There are three main traditional accounts of vagueness : one takes it as a genuinely metaphysical phenomenon, one takes it as a phenomenon of ignorance, and one takes it as a linguistic or conceptual phenomenon. In this paper I first very briefly present these views, especially the epistemicist and supervaluationist strategies, and shortly point to some well-known problems that the views carry. I then examine a 'statistical epistemicist' account of vagueness that is designed to avoid precisely these problems – it (...) will be a view that provides an account of the phenomenon of vagueness as coming from our linguistic practices, while insisting that meaning supervenes on use, and that our use of vague terms does yield sharp and precise meanings, which we ignore, thus allowing bivalence to hold. (shrink)

The purpose of this paper is to challenge some widespread assumptions about the role of the modal axiom S4 in a theory of vagueness. In the context of vagueness, S4 usually appears as the principle ‘If it is clear (determinate, definite) that A, then it is clear (determinate, definite) that it is clear (determinate, definite) that A’, or, more formally, CA → CCA. We show how in the debate over S4 two different notions of clarity are in play (Williamson-style "luminosity" (...) or self-revealing clarity and concealeable clarity) and what their respective functions are in accounts of higher-order vagueness. On this basis, we argue first that, contrary to common opinion, higher-order vagueness and S4 are perfectly compatible. This is in response to claims like that by Williamson that, if vagueness is defined with the help of a clarity operator that obeys S4, higher-order vagueness disappears. Second, we argue that, contrary to common opinion, (i) bivalence-preservers (e.g. epistemicists) can without contradiction condone S4, and (ii) bivalence-discarders (e.g. open-texture theorists, supervaluationists) can without contradiction reject S4. Third, we rebut a number of arguments that have been produced by opponents of S4, in particular those by Williamson. (The paper is pitched towards graduate students with basic knowledge of modal logic.). (shrink)

The paper presents a new theory of higher-order vagueness. This theory is an improvement on current theories of vagueness in that it (i) describes the kind of borderline cases relevant to the Sorites paradox, (ii) retains the ‘robustness’ of vague predicates, (iii) introduces a notion of higher-order vagueness that is compositional, but (iv) avoids the paradoxes of higher-order vagueness. The theory’s central building-blocks: Borderlinehood is defined as radical unclarity. Unclarity is defined by means of competent, rational, informed speakers (‘CRISPs’) whose (...) competence, etc., is indexed to the scope of the unclarity operator. The unclarity is radical since it eliminates clear cases of unclarity and, that is, clear borderline cases. This radical unclarity leads to a (bivalence-compatible, non-intuitionist) absolute agnosticism about the semantic status of all borderline cases. The corresponding modal system would be a non-normal variation on S4M. (shrink)

ABSTRACT: Recently a bold and admirable interpretation of Chrysippus’ position on the Sorites has been presented, suggesting that Chrysippus offered a solution to the Sorites by (i) taking an epistemicist position1 which (ii) made allowances for higher-order vagueness.2 In this paper I argue (i) that Chrysippus did not take an epistemicist position, but − if any − a non-epistemic one which denies truth-values to some cases in a Sorites-series, and (ii) that it is uncertain whether and how he made allowances (...) for higher-order vagueness, but if he did, this was not grounded on an epistemicist position. (shrink)

Most speakers experience unclarity about the application of predicates like tall and red to liminal cases. We formulate alternative psychological hypotheses about the nature of this unclarity, and report experiments that provide a partial test of them. A psychologized version of the ‘vagueness-as-ignorance’ theory is then advanced and defended.

I argue, pace Timothy Williamson, that one cannot provide an adequate account of what it is for a case to be borderline by appealing to facts about our inability to discriminate our actual situation from nearby counterfactual situations in which our language use differs in subtle ways. I consider the two most natural ways of using such resources to provide an account of what it is for a case to be borderline and argue that both face crippling defects. I argue (...) that the problems faced by these two accounts point to more general reasons to be skeptical of the claim that facts about semantic indiscriminability provide sufficient resources for an analysis of what it is for a case to be borderline. (shrink)

