Glanzberg at the University of Connecticut in 2002 (Wright 2003c, this volume, Chapter 9). A further opportunity for a more complete statement of it was provided by the invitation to contribute to the Library of Living Philosophers volume for Michael Dummett that was published in 2007 (Wright 2007, this volume, Chapter 11).
Michael’s graceful but incredulous reaction to my proposal in his Reply5 echoed, at least in point of incredulity, by Ian Rumfitt6—spurred me into thinking further about the question, what kind of semantics might be appropriate for a language containing vague expressions and the basic logical resources involved in the derivation of Sorites paradoxes, if the needed intuitionistic/logical distinctions, especially the potential contrast between the
5 ‘I am left, then, with admiration for the beautiful solution of the Sorites paradox advocated by Crispin Wright, clouded by a persistent doubt whether it is correct . . . I do not say that Wright’s proposed solution of the Sorites is wrong; I say only that we need a more far-going explanation than Wright has given us of why intuitionistic logic is the right logic for statements containing vague expressions before we can acknowledge it as correct. It is not enough to show that the Sorites paradox can be evaded by the use of intuitionistic logic: what is needed is a theory of meaning, or at least a semantics, for sentences containing vague expressions that shows why intuitionistic logic is appropriate for them rather than any other logic If Crispin Wright is to persuade us that he has the true solution to the Sorites paradox, he must give a more convincing justification of the use of intuitionistic logic for statements containing vague expressions: a justification namely, that does not appeal only to the ability of that logic to resist the Sorites slide into contradiction. We need a justification that would satisfy someone who was puzzled about vagueness but had never heard of the Sorites: a justification that would sketch a convincing semantics for sentences involving vague expressions’ (Dummett 2007, pp. 453–4).
6 Rumfitt does indeed proceed to offer an intricate semantics for vagueness that has some prospect for validating intuitionist logic (Rumfitt 2015, pp. 227ff). I have expressed reservations about its fitness for the philosophical purpose elsewhere (Wright 2020, pp. 378ff).
acceptability conditions of some kinds of sentence and those of their double negations, are to be semantically grounded? My recent contribution to the volume on the Sorites edited by Elia Zardini and Sergi Oms (this volume, Chapter 14) takes that question head on and also takes the opportunity to try to provide a more rounded and complete overview of the problems and treatment of vagueness from an intuitionistic point of view than was accomplished either in the Quandary paper, the article for Liars and Heaps, or the contribution to the Dummett festschrift.
Maybe a preface to a philosophical book may allowably make a purely philosophical point. If so, perhaps here is a place to emphasize that I do not myself resonate with the Dummett–Rumfitt thought that it is only after the provision of a satisfactory such semantics that the intuitionistic proposal will be ready for the philosophical market. The proposal, after all, recommends a revision of logic. So someone who thinks that it requires validation by a background semantics has to suppose also that logical principles generally stand on firm ground only when sustained by an appropriate semantic story about the logical operators involved. That thought is, in my opinion, by no means mandatory.
Notwithstanding my own sortie into the knowledge-theoretic semantics outlined in Chapter 14, I want to sound a note of reservation about the need for any such underpinning for proposed logical restrictions specifically in response to paradox. There are delicate questions in the vicinity here concerning what should count as a solution to a paradox—questions concerning how much, and what kind of, explaining of what is going wrong, one is required to accomplish.
It is useful to be mindful here of Stephen Schiffer’s distinction (1996, 2003) between ‘happy-face’ and ‘unhappy-face’ cases. In ‘happy-face’ cases, a paradox is successfully diagnosed as owing to a determinate mistake, or oversight, which is identifiable as such by standards of our practices that are in place before the paradox is considered. (The reader may re-attend here to the concluding sentence of the quote from Dummett in n. 5.) Regrettably, happy-face paradoxes have proved historically relatively rare. With a paradox of the latter, ‘unhappy-face’ kind by contrast, there may be little to offer by way of diagnosis and explanation other than to say that the paradox is spawned by concepts and conceptual practices that are in some way inherently incoherent, or otherwise objectionable, and that have become entrenched, and that the only solution is to modify them in ways which, perhaps because of their entrenchment, may have no independently arguable sanction.
