Philosophy Vagueness
by
Carl Ehrett
  • LAST REVIEWED: 08 October 2015
  • LAST MODIFIED: 29 June 2011
  • DOI: 10.1093/obo/9780195396577-0122

Introduction

Much, or perhaps all, of natural language is vague: the concepts expressed in natural language seem to have unclear boundaries. A central example is that of “heap”—as grains of sand are removed from a heap, at what point does it cease to be a heap? It seems that the removal of a single grain of sand can never, by itself, transform a heap into a non-heap; but applying that idea consistently would entail that a heap is still a heap when reduced to only a single grain, or zero grains. Such “sorites” paradoxes provoke troubling questions about the nature of vague languages. Some theorists (epistemicists) take vagueness to consist in speaker ignorance of certain semantic facts (such as the minimum number of grains in a heap); others take vagueness to consist in some form of semantic underdetermination (indeterminists), or in extremely subtle variations of context (contextualists). Other controversies include whether vagueness compels the adoption of a nonclassical logic, whether there is “ontic vagueness” (vagueness as a feature of the world itself rather than of representations), and whether ‘vagueness’ itself is vague.

General Overviews

Overviews of the literature on vagueness are available in varying levels of detail, and with varying focuses and aims. Sainsbury 2009 is intended as an introductory guide to various paradoxes; chapter three introduces the sorites paradox, describes the supervaluationist and degree-theoretic resolutions, and briefly discusses the notion of vague objects. Sorensen 2006 does likewise, but is a bit broader in scope, discussing contextualist responses; and it is freely available online. So too is Hyde 2005, which focuses on the sorites paradox itself (rather than vagueness generally), describing the many different versions of the paradox and its most common resolutions; Hyde also describes the ways the paradox interacts with core issues in the philosophy of language. A somewhat more detailed overview of the debate surrounding vagueness is the introduction of Keefe and Smith 1999. It includes more thorough discussion of the epistemicist solution to the sorites, and provides illuminating summaries of many of the most important writings on vagueness. Keefe 2000 includes a defense of supervaluationism and detailed descriptions and criticisms of the contemporary rivals of that view. Williamson 1994 is perhaps the most comprehensive single text on vagueness available; in addition to its defense of the epistemicist position and criticisms of rival theories, it includes three chapters on the history of vagueness and the sorites, both in ancient Greece and in analytic philosophy, as well as a chapter on ontic vagueness.

  • Hyde, D. “Sorites Paradox.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2005.

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    A comprehensive introduction to the sorites paradox. Includes discussion of the history of the paradox, explanations of its various forms, introductions to all major contemporary solutions to it, and its connections to related problems in the philosophy of language.

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    • Keefe, R. Theories of Vagueness. New York: Cambridge University Press, 2000.

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      Argues for a supervaluationist account of vagueness. Keefe includes detailed critiques of rival theories. Light on discussion of contextualism and ontic vagueness.

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      • Keefe, R., and P. Smith. “Introduction: Theories of Vagueness.” In Vagueness: A Reader. Edited by Rosanna Keefe and Peter Smith, 1–57. Cambridge, MA: MIT Press, 1999.

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        A detailed overview of the contemporary debate concerning the nature of vagueness. Provides illuminating summaries of many of the most important works on vagueness. Includes discussion of ontic vagueness.

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        • Sainsbury, R. M. “Vagueness: the Paradox of the Heap.” In Paradoxes. 3d ed. By R. M. Sainsbury, 40–68. New York: Cambridge University Press, 2009.

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          Useful for students who want a brief introduction to the sorites paradox and the problem of vagueness. Includes discussion of the most common sorts of resolutions of the problem, as well as the notion of vague objects.

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          • Sorensen, R. “Vagueness.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2006.

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            Good quick overview of the problem of vagueness, introducing all the major contemporary theories. A bit light on discussion of epistemicist accounts. This work is freely available online.

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            • Williamson, T. Vagueness. New York: Routledge, 1994.

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              Remains the most comprehensive single text on vagueness available, tracing its history and exploring all major contemporary theories, though it is light on contextualism. There is a chapter on ontic vagueness.

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              Anthologies

              The four anthologies listed here jointly collect much of the most important contemporary work on vagueness. Keefe and Smith 1999 is the only collection with historical breadth; it includes brief passages from Diogenes Laertius, Galen, and Cicero, and some works from the early 20th century, but is mostly constituted by works representative of the major contemporary positions on vagueness as semantic, epistemic, and ontic phenomena (although contextualism is not represented). Beall 2003 collects new essays more narrowly focused on semantic accounts of vagueness; contextualism is better represented here. Graff and Williamson 2002 is a very comprehensive collection of many of the most important works on all issues related to vagueness from the past few decades, with special attention to higher-order vagueness. Dietz and Morruzi 2010 is similarly wide-ranging but contains only original essays and is a more up-to-date representation of the various debates.

              • Beall, J. C., ed. Liars and Heaps: New Essays on Paradox. New York: Oxford University Press, 2003.