I motivate and briefly sketch a linguistic theory of vagueness, on which the notion of indeterminacy is understood in terms of the conventions of language: a sentence is indeterminate iff the conventions of language either forbid asserting it and forbid asserting its negation, under the circumstances, or permit asserting either. I then consider an objection that purports to show that if this theory (or, as far as I can see, any other theory of vagueness that deserved the label "linguistic" were (...) true, there would be no such thing as indeterminacy. I respond to this objection by arguing on independent grounds against its main premise, the widely-accepted claim that if it is indeterminate whether P, no human being knows whether P. I defend an alternative view according to which, when it is indeterminate whether P, it is often also indeterminate whether we know that P. (shrink)

In this paper I argue, first, that the only difference between Epistemicism and Nihilism about vagueness is semantic rather than ontological, and second, that once it is clear what the difference between these views is, Nihilism is a much more plausible view of vagueness than Epistemicism. Given the current popularity of certain epistemicist views (most notably, Williamson’s), this result is, I think, of interest.

According to the view I christen sharpism, when Joe says to his daughter in a perfectly ordinary context ‘The Earth is super-duper old’, his claim has an incredibly discriminating truth condition: although it’s true if the Earth is over 347,342,343 years, 2 days, and 17 nanoseconds old, if the Earth is even a nanosecond younger then his claim has some status other than “just plain true”—but we leave open what that new status might be: false, indeterminate, indeterminately indeterminate, meaningless, just (...) under 100% true, or whatnot. The material point is that the claim changes in truth status (“alethic status”) with a nanosecond change in the Earth’s age. The sharp cutoff might not be a cutoff separating the true from the false, but it is a sharp alethic cutoff nonetheless. It has this sensitive truth condition even though Joe has never made any relevant linguistic stipulations and doesn’t even know what a nanosecond is. Another example: when I say to a visiting speaker, ‘The auditorium where you’ll give your lecture is a short walk from here’, my claim is true if the auditorium is no more than 123 meters, 6 centimeters, and 16 nanometers away. Hence, if it turns out that we were 123 meters, 6 centimeters, and 17 nanometers away from the auditorium, my claim had some status other than true. (shrink)

Timothy Williamson’s potentially most important contribution to epistemicism about vagueness lies in his arguments for the basic epistemicist claim that the alleged cut-off points of vague predicates are not knowable. His arguments for this are based on so-called ‘margin for error principles’. This paper argues that these principles fail to provide a good argument for the basic claim. Williamson has offered at least two kinds of margin for error principles applicable to vague predicates. A certain fallacy of equivocation (on the (...) meaning of ‘knowable’) seems to underlie his justification for both kinds of principles. Besides, the margin for error principles of the first kind can be used in the derivation of unacceptable consequences, while the margin for error principles of the second kind can be shown to be compatible with the falsity of epistemicism, under a number of assumptions acceptable to the epistemicist. (shrink)

Timothy Williamson’s potentially most important contribution to epistemicism about vagueness lies in his arguments for the basic epistemicist claim that the alleged cut-off points of vague predicates are not knowable. His arguments for this are based on so-called ‘margin for error principles’. This paper argues that these principles fail to provide a good argument for the basic claim. Williamson has offered at least two kinds of margin for error principles applicable to vague predicates. A certain fallacy of equivocation (on the (...) meaning of ‘knowable’) seems to underlie his justification for both kinds of principles. Besides, the margin for error principles of the first kind can be used in the derivation of unacceptable consequences, while the margin for error principles of the second kind can be shown to be compatible with the falsity of epistemicism, under a number of assumptions acceptable to the epistemicist. (shrink)

Let us say that the proposition that p is transparent just in case it is known that p, and it is known that it is known that p, and it is known that it is known that it is known that p, and so on, for any number of iterations of the knowledge operator ‘it is known that’. If there are transparent propositions at all, then the claim that any man with zero hairs is bald seems like a good candidate. (...) We know that any man with zero hairs is bald. And it also does not seem completely implausible that we know that we know it, and that we know that we know that we know it, and so on. (shrink)

Let us say that the proposition that p is transparent just in case it is known that p, and it is known that it is known that p, and it is known that it is known that it is known that p, and so on, for any number of iterations of the knowledge operator ‘it is known that’. If there are transparent propositions at all, then the claim that any man with zero hairs is bald seems like a good candidate. (...) We know that any man with zero hairs is bald. And it also does not seem completely implausible that we know that we know it, and that we know that we know that we know it, and so on. Mario G´omez-Torrente (1997, p. 244) observes that if B(0) is transparent (where in general, B(n) stands for the claim that any man with n hairs is bald), then not all of the following of Timothy Williamson’s margin for error principles can be true. (shrink)