So in the present case. As far as the Sorites is concerned, I myself am compelled by the following train of thought:
• First, that we know the major premise of a Sorites is false because it is inconsistent, by absolutely elementary reasoning, with truths (the relevant polar verdicts);
• Second, that, in the usual run of examples, we clearly do not know that the hypothesis of a sharp cut-off has a witness in the relevant series of cases; and hence
• Third, that there, therefore, has to be something wrong with the reasoning—classical reasoning—that, granted plausible forms of closure of knowledge across entailment, forces us to deny the latter ignorance if we think we have the former knowledge. And this conclusion must stick before we identify a mishap—indeed, even if we cannot readily do so— in the double-negation elimination step that concludes in the postulation of a sharp cut-off.
In a recent seminar I attended in New York, a well-respected colleague was heard to say that what the Sorites teaches us is that we ‘just have to get used to’ the idea that there really are sharp boundaries in all vagueness-related Sorites series.7 I am vividly aware that a whole generation of (mostly Oxford-trained) professionals have indeed habituated themselves to that idea (or anyway profess that they have.) But I venture to suggest that, for most, the three-step train of thought articulated above will present an immensely more powerful, commonsensical appeal. If the reader concurs, they will see that the idea we need to get used to is not that of a crystalline world of unknowable sharp cutoffs that spares us Sorites paradoxes, but rather the idea of situations where classical logic lets us down.8 Why should we need a semantic theory before we can accept the charge that there is here a gap that classical reasoning evidently illicitly crosses?
Someone may of course, like Dummett, still insist that, if classical reasoning really is here unjustified, it must be recognizably so in the light of a proper account of how we already implicitly (when fully lucid and reflective)
7 Actually, some of those who take this line on the ‘regular’ Sorites propose that certain forms of modal Sorites—for example, what has come to be known as Chisholm’s paradox—require a different response. This is not the place to pursue the putative distinction.
8 A common rejoinder is that arguments for the revision of classical logic may rationally be discounted, since it has, after all, for approaching a century and a half, performed sterling service for us. No doubt it has—in mathematics and the exact sciences. But not at all as a logic of vagueness.
understand the logical operators involved. Semantic theory is therefore needed to give a correct account of that alleged prior implicit non-classical understanding. This insistence can be appropriate, however, only if we take the view that the Sorites-paradoxical reasoning, extended to the conclusion that there has to be a sharp cut-off, involves a mistake that is in principle recognizable as such by the lights of the understanding of the key notions involved that we already have. We have to be, in other words, in the territory of a possible ‘happy-face’ solution. And, if we are confident that that is so, there will now be a constraint on any semantic theory to be offered that it present a plausible account both of the antecedent understanding of ordinary thinkers and of why they are here inclined to misperceive its requirements. Good luck with that project. Revisionary proposals have historically sometimes been motivated by a sense that certain logical principles involve distortion of our understanding of the operators involved—‘relevance’ critiques of classical logic are one example. But the revisionism of the intuitionist is not of this character.
The intuitionist’s revisionism issues from a reformist stance. The semantic project is not to recover an account of extant distinctions which, if someone is seduced by the extended paradoxical reasoning, they overlook but, in the wake of already well-motivated revisions of classical logic, to propose a framework in which those revisions have an independent theoretical setting, so that we can restore a sense of knowing what we are doing in inferential practice and of how the suspect transitions may be conceived to fail. It is in this spirit that I offered the knowledge-theoretic clauses proposed in Chapter 14. However, if it is only for this purpose that it is useful, then semantic theory is precisely not needed to justify the relevant logical revisions, to persuade us of ‘the true solution to the Sorites paradox’. Instead, like Hegel’s Owl of Minerva, it spreads its wings of insight only with the coming of the dusk, when the day’s (revisionary) work is already done.
In arriving at and developing over the years the views offered in this volume, I have benefited from the published contributions of two colleagues in particular. First, the stability of the intuitionistic proposal, as argued for in the Quandary chapter, involving, as it does, maintaining a kind of agnosticism about sharp cut-offs simultaneously with the thesis of Evidential Constraint concerning a large class of vague predications, was challenged early on by Sven Rosenkranz. My original attempt to defuse his objections is contained in Chapter 8.9 Second, Stephen Schiffer’s writings have been especially
9 Rosenkranz pursues his criticisms in Rosenkranz (2009).
influential in persuading me of the importance of what I term the Characterization Problem: the challenge of saying what exactly a borderline case is. Too much of the literature has neglected this or proceeded on unexamined assumptions about the answer. But a satisfactory account of what the vagueness of a soritical predicate consists in is the essential first step both to understanding the place and significance of vagueness in natural language and to the dissolution of the Sorites. Schiffer’s own answer, latterly abandoned, is that the borderline cases of a vague predicate are those which distinctively excite a certain kind of partial belief in competent judges—and that vagueness is thus, in a certain sense, a psychological phenomenon. The relevant special notion of partial belief is, I believe, very difficult to substantiate in detail—some of the wrinkles are explored in Chapters 10 and 13—but, in thinking of borderline case as a status grounded in features of our judgemental psychology, rather than in the semantics of the relevant predicate, while simultaneously rejecting Bivalence, Schiffer made a key move in common with the intuitionistic proposal.