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                New essays focused on semantic accounts of vagueness and the liar paradox. A good collection for those interested in vagueness qua semantic paradox, but not the best anthology in this list for one looking for readings representative of the states of the major contemporary debates in general surrounding vagueness.

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                • Dietz, R., and S. Moruzzi, eds. Cuts and Clouds: Vagueness, Its Nature and Its Logic. Oxford and New York: Oxford University Press, 2010.

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                  New essays that constitute the most up-to-date representation of the state of the various debates concerning vagueness. Very comprehensive, representing all major aspects of vagueness, including ontic vagueness.

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                  • Graff, D., and T. Williamson, eds. Vagueness. The International Research Library of Philosophy. Aldershot, UK: Ashgate, 2002.

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                    Collects many of the most important works on vagueness from the past few decades. Very comprehensive in scope, and a good representation of the several active debates concerning vagueness. An unusual amount of attention is devoted here to higher-order vagueness.

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                    • Keefe, R., and P. Smith, eds. Vagueness: A Reader. Cambridge, MA: MIT Press, 1999.

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                      A collection with some historical breadth, emphasizing the 20th century. Contains works representative of all major contemporary positions on vagueness, except for contextualism.

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                      Historical Discussions

                      The sorites paradox seems to have been first employed by Eubulides of Miletus. It later became popular as a skeptical argument against Stoic epistemology; thereafter, the problem of vagueness (so conceived) received more or less sporadic attention until the birth of analytic philosophy. A very comprehensive history of the sorites is given in Williamson 1994 (cited under General Overviews). Williamson 1994 also identifies Chrysippus as employing an epistemicist solution to the paradox, but Bobzien 2002 argues that Chrysippus’ position is better understood as relying on truth-value gaps. Little is known about Eubulides’ use of the paradox, but Moline 1969 argues that he deployed it as an argument against Aristotle’s doctrine of the mean. Burnyeat 1982 explores at length the use of the sorites by skeptical opponents of Stoic epistemology. Levey 2002 discusses Leibniz’s take on the sorites paradox, arguing that his views changed widely throughout his professional life. Frege’s views on vagueness, and its relationship to his logic and ontology, are presented by van Heijenoort 1986. Russell 1923 characterizes vagueness as a feature of representations only (thereby repudiating the phenomenon of metaphysical vagueness), and argues that all language is vague—even logical operators—due to the vagueness of “true” and “false.”

                      • Bobzien, S. “Chrysippus and the Epistemic Theory of Vagueness.” Proceedings of the Aristotelian Society 102 (2002): 217–238.

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                        Argues, contra Williamson 1994 (also cited under General Overviews), that Chrysippus’ position on vagueness was not epistemicist; instead, Bobzien claims, he relied on truth-value gaps.

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                        • Burnyeat, M. F. “Gods and Heaps.” In Language and Logos: Studies in Ancient Greek Philosophy Presented to G. E. L. Owen. Edited by M. Schofield and M. C. Nussbaum, 315–338. New York: Cambridge University Press, 1982.

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                          This is the best available source exploring the use of the sorites paradox by skeptical opponents of Stoic epistemology.

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                          • Levey, S. “Leibniz and the Sorites.” Leibniz Review 12 (2002): 25–49.

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                            The most comprehensive and in-depth treatment available of Leibniz’s relationship to vagueness and the sorites. Levey shows that Leibniz’s views changed substantially over the course of his professional life; Levey offers convincing explanations of the changes.

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                            • Moline, J. “Aristotle, Eubulides and the Sorites.” Mind 78.311 (1969): 393–407.

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                              A rare attempt to uncover the purpose to which Eubulides of Miletus put the sorites paradox. Moline marshals evidence to show that Eubulides deployed it as an argument against Aristotle’s doctrine of the mean.

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                              • Russell, B. “Vagueness.” Australasian Journal of Philosophy 1.2 (1923): 84–92.

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                                Russell lays out with characteristic lucidity his view of vagueness. He is concerned here to establish that vagueness is a feature of representations only; he denies the possibility of ontic vagueness. He also argues for the unusual view that even logical operators are vague.

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                                • van Heijenoort, J. “Frege and Vagueness.” In Frege Synthesized: Essays on the Philosophical and Foundational Work of Gottlob Frege (Synthese Library). Edited by L. Haaparanta, 31–45. Boston: D. Reidel, 1986.

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                                  Very thorough exploration of Frege’s views on vagueness, and the way those views interacted with his logic and ontology. Helpfully contrasts Frege’s take on vagueness with the views of Russell and Quine.

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                                  • Williamson, T. Vagueness. New York: Routledge, 1994.

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                                    The most comprehensive available history of vagueness and the sorites. Williamson takes the reader from Eubulides of Miletus through to the contemporary debate over the nature of vagueness. He also argues that Chrysippus developed an epistemicist position.