If you’ve read the first five hundred pages of this book, you’ve read most of it (we assume that ‘most’ requires more than ‘more than half’). The set of natural numbers n such that the first n pages are most of this book is nonempty. Therefore, by the least number principle, it has a least member k. What is k? We do not know. We have no idea how to find out. The obstacle is something about the term ‘most’. It (...) is recognisably the same feature as the feature of ‘heap’ that prevents us from finding an answer to the question ‘How many grains make a heap?’ and the feature of many other expressions that prevents us from finding answers to similar questions involving them. Call this feature, whatever its underlying nature, vagueness. (shrink)

This paper presents an new epistemicist account of vagueness, one that avoids standard arbitrariness worries by exploiting a plenitudinous metaphysic.

Read as a comment on Crispin Wright's \"Vagueness: A Fifth Column Approach\", this paper defends a form of supervaluationism against Wright's criticisms. Along the way, however, it takes up the question what is really wrong with Epistemicism, how the appeal of the Sorities ought properly to be understood, and why Contextualist accounts of vagueness won't do.

Anyone who endorses Universalism and Four Dimensionalism owes us an argument for those controversial mereological theses. One may put forth David Lewis's and Ted Sider's arguments from vagueness. However, the success of those arguments depends on the rejection of the epistemic view of vagueness, and thus opens the door to a fatal confrontation with one particularly troubling version of The Problem of the Many. The alternative for friends of Universalism and Four Dimensionalism is to abandon those currently fashionable arguments in (...) favor of others which are consistent with the epistemic view of vagueness and with the elegant solution it furnishes to that problem. (shrink)

'Epistemic' theories of vagueness notoriously claim that (despite the appearances to the contrary) all of our vague terms have sharp boundaries, it's just that we can't know what they are. Epistemic theories are typically criticized for failing to explain (1) the source of the ignorance postulated, and (2) how our terms could come to have such precise boundaries. Both of these objections will, however, be shown to rest on certain 'presentist' assumptions about the relation between use and meaning, and if (...) allows that the meaning constitutive elements of our linguistic practices can extend into the future, the possibility of a new sort of 'normative epistemicism' emerges. (shrink)

Epistemicism about vagueness is the view that vagueness, or indeterminacy, is an epistemic matter. Truthmaker-gap epistemicism is the view that indeterminate truths are indeterminate because their truth is not grounded by any worldly fact. Both epistemicism in general and truthmaker-gap epistemicism originated in Roy Sorensen's work on vagueness. My aim in this paper is to give a characterization of truthmaker-gap epistemicism and argue that the view is incompatible with higher-order vagueness: vagueness in whether some case of the form ‘it is (...) determinate that A’ or ‘it is indeterminate whether A’ is true. Since it is highly likely that there is higher-order vagueness (and indeed, Sorensen is adamant that there is higher-order vagueness), truthmaker-gap epistemicism is in an uncomfortable position. (shrink)

The paper challenges Williamson’s safety based explanation for why we cannot know the cut-off point of vague expressions. We assume throughout (most of) the paper that Williamson is correct in saying that vague expressions have sharp cut-off points, but we argue that Williamson’s explanation for why we do not and cannot know these cut-off points is unsatisfactory. -/- In sect 2 we present Williamson's position in some detail. In particular, we note that Williamson's explanation relies on taking a particular safety (...) principle ('Meta-linguistic belief safety' or 'MBS') as a necessary condition on knowledge. In section 3, we show that even if MBS were a necessary condition on knowledge, that would not be sufficient to show that we cannot know the cut-off points of vague expressions. In section 4, we present our main case against Williamson's explanation: we argue that MBS is not a necessary condition on knowledge, by presenting a series of cases where one's belief violates MBS but nevertheless constitutes knowledge. In section 5, we present and respond to an objection to our view. And in section 6, we briefly discuss the possible directions a theory of vagueness can take, if our objection to Williamson's theory is taken on board. (shrink)