One other issue is prominent in the chapters that follow. Anyone thinking about vagueness needs to address the putative phenomenon of higher-order vagueness: the apparent fact that the distinction between the clear cases of a predicate and its borderline cases itself seems to have no sharp boundary, and that the point must reiterate in vertiginous fashion. The apparent fact troubled me for a long time before I hit on the argument of Chapter 5 that higherorder vagueness is itself distinctively soritical. A different argument to the conclusion that the very notion of higher-order vagueness is intrinsically incoherent is given in Chapter 9. But a third argument, gratifyingly endorsed by Riki Heck in their introductory chapter and drawing directly on my preferred approach to the Characterization Problem, has to wait until Chapter 12. If it is correct, there is simply no such thing as higher-order vagueness, as usually conceived, and it is accordingly no constraint on a satisfactory account of vagueness that it accommodate, still less explain it.
Forty-five years thinking about these matters has built up large enough debts to completely swamp my own investments. Besides the conversations at the beginning with Aidan Sudbury and Michael Dummett, my work in this area has probably benefited more extensively than even I realize from inputs from so many sources over the years. But the following deserve special mention. In the late 1990s, when I was fortunate enough to be awarded a Leverhulme Personal Research Professorship for, inter alia, a project on Vagueness, I enjoyed countless valuable discussions with my former St Andrews research students Patrick Greenough and Sven Rosenkranz, who were then working
xiv Preface and a cknowledgements
on doctoral theses concerning, respectively, vagueness and agnosticism. Later, I learned a huge amount from the regular seminar sessions with the participants in the 2003–6 AHRC-supported Arché Vagueness project: Agustin Rayo, Stewart Shapiro, Richard Dietz, Sebastiano Morruzzi, Elizabeth Barnes, and Elia Zardini. Elia, in particular, has, then and since, given me invaluable detailed written feedback on early drafts of several of the papers reprinted here. I also have profited greatly from interactions with my NYU colleagues Hartry Field, Kit Fine, and especially Stephen Schiffer, with whom I taught an exceptionally interesting graduate seminar on the topic in 2007. In more recent times I have enjoyed very helpful conversations with Susanne Bobzien and Ian Rumfitt, whose excellent, recently published co-authored paper (Bobzien and Rumfitt 2020) has important points of affinity with the views developed here.
My thanks to Dirk Kindermann, and Yu Guo for help at different times with the Bibliography, and to Yu again and Sergi Oms for extensive and meticulous work in preparing corrected copy for the Press. Special thanks to Peter Momtchiloff for his usual tact and patience while I repeatedly put off the work necessary to compile the final manuscript.
Finally, I am beyond grateful to Riki Heck for their patient, searching, and perceptive critical reconstruction of my journey. Early in their essay, they advise readers not to attempt the chapters that follow before they have assimilated Dummett’s ‘Wang’s Paradox’. I strongly endorse the same advice about Riki’s Introduction.
Crispin Wright Kemback, Fife June 2020
Origins of the Essays
The Introduction by Richard Kimberly Heck was specially written for this volume.
Chapter 1 ‘On the Coherence of Vague Predicates’, was first published in Synthese, 30 (1975), 325–65. It is reprinted here by kind permission of Springer Nature.
Chapter 2 ‘Language-Mastery and the Sorites Paradox’ was first published in G. Evans and J. McDowell (eds.), Truth and Meaning (Oxford University Press, 1976), 223–47.
Chapter 3 ‘Hairier than Putnam Thought’ (co-authored with Stephen Read) was first published in Analysis, 45 (1985) (Oxford University Press), 56–8.
Chapter 5 ‘Is Higher-Order Vagueness Coherent’ was first published in Analysis, 52 (1992) (Oxford University Press), 129–39.
Chapter 6 ‘The Epistemic Conception of Vagueness’ was first published in Southern Journal of Philosophy, 33 (1995), special number on Vagueness, 133–59. It is reprinted here by kind permission of John Wiley and Sons Inc.