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                                    Epistemicist Solutions

                                    The epistemicist account of vagueness holds that it is at root an epistemic phenomenon, explained by our ignorance of the extensions of our vague terms. Epistemicism retains classical logic and bivalence at the price of postulating counterintuitively sharp boundaries for vague terms. Williamson 1992 argues that our ignorance is explained by reliabilist constraints on knowledge; one cannot reliably locate a borderline bald man in relation to the boundary between bald and non-bald. Williamson 1994 contains a more detailed exploration of that view, as well as detailed critiques of rival accounts of vagueness. Horwich 1997 agrees that we are ignorant of precise, classical extensions for vague terms, but he argues contra Williamson 1994 that ignorance should be explained by our unwillingness to issue judgments about borderline cases, not vice versa. Sorensen 2001 defends a similar position. He argues that the view of Williamson 1992 and Williamson 1994 cannot account for what Sorensen takes to be a central feature of vagueness: that ignorance of the extension of a vague term is a necessary condition for competence with respect to that term. Wright 1994 argues that neither Sorensen 2001 nor Williamson 1994 adequately demonstrates that vague terms can and must have absolutely precise boundaries. Burgess 2001 takes epistemicism to task for failing to provide a reasonable metaphysics of content—to explain how it could be that absolutely precise semantic facts supervene on our rough usage of vague terms. Edgington 1999 argues that taking vagueness to consist in ignorance leads to bad predictions about our preferences and behavioral responses with respect to borderline cases. Merricks 2001 argues that all vagueness must be either metaphysical or epistemic (and not linguistic).

                                    • Burgess, J. A. “Vagueness, Epistemicism and Response-Dependence.” Australasian Journal of Philosophy 79.4 (2001): 507–524.

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                                      Presses epistemicism on one of its two major problems—the task of providing a reasonable explanation of how absolutely precise semantic facts supervene on our rough usage. Criticizes Williamson’s attempts to resolve this problem.

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                                      • Edgington, D. “Vagueness by Degrees.” In Vagueness: A Reader. Edited by R. Keefe and P. Smith, 294–316. Cambridge, MA: MIT Press, 1999.

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                                        Articulates the worry that ignorance cannot be what vagueness consists in, because such an account would misrepresent the phenomenology of borderline cases.

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                                        • Horwich, P. “The Nature of Vagueness.” Philosophy and Phenomenological Research 57.4 (1997): 929–935.

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                                          Helps to clarify a common complaint against the position taken in Williamson 1994: that ignorance cannot plausibly play the explanatory role that Williamson affords it.

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                                          • Merricks, T. “Varieties of Vagueness.” Philosophy and Phenomenological Research 62.1 (2001): 145–157.

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                                            The major rival positions to epistemicism take vagueness to be a linguistic phenomenon consisting in some sort of semantic underdetermination. This piece is illuminating in its clarification of the relationship between such theories and theories of vagueness as an epistemic phenomenon. Merricks argues that vagueness must be either epistemic or metaphysical.

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                                            • Sorensen, R. A. Vagueness and Contradiction. Oxford: Clarendon, 2001.

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                                              Develops Sorensen’s epistemicist position on vagueness. Sorensen’s account is friendlier than Williamson 1992 and Williamson 1994 to the notion that there is in some limited sense no fact of the matter as to the status of borderline cases of vague terms.

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                                              • Williamson, T.“Vagueness and Ignorance.” Proceedings of the Aristotelian Society supp. 66 (1992): 145–162.

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                                                Williamson argues for his epistemicist resolution of vagueness, holding that reliabilist constraints on knowledge explain our ignorance of the status of borderline cases.

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                                                • Williamson, T. Vagueness. New York: Routledge, 1994.

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                                                  A book-length defense of the position found in Williamson 1992, including discussion of the history of the problem and extended critiques of all major rival positions.

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                                                  • Wright, C. “The Epistemic Conception of Vagueness.” Southern Journal of Philosophy 33 (1994): 133–159.

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                                                    A particularly good presentation of the two major objections to epistemicism—that epistemicism lacks explanations of both the semantic fact of sharp boundaries and the epistemic fact of our ignorance of those boundaries.

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                                                    Supervaluationist Solutions

                                                    Supervaluationism takes a vague sentence to be true just in case it is supertrue—true on all precisifications, all ways of sharpening the claim to make it precise. Fine 1975 is a very influential piece arguing that supervaluations preserve classical logic, including the law of excluded middle (LEM). Keefe 2000 is a more recent and detailed defense of supervaluationism that incorporates responses to the major criticisms of the view, as well as Keefe’s objections to rival theories. Fodor and Lepore 1996 argues contra supervaluationism that we’ve no reason to take precisifications of a language as informative about the semantics of that language. Williamson 1994 argues that although supervaluationism preserves classical tautologies, it does not preserve classical consequence; and Williamson questions the equation of truth with supertruth. Many writers attempt to salvage a broadly supervaluationist framework while jettisoning one or more central commitments of the theory—Braun and Sider 2007 agrees with the criticism that truth is not supertruth but adapts a supervaluationist framework to flesh out the idea that vague assertions can be “approximately true” in contexts where vagueness is ignored. Eklund 2001 argues that the supervaluationist is wrong to conceive of vagueness as consisting in semantic underdetermination, and that “vagueifiers” (like the term roughly) introduce vagueness instead through semantic overdetermination. Tye 1989 and Burgess and Humberstone 1987 both agree that although the validation of the law of excluded middle is often listed as a strength of supervaluationism, in fact such validation might be undesirable. The former paper is a very concise argument that Fine 1975 should lead us to reject LEM. Burgess and Humberstone 1987 presents concerns at greater length about LEM, and develops a family of logics that allow LEM to fail.