We argue that in the decision making process required for selecting assertible vague descriptions of an object, it is practical that communicating agents adopt an epistemic stance. This corresponds to the assumption that there exists a set of conventions governing the appropriate use of labels, and about which an agent has only partial knowledge and hence significant uncertainty. It is then proposed that this uncertainty is quantified by a measure corresponding to an agent’s subjective belief that a vague concept label (...) can be appropriately used to describe a particular object. We then apply Bayesian networks to investigate, in the case when knowledge of labelling conventions is represented by an ordering or ranking of the labels according to their appropriateness, how measure values allocated to basic labels can be used to directly infer the appropriateness measure of compound expressions. (shrink)

Timothy Williamson famously offered an argument from these Tarskian principles in favor of bivalence. I show, dwelling on (Andjelkovic & Williamson, 2000), that the argument depends on a contentious formulation of the Tarskian principles about truth (and falsity), which the supervaluationist can reject without jeopardizing the Tarskian insight. In the mentioned paper, Adjelkovic and Williamson argue that, even if the appropriate formulation seems to make room for failure of bivalence in borderline cases, this appearance is illusory, once one grants an (...) independent further principle involving biconditionals. I finally argue that the formulation of this is, however, contentious in a similar manner. (shrink)

It is taken for granted in much of the literature on vagueness that semantic and epistemic approaches to vagueness are fundamentally at odds. If we can analyze borderline cases and the sorites paradox in terms of degrees of truth, then we don’t need an epistemic explanation. Conversely, if an epistemic explanation suffices, then there is no reason to depart from the familiar simplicity of classical bivalent semantics. I question this assumption, showing that there is an intelligible motivation for adopting a (...) many-valued semantics even if one accepts a form of epistemicism. The resulting hybrid view has advantages over both classical epistemicism and traditional many-valued approaches. (shrink)

We argue that the epistemic theory of vagueness cannot adequately justify its key tenet-that vague predicates have precisely bounded extensions, of which we are necessarily ignorant. Nor can the theory adequately account for our ignorance of the truth values of borderline cases. Furthermore, we argue that Williamson’s promising attempt to explicate our understanding of vague language on the model of a certain sort of “inexact knowledge” is at best incomplete, since certain forms of vagueness do not fit Williamson’s model, and (...) in fact fit an alternative model. Finally, we point out that a certain kind of irremediable inexactitude postulated by physics need not be-and is not commonly-interpreted as epistemic. Thus, there are aspects of contemporary science that do not accord well with the epistemicist outlook. (shrink)

There are two natural objections to epistemicist accounts of vagueness that one frequently encounters in conversation (objections that are frequently run together).2 One objection is that it is hard to live without an informative answer to the question as to how the nonsemantic facts – non-relational and relational -- about a given individual determine the semantic profile of that individual. Let us call this the Bruteness worry. A second objection is that it seems metaphysically arbitrary that just one of the (...) many candidate cutoffs for, say, baldness, should be associated with some ordinary pattern of use of the term ‘bald’. Let us call this the Arbitrariness worry. In this paper we sketch an epistemicist approach to vagueness that provides a distinctive perspective on these foundational issues. On the one hand, it affords epistemicism protection against the Arbitrariness worry, by combining the even-handedness typically associated with supervaluationism with the logic and semantics distinctive of epistemicism.3 And on the other, it renders the Bruteness objection less pressing by assimilating it to a kind of bruteness that many of us have already learned to live with. Our view relies on a plenitudinous metaphysics to which we are antecedently sympathetic. Like any metaphysically laden view, it is inevitably somewhat tendentious. But its unity and elegance make it at least deserving of a place on the philosophical map. (shrink)

Timothy Williamson, in various places, has put forward an argument that is supposed to show that denying bivalence is absurd. This paper is an examination of the logical force of this argument, which is found wanting.

According to Timothy Williamson's epistemic view, vague predicates have precise extensions, we just don't know where their boundaries lie. It is a central challenge to his view to explain why we would be so ignorant, if precise borderlines were really there. He offers a novel argument to show that our insuperable ignorance ``is just what independently justified epistemic principles would lead one to expect''. This paper carefully formulates and critically examines Williamson's argument. It is shown that the argument (...) does not explain our ignorance, and is not really apt for doing so. Williamson's unjustified commitment to a controversial and crucial assumption is noted. It is also argued in three different ways that his argument is, in any case, self-defeating – the same principles that drive the argument can be applied to undermine one of its premises. Along the way, Williamson's unstated commitment to a number of other controversial doctrines comes to light. (shrink)