Chapter 7 ‘On Being in a Quandary: Relativism, Vagueness, Logical Revisionism’ was first published in Mind, 110 (2001) (Oxford University Press), 45–98.
Chapter 8 ‘Rosenkranz on Quandary, Vagueness, and Intuitionism’ was first published in Mind, 112 (2003) (Oxford University Press), 465–74.
Chapter 9 ‘Vagueness: A Fifth Column Approach’ was first published in J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox (Oxford University Press, 2004), 84–105.
Chapter 10 ‘Vagueness-Related Partial Belief and the Constitution of Borderline Cases’ was first published in a Book Symposium on Stephen Schiffer’s The Things We Mean in Philosophy and Phenomenological Research, 73/1 (Wiley, 2007), 225–32. It is reprinted here by kind permission of John Wiley and Sons Inc.
Chapter 11 ‘ “Wang’s Paradox”’ was first published in The Library of Living Philosophers, volume XXI, The Philosophy of Michael Dummett, edited by Randall E. Auxier and Lewis Edwin Hahn (Open Court, 2007), 415–45. It is reprinted here by kind permission of the Open Court Publishing Company.
Chapter 12 ‘The Illusion of Higher-Order Vagueness’ was first published in Richard Dietz and Sebastiano Morruzzi (eds.), Cuts and Clouds (Oxford University Press, 2010), 523–49.
Chapter 13 ‘On the Characterization of Borderline Cases’ was first published in Gary Ostertag (ed.), Meanings and Other Things: Themes from the Work of Stephen Schiffer (Oxford University Press, 2016), 190–210.
The present volume collects most of Crispin Wright’s published papers on vagueness.1 These papers represent the fruits of a career-long investigation of the Sorites paradox and its significance for our understanding of language and cognition. Anyone at all familiar with the literature on vagueness will already know the early papers, as many of them have long featured as standard fare in any course on the subject. The more recent material, too—meaning the papers published in the twenty-first century—will be known to aficionados. But having recently read through the corpus myself, it seems safe to say that reading these papers together allows for an understanding of the development of Wright’s thought, and of the connections between the papers, that repays the effort required many times over. The relations between the earlier papers and the later ones are especially intriguing. What I want to do in this introduction is to provide a sense of what I have learned from my own recent rereading, and of the issues that seem to me most to need attention. I will begin, however, by providing a high-level guide to the papers themselves: to their content and their context. Afterwards, I will take up five themes that seem to me of particular interest.
An Overview
By his own account,2 Wright’s interest in vagueness—like that of many other philosophers at the time—was inspired by Sir Michael Dummett’s now classic
1 First, let me thank Crispin for asking me to write this introduction. It is an honour to do so. Though I have been reading and thinking about his papers on vagueness for almost as long as I have been doing philosophy—‘Language-Mastery’ was an early favourite—the reflections to follow grew most immediately out of a graduate seminar on vagueness that I gave at Brown University in the autumn of 2008. (This is what accounts for the fact that references to more recent literature are mostly lacking.) Thanks to all the members of that seminar for their contributions, but especially to my colleagues David Christensen and Josh Schechter, whose participation was purely optional and most welcome.
2 See the introductory remarks to ‘ “Wang’s Paradox” ’ (this volume, Chapter 11).
paper ‘Wang’s Paradox’ (Dummett 1978: 248–68).3 Perhaps the most important contribution of Dummett’s paper, as Wright again notes, was to make it plain, first, what meagre logical resources are required to generate the Sorites paradox and, second, how utterly plausible the major premise of the paradox is, at least in central cases. This is especially so in the observational case, which Dummett clearly sees as critical: if two patches of colour are visually indiscriminable by me, how could it possibly be that one of them looked red to me but the other did not? But, if visually indiscriminable patches must both look to be red if either does, then quite simple reasoning—reasoning that does not, in particular, need to appeal to induction—will lead quickly to absurd results. Dummett’s own dramatic conclusion was more or less Frege’s: that observational predicates in particular, and vague predicates more generally, are logically incoherent, so that no proper semantics for them is possible.
One reaction to Dummett’s argument is to attempt to provide the semantics Dummett had claimed could not be given.4 Wright’s reaction was different. It was to see Dummett’s argument as revealing an incoherence in certain very general assumptions about the nature of language use. Given those premises, Wright’s thought was, Dummett’s argument was correct, and what had to go were therefore the very general assumptions with which the argument began.5 It is the burden of ‘On the Coherence of Vague Predicates’ (this volume, Chapter 1)6 to make that argument.