                                                    • Braun, D., and T. Sider. “Vague, So Untrue.” Noûs 41.2 (2007): 133–156.

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                                                      Shows that supervaluationism can retain much of its force as an account of vagueness—and can escape some troubling difficulties for the standard version of the theory—by dropping the idea that truth is supertruth.

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                                                      • Burgess, J. A., and I. L. Humberstone. “Natural Deduction Rules for a Logic of Vagueness.” Erkenntnis 27.2 (1987): 197–229.

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                                                        Provides more logical detail than Tye 1989, developing a supervaluationism-inspired family of logics which, unlike the standard versions of the theory, avoid a commitment to the validity of LEM.

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                                                        • Eklund, M. “Supervaluationism, Vagueifiers, and Semantic Overdetermination.” Dialectica 55.4 (2001): 363–378.

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                                                          This revisionary supervaluationist position is helpful in illuminating the supervaluationist view of what vagueness consists in. Eklund argues that vagueness may consist in semantic overdetermination rather than semantic underdetermination.

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                                                          • Fine, K. “Vagueness, Truth and Logic.” Synthese 30.3–4 (1975): 265–300.

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                                                            This is probably the single most influential and oft-cited paper describing and supporting the supervaluationist approach to vagueness.

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                                                            • Fodor, J. A., and E. Lepore. “What Cannot Be Evaluated Cannot Be Evaluated and It Cannot Be Supervalued Either.” Journal of Philosophy 93.10 (1996): 516–535.

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                                                              One of the central debts of supervaluationism is to explain why precisifications of a language should be informative for the semantics of that language. Fodor and Lepore lucidly press this problem against supervaluationism.

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                                                              • Keefe, R. Theories of Vagueness. New York: Cambridge University Press, 2000.

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                                                                A clear, relatively up-to-date, and compelling case for supervaluationism. Keefe includes evaluations of rival theories, and has a particularly good treatment of higher-order vagueness.

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                                                                • Tye, M. “Supervaluationism and the Law of Excluded Middle.” Analysis 49.3 (1989): 141–143.

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                                                                  Argues that the reasoning of Fine 1975 should lead us (contra Fine) to reject LEM.

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                                                                  • Williamson, T. “Supervaluations.” In Vagueness. By T. Williamson, 142–164. New York: Routledge, 1994.

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                                                                    Explains and critiques supervaluationism. Of particular interest is Williamson’s presentation of a common complaint against supervaluationism: that supervaluationist logic fails to validate certain central forms of classical inference.

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                                                                    Contextualist Solutions

                                                                    Almost all participants in the debate over vagueness agree that the extensions of vague terms vary from context to context; the unique contribution of the contextualist account of vagueness is the idea that this phenomenon of context-sensitivity may, in and of itself, explain and resolve the problems generated by vagueness. Kamp 1981 pioneers the contextualist position, arguing that in a sorites argument relying on a series of conditional statements, the antecedent of each conditional subtly alters the context of evaluation for the consequent. Raffman 1994 and Raffman 1996 should be read as two installments of the same project; Raffman fleshes out a Kamp-style contextualism by arguing that in addition to ordinary notions of context, a sentence must be evaluated in a subpersonal “internal” context—which is partially a function of brute mechanical variations in our cognition. Shapiro 2006 is a similar contextualist account, but where Raffman 1994 and Raffman 1996 explain the relevant variety of contextual variation as originating in subpersonal events, Shapiro explains them in terms of social events, using the notion of a “conversational score.” Part 3 of Soames 1999 argues for a contextualist account of vagueness that takes speakers to have the discretion to adjust the extension and anti-extension of their terms. Whereas the projects of Shapiro 2006 and Raffman 1994 and Raffman 1996 are best conceived as attempts to explain the semantic phenomenon of contextualism, Soames 1999 concentrates on offering a semantic argument that a contextualist solution is mandated for vagueness. Graff 2000, though not strictly contextualist, is very much in the spirit of contextualist resolutions of vagueness; Graff relies on the notion of interest-relativity rather than context-sensitivity, holding that vague terms express interest-relative properties. Stanley 2003 argues that contextualist accounts of vagueness run afoul of the linguistic facts; context is, demonstrably, invariant under verb phrase ellipsis, although contextualist resolutions of vagueness must predict otherwise. Graff 2000 escapes these criticisms, but Stanley argues that Graff’s interest-relativity requires imputing implausibly complex properties to our ordinary vague utterances and beliefs.