Minimalists about truth say that the important properties of the truth predicate are revealed in the class of T -biconditionals. Most minimalists demur from taking all of the T -biconditionals of the form “ p is true if and only if p”, to be true, because to do so leads to paradox. But exactly which biconditionals turn out to be true? I take a leaf out of the epistemic account of vagueness to show how the minimalist can avoid giving a (...) comprehensive answer to that question. I also show that this response is entailed by taking minimalism seriously, and that objections to this position may be usefully aided and abetted by Gupta and Belnap’s revision theory of truth. (shrink)

This paper considers the connections between semantic shiftiness (plasticity), epistemic safety and an epistemic theory of vagueness as presented and defended by Williamson (1996a, b, 1997a, b). Williamson explains ignorance of the precise intension of vague words as rooted in insensitivity to semantic shifts: one’s inability to detect small shifts in intension for a vague word results in a lack of knowledge of the word’s intension. Williamson’s explanation, however, falls short of accounting for ignorance of intension.

No one can know the threshold of a vague predicate. For instance, given that ten is the last small number, no one can know that ten fits the description `the last small number’.1 Knowledge that ten is the last small number requires knowledge of the weaker proposition that ten is a small number. Knowledge of thresholds entails knowledge at thresholds. Timothy Williamson ingeniously argues that this weaker knowledge is always impossible. On that basis he claims to have explained our ignorance (...) of the thresholds of vague predicates. (shrink)

Rorty goes on to connect the sorites paradox to analytic philosophy’s long standing concern with the correspondence theory of truth. How do our words hook up with reality? Do our categories map pre-existing contours? The nominalist answers that “facts” are just projections of our forms of speech. Rorty characterizes epistemicism as a hyper-realist backlash. In addition to thinking that our scientific terminology cuts nature at the joint, the epistemicist asserts that even the vague vocabulary of common sense has sharp thresholds.

Epistemicists say there is a last positive instance in a sorites sequence-we just cannot know which is the last. Timothy Williamson explains that knowledge requires a margin for error and this ensures that the last heap will not be knowable as a heap. However, there is a class of disjunctive predicates for which knowledge at the thresholds is possible. They generate sorites paradoxes that cannot be diagnosed with the margin for error principle.

Roy Sorenson offers a unique exploration of an ancient problem: vagueness. Did Buddha become a fat man in one second? Is there a tallest short giraffe? According to Sorenson's epistemicist approach, the answers are yes! Although vagueness abounds in the way the world is divided, Sorenson argues that the divisions are sharp; yet we often do not know where they are. Written in Sorenson'e usual inventive and amusing style, this book offers original insight on language and logic, the way world (...) is, and our understanding of it. (shrink)

Poindexter points and asserts `That is Clinton''. But it is vague as to whether he pointed at Clinton or pointed at the more salient man, Gore. Since the vagueness only occurs at the level of reference fixing, the content of the identity proposition is precise. Indeed, it is either a necessary truth or a necessary falsehood. Since Poindexter''s utterance has a hidden truth value by virtue of vagueness, it increases the plausibility of epistemicism. Epistemicism says that vague statements have hidden (...) truth values. If a precise statement can have a hidden truth value conferred indirectly by vaguesness, then a vague statement can have a hidden truth value directly by its own vagueness. (shrink)

Vagueness theorists tend to think that evolutionary theory dissolves the riddle "Which came first, the chicken or the egg?". After all, 'chicken' is vague. The idea is that Charles Darwin demonstrated that the chicken was preceded by borderline chickens and so it is simply indeterminate as to where the pre-chickens end and the chickens begin.

Sorensen here offers a unified solution to a large family of philosophical puzzles and paradoxes through a study of "blindspots": consistent propositions that cannot be rationally accepted by certain individuals even though they might by true.