The general assumptions in question are ones any philosopher working in Oxford in the mid-1970s might well have conceived as orthodoxy. Hence, Wright styles them the ‘governing view’. The first thought behind the governing view is that language use is, by and large, an activity governed by rules, rules that competent speakers really do follow, not so much in the sense that they consciously appeal to those rules—it is, familiarly, often a difficult matter to say what the rules are—but rather in the sense that these rules are norms that govern our linguistic behaviour, which is partly to be explained in terms of our allegiance to them. The second thought then follows more or less
3 Anyone who is considering reading Wright’s papers but who has not already read Dummett’s is hereby strongly advised to read Dummett first.
4 This is the reaction of supervaluationists (Fine 1975) and degree theorists (Goguen 1969; Peacocke 1981). Contextualism and its variants (Raffman 1996; Soames 1999; Fara 2000; Shapiro 2006: ch. 7) represent another class of responses. Wright pays very little attention to the latter. As it happens, I also tend to think that such views miss the point. See n. 29 for my reasons.
5 An argument with much the same structure is given by Jamie Tappenden (1995), though his target is Timothy Williamson’s argument (1994) for the epistemic conception of vagueness. Tappenden’s conclusions are very different from Wright’s.
6 And of the shortened version, ‘Language-Mastery and the Sorites Paradox’ (this volume, Chapter 2).
naturally. If these rules really are ones we follow, if our behaviour is supposed to be guided by these norms, then their content ought in some sense to be available to reflection.7
It should be clear enough that Dummett himself is committed to the governing view, and it may well be true that it plays an essential role in the argument of ‘Wang’s Paradox’. What is less clear is whether the governing view really can enforce the key premise of the Sorites paradox, as both Dummett and Wright require. We will worry about this below.
Not long after the two papers so far mentioned were published, however, Wright himself became somewhat dissatisfied with his treatment of vagueness. The paper ‘Further Reflections on the Sorites Paradox’ (this volume, Chapter 4) is his effort to do better. That is a long paper, and there is a lot in it. Much of it consists of critical reaction to a paper by Christopher Peacocke (1981) that had sought to defend the governing view against Wright’s criticisms by developing a degree-theoretic account of vagueness. Wright answers some of Peacocke’s arguments against his own proposals, and he adds new criticisms of degree-theoretic semantics. Many of these criticisms still seem to me quite damaging. But, all these many years on, the engagement with Peacocke is a sideshow, and what is most important in the paper lies elsewhere. Perhaps the most important contribution is Wright’s analysis of what he calls the ‘Tachometer paradox’, which I will discuss in some detail below. Almost as important, however, is Wright’s isolation of what he calls the ‘No Sharp Boundaries paradox’.
As usually formulated, the key premise of the Sorites paradox is, for example, that, if a patch is red, then any patch pairwise indistinguishable from it (in colour, of course) must also be red. The key premise is thus a universally quantified conditional: (1)
7 This second condition is what characterizes ‘implicit’ knowledge, in the sense in which that notion figures in Dummett’s writings on the philosophy of language between roughly 1974 and 1985. One can know something implicitly and not be able, at that time, to articulate it; but one is supposed, in principle, to be able to articulate it. Note how this contrasts with what has come to be called tacit knowledge. One’s knowledge of the rules of syntax—of universal grammar, for example—is in no sense supposed to be available to reflection. It is nowadays a common alternative to regard our ‘knowledge’ of whatever rules might be involved in concept-use as merely tacit. It would be worth investigating the significance of this point for present concerns. Some of what follows contributes to such a project, though it hardly completes it.
where ∼ expresses indistinguishability.8 The intuitive justification for this premise is supposed to be that redness is tolerant of small changes, and that is very much how the Sorites paradox is usually presented: surely removing one grain, plucking one hair, and so on, cannot ‘make a difference’. The No Sharp Boundaries paradox, on the other hand, proceeds from the assumption that there cannot be two indistinguishable patches one of which is red while the other is not. This premise, then, is the negation of an existential:
And the justification for this claim is supposed to be that the very vagueness of ‘red’ requires that there not be a sharp boundary between the red and the not-red. For this reason, Wright suggests, the No Sharp Boundaries paradox threatens to disclose an incoherence in the very notion of vagueness (this volume, Chapter 4, 143–145).