                                                                    • Graff, D. “Shifting Sands: An Interest-Relative Theory of Vagueness.” Philosophical Topics 28 (2000): 45–81.

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                                                                      The best available attempt to pry the structure of a contextualist resolution of vagueness apart from the notion of context-sensitivity. Instead of context, Graff argues that vague terms express interest-relative properties.

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                                                                      • Kamp, H. “The Paradox of the Heap.” In Aspects of Philosophical Logic. Edited by U. Mönnich, 225–277. Dordrecht, The Netherlands: Reidel, 1981.

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                                                                        Influential work that pioneered the contextualist position on vagueness. Holds that context shifts in unsurveyably subtle ways. Reprinted in Graff and Williamson 2002 (cited under Anthologies).

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                                                                        • Raffman, D. “Vagueness without Paradox.” Philosophical Review 103.1 (1994): 41–74.

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                                                                          Together with Raffman 1996, this work develops a Kamp-style contextualist position that explains the extremely subtle shifts of context in terms of a subpersonal, “internal” psychological context of evaluation for vague terms.

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                                                                          • Raffman, D. “Vagueness and Context-Relativity.” Philosophical Studies 81.2–3 (1996): 175–192.

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                                                                            Should be read alongside Raffman 1994. Here, Raffman provides a detailed explanation of how her contextualist account handles a “forced march” sorites, and fills out some details about the nature of a fixed total context.

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                                                                            • Shapiro, S. Vagueness in Context. Oxford and New York: Oxford University Press, 2006.

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                                                                              Excellent in clarifying the contextualist position. Shapiro describes the motivations and content of his contextualism, and develops at length a contextualist logic of vagueness.

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                                                                              • Soames, S. “Part III: Extensions.” In Understanding Truth. By Scott Soames, 203–262. New York: Oxford University Press, 1999.

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                                                                                Pairs an argument for deflationism about truth with a semantic argument for a contextualist account of vagueness.

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                                                                                • Stanley, J. “Context, Interest-Relativity and the Sorites.” Analysis 63.4 (2003): 269–280.

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                                                                                  A trenchant critique of contextualist accounts of vagueness, as Stanley shows that such positions seem to generate demonstrably false predictions about context-variation. Separate and powerful problems are presented for Graff 2000.

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                                                                                  Multivalent Logics

                                                                                  Since the epistemicist’s commitment to bivalence is seen as untenable by many, a number of multivalent logics have been proposed for use with vague languages. Most of these are either three-valued logics, or else degree-theoretic logics, in which a sentence’s truth-value can be any real number in the interval [0,1]. In addition, “fuzzy logics” have been proposed, which combine degree-theoretic logic with fuzzy set theory to define “fuzzy” truth values (such as “very true,” “not quite false,” etc.). Freely available online, Gottwald 2009 is an accessible but comprehensive discussion of multivalent logics, including both 3- and 4-valued logics as well as degree-theoretic logics, along with their applications (of which vagueness is one). Field 2003 is an attempt to account for vagueness with a 3-valued logic that rejects the law of excluded middle; Field includes critiques of epistemicism, supervaluationism, and paraconsistent logics for vagueness. Machina 1976 and Edgington 1999 each offer degree-theoretic logics for vagueness; the former retains a commitment to truth-functionality, while the latter rejects it. Edgington also includes an argument against taking the degrees of such a calculus to be epistemic rather than semantic. Wright 1987 includes six objections to the use of degree-theoretic logics for vagueness. Weatherson 2005 argues that central flaws in the many-valued approach to vagueness stem from the common assumption that degrees of truth are linearly ordered; Weatherson fleshes out a many-valued semantics that drops this assumption. Hajek 2010 is an overview of the nature and applications of fuzzy logic, and is freely available online. Zadeh 1975 offers a fuzzy logic for vagueness that defines a small, finite set of fuzzy truth-values that can support a notion of “approximate validity.”

                                                                                  • Edgington, D. “Vagueness by Degrees.” In Vagueness: A Reader. Edited by R. Keefe and P. Smith, 294–316. Cambridge, MA: MIT Press, 1999.

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                                                                                    Like Machina 1976, this is an argument for a degree-theoretic logic of vagueness, but unlike Machina, Edgington rejects truth-functionality. Readers may find helpful her explanation of why degrees of “verity” should not be understood epistemically.

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                                                                                    • Field, H. “No Fact of the Matter.” Australasian Journal of Philosophy 81.4 (2003): 457–480.

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                                                                                      An illuminating case for a 3-valued logic of vagueness. Field helps clarify the discursive terrain by motivating his view with critiques of epistemicism, supervaluationism, and paraconsistent logics of vagueness.

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                                                                                      • Gottwald, S. “Many-Valued Logic.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2009.

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                                                                                        A very accessible introduction to various types of multivalent logics, along with discussion of their motivations and applications, including vagueness. Freely available online.

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                                                                                        • Hajek, P. “Fuzzy Logic.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2010.