Let us begin with a word about what our topic is not. There is a familiar kind of argument for an epistemic view of vagueness in which one claims that denying bivalence introduces logical puzzles and complications that are not easily overcome. One then points out that, by ‘going epistemic’, one can preserve bivalence—and thus evade the complications. James Cargile presented an early version of this kind of argument [Cargile 1969], and Tim Williamson seemingly makes a similar point in his (...) paper, ‘Vagueness and Ignorance’, when he says that ‘classical logic and semantics are vastly superior to . . . alternatives in simplicity, power, past success, and integration with theories in other domains’, and contends that this provides some grounds for not treating vagueness in this way [Williamson 1996: 279].2 Obviously an argument of this kind invites a rejoinder about the puzzles and complications that the epistemic view introduces. Here are two quick examples. First, postulating, as the epistemicist does, linguistic facts no speaker of the language could possibly know, and which have no causal link to actual or possible speech behaviour, is accompanied by a litany of disadvantages—as the reader can imagine. Second, since Williamson’s preferred explanation of our failure to know the exact boundary of vague predicates is precisely that speakers don’t know the relevant linguistic facts—and this because they might be changing constantly, in subtle ways—we risk preserving classical logic at the cost of giving up its usefulness. If this epistemic view is correct, then for all we know, every supposedly valid argument might involve a fallacy of equivocation between what the words in the premises mean and, milliseconds later, what the (homophonous but not known-to-be synonymous) words in the conclusion mean. It looks, then, like there will be complexities whichever way one goes. In which case, whether this familiar kind of argument ends up being compelling depends very much on detailed issues concerning which complications are introduced by which logics for vagueness.. (shrink)

Derek Parfit’s combined-spectrum argument seems to conflict with epistemicism, a viable theory of vagueness. While Parfit argues for the indeterminacy of personhood, epistemicism denies indeterminacy. But, we argue, the linguistically based determinacy that epistemicism supports lacks the sort of normative or ontological significance that concerns Parfit. Thus, we reformulate his argument to make it consistent with epistemicism. We also dispute Roy Sorensen’s suggestion that Parfit’s argument relies on an assumption that fuels resistance to epistemicism, namely, that “the magnitude of a (...) modification must be proportional to its effect.”. (shrink)

John Burgess has recently argued that Timothy Williamson’s attempts to avoid the objection that his theory of vagueness is based on an untenable metaphysics of content are unsuccessful. Burgess’s arguments are important, and largely correct, but there is a mistake in the discussion of one of the key examples. In this note I provide some alternative examples and use them to repair the mistaken section of the argument.

Book Information Vagueness and Contradiction. Vagueness and Contradiction Roy Sorensen Oxford Clarendon Press 2001 208 £25 By Roy Sorensen. Clarendon Press. Oxford. Pp. 208. £25.

In their paper “Vagueness, Ignorance, and Margins for Error” Kenton Machina and Harry Deutsch criticize the epistemic theory of vagueness. This paper answers their objections. The main issues discussed are: the relation between meaning and use; the principle of bivalence; the ontology of vaguely specified classes; the proper form of margin for error principles; iterations of epistemic operators and semantic compositionality; the relation or lack of it between quantum mechanics and theories of vagueness.

According to the principle of bivalence, truth and falsity are jointly exhaustive and mutually exclusive options for a statement. It is either true or false, and not both, even in a borderline case. That highly controversial claim is central to the epistemic theory of vagueness, which holds that borderline cases are distinguished by a special kind of obstacle to knowing the truth-value of the statement. But this paper is not a defence of the epistemic theory. If bivalence holds, it presumably (...) does so as a consequence of what truth and falsity separately are. One may therefore expect bivalence to be derivable from a combination of some principles characterizing truth and other principles characterizing falsity. Indeed, such derivations are easily found. Their form will of course depend on the initial characterizations of truth and falsity, and not all such characterizations will permit bivalence to be derived. In this paper we focus on the relation between its derivability and some principles about truth and falsity. We will use borderline cases for vague expressions as primary examples of an urgent.. (shrink)

On the epistemic view of vagueness, a vague expression has sharp boundaries whose location speakers of the language cannot recognize. The paper argues that one of the deepest sources of resistance to the epistemic view is the idea that all truths are cognitively accessible from truths in a language for natural science, conceived as precise, in a sense explained. The implications of the epistemic view for issues about the relations between vague predicates and scientific predicates are investigated.

Vagueness provides the first comprehensive examination of a topic of increasing importance in metaphysics and the philosophy of logic and language. Timothy Williamson traces the history of this philosophical problem from discussions of the heap paradox in classical Greece to modern formal approaches such as fuzzy logic. He illustrates the problems with views which have taken the position that standard logic and formal semantics do not apply to vague language, and defends the controversial realistic view that vagueness is a kind (...) of ignorance--that there really is a grain of sand whose removal turns a heap into a non-heap, but we cannot know which one it is. (shrink)