Of course, (1) and (2) are classically equivalent. But they are not intuitionistically equivalent, and this observation was part of what fuelled a proposal, due to Hilary Putnam (1983a), that intuitionistic logic might help here. This suggestion is criticized in ‘Hairier than Putnam Thought’ (this volume, Chapter 3), for which Stephen Read joined Wright. That is actually where the No Sharp Boundaries paradox originally appears.
It is worth pausing to appreciate, before we get too far along to notice such things, just how different these two premises feel, even if they are classically equivalent. Perhaps what is most important here is that (2) is weaker than (1) in the absence of excluded middle. For how might one argue for (1) given (2)? Well, suppose Rx and xy ∼ . By (2), it cannot be that Ry¬ . So, given excluded middle (or double-negation elimination), Ry . But, without excluded middle, we cannot draw that conclusion. And it is a common move in discussions of vagueness to reject the law of excluded middle—or, at least, the law of Bivalence—for borderline cases.9
8 Not many years later, Timothy Williamson (1990) would launch a wide-ranging investigation of the notion of indiscriminability and, shortly thereafter, revolutionize the study of vagueness in ways we will consider below (Williamson 1994).
9 Let me quickly pet a peeve. It is often said that intuitionists cannot deny the law of Bivalence. This is simply false. What is true is that an intuitionist cannot deny any particular instance of the law of Bivalence. That law, understood as (()()) xTxTx∀∨¬ —that is, as the universal quantification of an instance of excluded middle—can consistently be denied. And that is so even though (()())xTxTx¬∃¬∨¬ is a logical truth. This is a consequence of the fact that ()xFx¬∀ and ()xFx¬∃¬ are intuitionistically consistent.
We will return to this issue below. For now, let me mention a third contribution made in ‘Further Reflections’: the paradox of higher-order vagueness. It is, as was just said, natural to respond to the Sorites by saying that the paradox somehow assumes Bivalence—for example, that every statement of the form ‘Patch �� is red’ is either true or false. Rather, the thought is, there are some patches to which redness can neither truly nor falsely be ascribed; it is objectively indeterminate whether they are red. The now standard objection is that we are no more able to identify the boundary between the red and the indeterminate than we are to identify the boundary between the red and the not-red. So these intermediate boundaries are themselves vague, and that in turn gives rise to what is nowadays called ‘higher-order vagueness’. One sometimes has the sense that higher-order vagueness is ultimately what makes the problem of vagueness so utterly intractable. What Wright suggests, first in Section 6 of ‘Further Reflections’ and then again in ‘Is Higher-Order Vagueness Coherent?’ (this volume, Chapter 5), is that higher-order vagueness itself suffers from a special kind of incoherence. We will discuss this in some detail in Section 6.
About the same time Wright published ‘Is Higher-Order Vagueness Coherent?’ Timothy Williamson published his first paper defending an epistemic account of vagueness: Williamson (1992b) argued that there really is a last hair before baldness, a last red patch in the series; it is just that we do not know, and perhaps cannot know, which one it is. This view had previously been held,10 but it had never been defended with such resourcefulness. It suffices to say that things have not been the same since. Wright’s initial reaction appears in ‘The Epistemic Conception of Vagueness’ (this volume, Chapter 6).11 Wright’s central criticism is the same as that of many others: the epistemic conception has no plausible account of what determines the sharp cut-offs of whose existence it assures us, and yet whose location it insists we cannot know.12 But Wright makes other criticisms of the epistemic view, too, and these will repay close study. We will discuss one below. What is most intriguing about this paper, however, is that Wright does not disagree with Williamson’s claim that vagueness is an epistemic phenomenon. On the contrary, Wright concedes that the notion of a borderline case is to be
10 Williamson (1994) discusses the history.
11 For Williamson’s response, see his paper ‘Wright on the Epistemic Conception of Vagueness’ (Williamson 1996b).
12 My own version of this worry appears in ‘Semantic Accounts of Vagueness’ (Heck 2003: sect. 1), which was itself a commentary on what became Wright’s paper ‘Vagueness: A Fifth Column Approach’ (this volume, Chapter 9).
explained in broadly cognitive terms. What distinguishes Wright’s view is his insistence that vagueness is not just an epistemic matter but also a semantic one.