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                                                                                          A relatively accessible introduction to, and overview of, fuzzy logics. Freely available online, this entry clarifies the nature of fuzzy logic; its applications and motivations are not discussed at length. Readers without much logic background may wish to read Gottwald 2009 first.

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                                                                                          • Machina, K. F. “Truth, Belief and Vagueness.” Journal of Philosophical Logic 5.1 (1976): 47–78.

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                                                                                            Develops and motivates a degree-theoretic logic of vagueness that (unlike Edgington 1999) allows the law of excluded middle and the law of non-contradiction to have less than full truth, and which preserves truth-functionality.

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                                                                                            • Weatherson, Brian. “True, Truer, Truest.” Philosophical Studies 123.1/2 (2005): 47–70.

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                                                                                              Critiques degree-theoretic logics of vagueness and proposes a many-valued semantics for classical logic that drops the assumption that the “truer-than” relation is linearly ordered (and so holds that degrees of truth cannot be represented by the real numbers).

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                                                                                              • Wright, C. “Further Reflections on the Sorites Paradox.” Philosophical Topics 15.1 (1987): 227–290.

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                                                                                                This is a long article with several very well-argued moving parts; one of its aims is to provide six trenchant objections to the use of degree-theoretic logics for vagueness.

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                                                                                                • Zadeh, L. A. “Fuzzy Logic and Approximate Reasoning.” Synthese 30.3–4 (1975): 407–428.

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                                                                                                  The piece describes a fuzzy logic intended for use with vagueness. Readers lacking relevant background in multivalent logic may wish to read Gottwald 2019 first.

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                                                                                                  Ontic Vagueness

                                                                                                  It is sometimes assumed without argument that vagueness must be either a linguistic or an epistemic phenomenon, but we might also wonder whether the world can actually be vague. This has spawned two broad questions about so-called ontic vagueness (also referred to as metaphysical vagueness, ontological vagueness, worldly vagueness, and vagueness in the world). One question is whether vague identity is possible; that is, whether the identity relation is such that there may be two objects A and B such that it is indeterminate whether A = B. Secondly, there are questions of vague composition—cases in which it seems indeterminate whether A has B as a proper part. Unger 1980 presents the problem of vague composition in an especially troubling way as the “problem of the many”; he defends a nihilistic solution, holding that there exist no objects corresponding to vague terms (like “table,” “person,” etc.). Weatherson 2009 provides a comprehensive overview of the discussion of the problem of the many, including a survey of available solutions. Weatherson 2003 defends supervaluationist solutions to the problem of the many, arguing that the relevant vagueness should be taken as semantic rather than ontic. It is little exaggeration to say that all contemporary discussion of vague identity has its basis in Evans 1978, a single-page argument against the possibility of indeterminate identity. Parsons 2000 is an extended discussion and defense of worldly indeterminacy of both identity and composition; Parsons responds at length to Evans 1978. Morreau 2002 does not challenge the conclusion of Evans 1978, but argues that even without indeterminate identity, there may exist vague objects due to indeterminate composition. Barnes and Williams 2009 does not challenge Evans 1978 with respect to de re indeterminate identity, but the authors do argue, specifically contra Weatherson 2003, that referential vagueness can still deliver genuine worldly indeterminacy. Burgess 1990 offers a comprehensive treatment of ontological issues related to vagueness; he ultimately argues for the unusual position that it is an empirical matter whether the world is vague.

                                                                                                  • Barnes, E., and J. R. Williams. “Vague Parts and Vague Identity.” Pacific Philosophical Quarterly 90.2 (2009): 176–187.

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                                                                                                    Partially endorsing the conclusion of Evans 1978, the authors hope to limit the scope of that conclusion. They argue that only de re indeterminate identity is impossible; genuine worldly indeterminacy is still possible, and can be understood on the model of referential vagueness.

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                                                                                                    • Burgess, J. A. “Vague Objects and Indefinite Identity.” Philosophical Studies 59.3 (1990): 263–287.

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                                                                                                      Attempts a comprehensive treatment of ontological issues related to vagueness and argues that it is an empirical matter whether the world is vague, and hence that worldly vagueness is both intelligible and metaphysically possible.

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                                                                                                      • Evans, G. “Can There Be Vague Objects?” Analysis 38.4 (1978): 208.

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                                                                                                        This extremely influential, single-page argument remains central to debates over vague identity.

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                                                                                                        • Morreau, M. “What Vague Objects Are Like.” Journal of Philosophy 99.7 (2002): 333–361.

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                                                                                                          Unlike Parsons 2000, Morreau does not challenge the conclusion of Evans 1978; instead, Morreau argues that vague objects are compatible with Evans’s conclusions, and should be understood on the model of vague composition.

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                                                                                                          • Parsons, T. Indeterminate Identity: Metaphysics and Semantics. New York and Oxford: Clarendon, 2000.

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                                                                                                            An extensive and wide-ranging discussion and defense of the notion of indeterminate identity. Evans 1978 is addressed at length.

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                                                                                                            • Unger, P. “The Problem of the Many.” Midwest Studies in Philosophy 5 (1980): 411–467.

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                                                                                                              Extremely influential paper in which Unger introduces the problem of the many and defends a nihilistic solution.

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                                                                                                              • Weatherson, B. “Many Many Problems.” Philosophical Quarterly 53.213 (2003): 481–501.

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                                                                                                                Argues that the problem of the many should be resolved with supervaluations, and that it is therefore a problem of vague language, not ontic vagueness.

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                                                                                                                • Weatherson, B. “The Problem of the Many.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2009.

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                                                                                                                  This entry, freely available online, is an accessible but thorough description of the problem of the many and the available range of solutions.

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                                                                                                                  Higher-Order Vagueness

                                                                                                                  Just as for a vague predicate F there will be borderline cases (items that are neither clearly F nor clearly not F), so too it is generally accepted that there will be borderline borderline cases (items such that they are neither clearly borderline cases of F nor clearly not borderline cases of F), and borderline borderline borderline cases, and so on. This is the phenomenon of higher-order vagueness (HOV). Shapiro 2006 explores a variety of options available for resolving HOV in his contextualist framework; these options are mostly transferable to noncontextualist solutions as well. Greenough 2005 critiques Shapiro’s options for HOV, and suggests instead an adaptation of a resolution of HOV first applied by Rosanna Keefe to supervaluationism. Most theorists take there to be either no such thing as HOV, or else infinite levels of HOV. Burgess 1990 and Soames 2003 both take the uncommon position that there are nonzero but finite levels of HOV. Burgess argues that the exact number of levels of HOV is an empirical matter; Soames argues that there is exactly one level of HOV. An argument for the existence of HOV is critiqued by Varzi 2003. Sainsbury 1991 likewise argues that the phenomenon of HOV is real enough but should not be understood as a hierarchy of levels of vagueness. He also argues that Crispin Wright’s position on vagueness commits him (implausibly) to a sharply tripartite semantics for vague terms. Wright 1992 attempts to answer Sainsbury’s objections and argues for the incoherence of the notion of HOV. Edgington 1993 responds to Wright 1992 to defend the coherence of HOV, and to argue (contra Sainsbury) that the objections of Sainsbury 1991 help motivate a degree-theoretic account of vagueness.

                                                                                                                  • Burgess, J. A. “The Sorites Paradox and Higher-Order Vagueness.” Synthese 85.3 (1990): 417–474.

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                                                                                                                    Challenges the common supposition that either there is no HOV, or infinite orders of HOV. Burgess offers an account of the nature of HOV that plausibly explains how it could terminate at low, finite levels.

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                                                                                                                    • Edgington, D. “Wright and Sainsbury on Higher-Order Vagueness.” Analysis 53.4 (1993): 193–200.

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                                                                                                                      Defends the coherence of HOV against Wright 1992 by attacking part of his conception of definiteness. Edgington agrees with Sainsbury 1991 about the inadequacy of most accounts of vagueness with respect to HOV but argues that rather than pointing to Sainsbury’s boundaryless concepts, this should lead us to her degree-theoretic account.

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                                                                                                                      • Greenough, P. “Contextualism about Vagueness and Higher-Order Vagueness.” Proceedings of the Aristotelian Society supp. 79.1 (2005): 167–190.

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                                                                                                                        Instructively critiques the options available to Shapiro 2006 in handling HOV and argues in favor of a solution originally developed by Rosanna Keefe for her supervaluationist theory.

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                                                                                                                        • Sainsbury, R. M. “Is There Higher-Order Vagueness?” Philosophical Quarterly 41.163 (1991): 167–182.

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                                                                                                                          Illuminating in its attack on near-universal suppositions about the nature of HOV; the phenomenon of HOV is real enough, Sainsbury holds, but it is not hierarchical.

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                                                                                                                          • Shapiro, S. “Refinements and Extensions I: So-Called ‘Higher-Order Vagueness.’” In Vagueness in Context. By Stewart Shapiro, 125–164. New York: Oxford University Press, 2006.

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                                                                                                                            Explores potential positions on HOV to accompany Shapiro’s contextualist framework, but his discussion here has interest beyond contextualism. The options he considers can largely be adapted or straightforwardly applied to other views of vagueness.

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                                                                                                                            • Soames, S. “Higher-Order Vagueness for Partially Defined Predicates.” In Liars and Heaps: New Essays on Paradox. Edited by J. C. Beall, 128–150. New York: Oxford University Press, 2003.

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                                                                                                                              Like Burgess 1990, Soames thinks that HOV exists but not at infinitely many levels, but whereas Burgess takes the exact number of levels to be an empirical matter, Soames argues that there must be exactly one level of HOV.

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                                                                                                                              • Varzi, A. C. “Higher-Order Vagueness and the Vagueness of ‘Vague.’” Mind 112.446 (2003): 295–299.