Anyone familiar with Dummett’s writings on anti-realism should see quickly both where this is headed and where it comes from. Anti-realism, as Dummett understands it, is founded on two thoughts:13 First, the semantic properties of our words must supervene on the way we use them; second, our use of at least certain parts of our language is inadequate to secure classical truth conditions for our utterances. The familiar worry about epistemicism is strikingly similar: our use of such predicates as ‘red’, ‘heap’, and the like cannot fix determinate classical extensions for them. To be sure, and importantly, one does not have to agree with Dummett’s arguments for anti-realism about mathematics (say) to share this worry about vague predicates. But the similarities are nonetheless suggestive, and that makes a broadly intuitionistic approach to vagueness worth exploring.14
It is with the exploration of such an approach that Wright’s twenty-firstcentury writings on vagueness have largely been concerned. There are hints of it in ‘The Epistemic Conception of Vagueness’, but it first emerges with clarity in ‘On Being in a Quandary’ (this volume, Chapter 7). That paper is not exclusively concerned with vagueness. Indeed, in some ways, it is more concerned to elaborate and defend aspects of the approach to questions about realism that are explored in Wright’s book Truth and Objectivity (1992b)15 and to reinforce a certain form of argument for rejection of the law of excluded middle. And that, to my mind, is precisely what makes Wright’s latest approach to problems about vagueness so interesting: he has always seen issues about vagueness as intertwined with some of the deepest issues in philosophy.
The question Wright makes central in his most recent work is how socalled borderline cases are to be characterized. Until recently, it was widely taken for granted that borderline cases of a vague predicate are ones in which
13 For Dummett’s view, see, e.g., The Logical Basis of Metaphysics (Dummett 1991). The introductory essay to Wright’s earlier collection, Realism, Meaning, and Truth (Wright 1993), remains an excellent survey.
14 For someone attracted to this sort of argument against epistemicism, these reflections should raise the question where exactly they want to get off Dummett’s train, and why. Why, that is to say, should these sorts of reflections undermine epistemicism but not undermine mathematical realism? One can understand Williamson as pushing precisely this line of argument, but from the other side: At least some of the objections to epistemicism threaten to commit one to a much broader antirealism (about mathematics, the past, etc).
15 Indeed, ‘On Being in a Quandary’ has previously been reprinted in Wright’s book Saving the Differences: Essays on Themes from Truth and Objectivity (Wright 2003b).
the predicate can neither truly nor falsely be ascribed. But Wright now agrees with Williamson (1992b: sect. I), as he did not in ‘The Epistemic Conception’, that such a description is flatly inconsistent: in the presence of the so-called Tarski biconditionals (or T-sentences), such a description leads to contradiction, and Wright, like Williamson, regards the Tarski biconditionals as nonnegotiable. Unlike Williamson, however, Wright does not conclude that Bivalence must therefore hold. Rather, he concludes, again following the intuitionists, that borderline cases cannot be ones in which Bivalence fails, in the sense that they are cases in which a particular statement is neither true nor false. Bivalence fails only in the sense that we cannot endorse it:16 we cannot say, affirmatively, that the statement in question is either true or false, but nor can we preclude its being either true or false. We must remain agnostic about the matter. We must, in particular, remain open-minded about the question whether, say, a particular patch midway between red and orange might be red. We might have no idea how the question might be decided. We might even, Wright suggests, have no idea whether it is even metaphysically possible for the question to be decided. Still, Wright’s claim is, we cannot, on pain of contradiction, say that the question can have no answer. If that seems puzzling—if one finds oneself wondering how borderline cases so much as could be resolved—well, let me simply assure the reader that they are not alone.
So, that is the broad sweep of Wright’s discussion. Let us now discuss some of these issues in more detail. We will start by exploring some themes from the early papers.
Tolerance as Putatively a priori
In his own later work, Wright often characterizes the conclusion of the early papers, ‘On the Coherence of Vague Predicates’ and ‘Language-Mastery’, as being that our use of vague predicates is not governed by ‘rules’. In fact, however, the ‘governing view’ he wants to challenge has two components: first, that the use of language is rule governed and, second, that, in uncovering the rules in question, ‘we may legitimately approach our use of language from within, that is, reflectively as self-conscious masters of it, rather than externally, equipped only with behavioural notions’ (this volume,
16 Cannot endorse it for any particular case. As I mentioned earlier, in n. 8, there is no general reason to suppose that we cannot deny Bivalence, as a general principle.
Chapter 1, 43). I think it is clear, on a careful reading, that the arguments of the early papers are directed not so much at the first thesis as at the second one. The claim is that, if the rules governing our use of vague predicates have to be discernible ‘from within’, then those rules will, among other things, commit us to the major premise of the Sorites and so to the incoherence of vague predicates. In particular, it will prove to follow from (if not just to be among) the rules that govern our use of vague predicates that their application is, as Wright puts it, tolerant of small changes to the objects to which they apply. But lots of small changes add up to big ones, and so paradox ensues.
One point that deserves emphasis is that, on Wright’s treatment, as on Dummett’s, the tolerance of vague predicates is supposed to be a putatively conceptual truth, since, as was said, tolerance is supposed to emerge from the rules that govern our use of vague predicates. And if one reflects upon the sort of paradox most commonly discussed under the heading ‘Sorites’, this should seem obviously correct: in so far as one is attracted to the claim that, say, anything that is pairwise indistinguishable from a red thing is red, it is on broadly a priori grounds. Indeed, it is hard to see how Dummett’s conclusion that observational predicates as such are incoherent could be reached except via some general, a priori argument from observationality to tolerance, and any such argument is going to make tolerance a conceptual matter.
This point, it seems to me, has been lost in some work that exploits Soritesstyle reasoning.17 There is a tendency nowadays to regard almost all expressions of natural language as vague. Now, perhaps they are all in some sense imprecise. But it is far from obvious that expressions such as ‘dog’, or even ‘chair’, are vague in the way that predicates like ‘red’ are vague. I occasionally encounter a general argument to the contrary, one that purports to demonstrate the Sorites susceptibility of (almost?) every property. It goes like this: anything that is a chair (dog, flea) will still be a chair (dog, flea) if one removes just one atom from it; quod erat demonstrandum 18 Or again: anything that is a molecule of water will still be a molecule of water if one moves its constituent atoms some incredibly minute distance further apart from one another than they originally were. But, for one thing, neither of these claims is clearly true. As far as the chair is concerned, removing the crucial atom that
17 It is emphasized, however, by Elia Zardini (2008).
18 An argument of this general sort is used by Theodore Sider (2001) to establish unrestricted mereological composition, and it may be that he is responsible for the popularity of this argument. But it is not clear to me whether Sider’s own use of the argument is vulnerable to the criticisms I am making here.
was holding the back onto the seat will not leave one with a chair.19 And, as far as water molecules are concerned, I would have thought that there would come a point at which one had moved the atoms far enough apart to break the atomic bonds holding the molecule together, at which point the atoms would proceed to go their separate ways. But, for present purposes, it does not matter at all whether I am right about this. My point is simply that, even if they are true, neither of these ‘Sorites premises’ has any chance whatsoever of being a conceptual truth. And it is in this respect that the vagueness of ‘red’ is very different from the vagueness of ‘dog’, if, indeed, ‘dog’ is vague (a question about which I hereby register serious doubt). Whether ‘chair’ is vague in the privileged sense, I am not sure. Maybe a series based upon a slow change of shape, or of ability to support weight, would give rise to a Sorites premise whose appeal was relatively a priori. But, if so, then that will be because having a distinctive shape and being able to support weight are, as Frege would have put it, ‘marks’ of the concept of a chair.
In any event, it is important to recognize that the target of Wright’s argument, in the early papers, is really much broader than a certain conception (then locally popular) of how the philosophy of language ought to be done.
To see this, we need to ask how precisely the governing view is supposed to conceive the way that ‘self-conscious reflection’ is supposed to inform the theoretical account of our use of language. Is it supposed to be distinctive of the governing view that it permits self-conscious reflection as an analytical tool? To put it differently: is it enough to make one an adherent of the governing view that one is prepared to reflect self-consciously on what justifies the application of a particular predicate as one attempts to uncover the rules that control our use of it? Or does the governing view instead assume that the character of these rules must be wholly available to self-conscious reflection? Much of what Wright has to say suggests the former interpretation, but only the latter will serve his purposes. Justifying this claim would require more careful, and more lengthy, exposition of Wright’s discussion than I can undertake here. So let me just say that I believe careful study will support this reading.20 If so, however, then the governing view might well be redescribed simply as one that is committed to the possibility of a priori conceptual
19 I owe this quip to Josh Schechter.
20 Consider this remark: ‘[C]onclusions [about the content of semantic rules] are, apparently, not to be drawn by reference to the character of our response to a predication of “looks red” when, by ordinary criteria, all the provisos are fulfilled. And in order for this exclusion to be legitimate, the dictates of the semantic rules for “looks red” have to be constituted independently of such responses’ (this volume, Chapter 4, 159; emphasis in original). Note that Wright speaks here of the governing view as excluding certain sorts of data that would not be available to self-conscious reflection.