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                                                                                                                                Offers a lucid and instructive defense of the position that there is no HOV.

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                                                                                                                                • Wright, C. “Is Higher Order Vagueness Coherent?” Analysis 52.3 (1992): 129–139.

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                                                                                                                                  Wright defends his view that the notion of HOV is incoherent. He specifically responds to the arguments of Sainsbury 1991.

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                                                                                                                                  Coherence and Phenomenal Continuity

                                                                                                                                  Vagueness seems to be especially troubling in cases of phenomenal continuity—when a sorites can be constructed for an observational term (e.g., “red”) in which adjacent members of the series cannot be perceptually distinguished by direct observation. Such cases are sometimes used to support the idea that vague language is incoherent. Wright 1975 takes this position, arguing from phenomenally continuous sorites series that vague language cannot be taken to be rule-governed, given plausible constraints on linguistic rules. Dummett 1975 takes a different lesson from phenomenally continuous sorites series: not that language is not rule-governed, but rather that it is governed by inconsistent rules. Burns 1986 responds to both Wright 1975 and Dummett 1975 in defense of the view that vague language is governed by consistent semantic rules. Peacocke 1981 argues, primarily contra Wright 1975, that a degree-theoretic logic can resolve the phenomenally continuous sorites. Wright 1987 extends his earlier position; he includes three objections to Dummett’s view of vague language (Dummett 1975) as governed by inconsistent rules and six objections to the use of degree-theoretic logics for vagueness. Eklund 2005 agrees with Wright 1987 that vagueness cannot consist in inconsistent rules of application but argues that it does consist in a disposition to accept such rules; this position is more amenable than Wright’s to the application of logic to vague language. Sainsbury 1999, like Wright 1975 and Wright 1987, rejects the application of classical logic to vague terms; but whereas Wright attributes this to the nonobjectivity of meaning, Sainsbury attributes it to a misunderstanding of the structure of meaning. Vague terms are boundaryless, he argues, perhaps best understood via prototype theory, and requiring some nonclassical logic. Finally, Graff 2001 questions a central assumption of much of these debates, arguing that phenomenal nondiscriminability is in fact transitive after all.

                                                                                                                                  • Burns, L. “Vagueness and Coherence.” Synthese 68.3 (1986): 487–513.

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                                                                                                                                    Resists the conclusions of Wright 1975 and Dummett 1975 by arguing that the “tolerance” of vague language is a more complex phenomenon than either Dummett or Wright appreciates.

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                                                                                                                                    • Dummett, M. “Wang’s Paradox.” Synthese 30.3–4 (1975): 301–324.

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                                                                                                                                      Dummett’s influential treatment of the phenomenally continuous sorites takes vague language to be governed by inconsistent rules.

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                                                                                                                                      • Eklund, M. “What Vagueness Consists In.” Philosophical Studies 125.1 (2005): 27–60.

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                                                                                                                                        Defends a middle path between the positions of Dummett 1975, Wright 1975, and Wright 1987. Eklund agrees with Wright that vagueness cannot consist in inconsistent linguistic rules but argues that it does consist in a disposition to accept inconsistent rules. He discusses the sort of semantics that should accompany such a view.

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                                                                                                                                        • Graff, D.. “Phenomenal Continua and the Sorites.” Mind 110.440 (2001): 905–935.

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                                                                                                                                          In the debate between Wright and Dummett, as in most of the relevant literature, phenomenal nondiscriminability is taken to be obviously intransitive. Graff challenges this assumption, arguing nondiscriminability to be transitive.

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                                                                                                                                          • Peacocke, C. “Are Vague Predicates Incoherent?” Synthese 46.1 (1981): 121–141.

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                                                                                                                                            Responds to Wright 1975, arguing that Wright’s extreme solution is not needed; a degree-theoretic logic can resolve the phenomenally continuous sorites.

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                                                                                                                                            • Sainsbury, R. M. “Concepts without Boundaries.” In Vagueness: A Reader. Edited by R. Keefe and P. Smith, 251–264. Cambridge, MA: MIT Press, 1999.

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                                                                                                                                              Skeptical responses to the sorites have sometimes involved drastic error theories about the use of vague language. Sainsbury’s position helpfully clarifies a path that imputes error not to ordinary ways of speaking but to fundamental aspects of traditional semantics.

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                                                                                                                                              • Wright, C. “On the Coherence of Vague Predicates.” Synthese 30.3–4 (1975): 325–365.

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                                                                                                                                                An influential paper in which Wright relies on phenomenally continuous sorites series to argue against the view of vague language as rule-governed, given ordinary constraints on linguistic rules. Therefore there is no logic for vague language; vague language is incoherent.

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                                                                                                                                                • Wright, C. “Further Reflections on the Sorites Paradox.” Philosophical Topics 15.1 (1987): 227–290.

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                                                                                                                                                  Responds both to Dummett 1975 and to Peacocke 1981. This paper is helpful in clarifying the nature of the disagreement among the three theorists, and what hangs on the choice of answers to the relevant questions.

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