In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, (...) which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases. (shrink)

I propose that the meanings of vague expressions render the truth conditions of utterances of sentences containing them sensitive to our interests. For example, 'expensive' is analyzed as meaning 'costs a lot', which in turn is analyzed as meaning 'costs significantly greater than the norm'. Whether a difference is a significant difference depends on what our interests are. Appeal to the proposal is shown to provide an attractive resolution of the sorites paradox that is compatible with classical logic and semantics.

The Rachels–Temkin spectrum arguments against the transitivity of better than involve good or bad experiences, lives, or outcomes that vary along multiple dimensions—e.g., duration and intensity of pleasure or pain. This paper presents variations on these arguments involving combinations of good and bad experiences, which have even more radical implications than the violation of transitivity. These variations force opponents of transitivity to conclude that something good is worse than something that isn’t good, on pain of rejecting the good altogether. That (...) is impossible, so we must reject the spectrum arguments. (shrink)

According to what I will call a contextualist solution to the sorites paradox, vague terms are context-sensitive, and one can give a convincing dissolution of the sorites paradox in terms of this context-dependency. The reason, according to the contextualist, that precise boundaries for expressions like “heap” or “tall for a basketball player” are so difficult to detect is that when two entities are sufficiently similar (or saliently similar), we tend to shift the interpretation of the vague expression so that if (...) one counts as falling in the extension of the property expressed by that expression, so does the other. As a consequence, when we look for the boundary of the extension of a vague expression in its penumbra, our very looking has the effect of changing the interpretation of the vague expression so that the boundary is not where we are looking. This accounts for the persuasive force of sorites arguments. (shrink)

We derive a probabilistic account of the vagueness and context-sensitivity of scalar adjectives from a Bayesian approach to communication and interpretation. We describe an iterated-reasoning architecture for pragmatic interpretation and illustrate it with a simple scalar implicature example. We then show how to enrich the apparatus to handle pragmatic reasoning about the values of free variables, explore its predictions about the interpretation of scalar adjectives, and show how this model implements Edgington’s Vagueness: a reader, 1997) account of the sorites paradox, (...) with variations. The Bayesian approach has a number of explanatory virtues: in particular, it does not require any special-purpose machinery for handling vagueness, and it is integrated with a promising new approach to pragmatics and other areas of cognitive science. (shrink)

The goal of this paper is a comprehensive analysis of basic reasoning patterns that are characteristic of vague predicates. The analysis leads to rigorous reconstructions of the phenomena within formal systems. Two basic features are dealt with. One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-increment of a walking distance is a walking distance). The other is the existence of borderline cases. The paper shows why these should be treated as different, though (...) related phenomena. Tolerance is formally reconstructed within a proposed framework of contextual logic, leading to a solution of the Sorites paradox. Borderline-vagueness is reconstructed using certain modality operators; the set-up provides an analysis of higher order vagueness and a derivation of scales of degrees for the property in question. (shrink)

This paper introduces a novel theory of vagueness. Its main aim is to show how naïve judgments about tolerance and indeterminacy can be preserved while departing from classical logic only in ways which are independently motivated. -/- The theory makes use of a bilateral approach to acceptance and rejection. Combined with a standard account of validity, this approach gives rise to an entailment relation which is non-transitive. I argue that this is desirable: it is both pre-theoretically plausible and provides a (...) compelling solution to a number of longstanding puzzles. The theory aims to preserve the naïve picture of vagueness. It combines tools from expressivist and dynamic treatments of information in conversation to show how principles which are generally assumed to be in tension are compatible. The resulting logic departs from classical logic in precisely those places we should expect it to. -/- . (shrink)

In this paper I offer a critique of the recent popular strategy of giving a contextualist account of vagueness. Such accounts maintain that truth-values of vague sentences can change with changes of context induced by confronting different entities (e.g. different pairs through a sorites series). I claim that appealing to context does not help in solving the sorites paradox, nor does it give us new insights into vagueness per se. Furthermore, the contextual variation to which the contextualist is committed is (...) problematic in various ways. For example, it yields the consequence that much of our everyday (non-soritical) reasoning is fallacious, and it renders us ignorant of what we and others have said. (shrink)

The sorites paradox is the name given to a class of paradoxical arguments, also known as little by little arguments, which arise as a result of the indeterminacy surrounding limits of application of the predicates involved. For example, the concept of a heap appears to lack sharp boundaries and, as a consequence of the subsequent indeterminacy surrounding the extension of the predicate ‘is a heap’, no one grain of wheat can be identified as making the difference between being a heap (...) and not being a heap. Given then that one grain of wheat does not make a heap, it would seem to follow that two do not, thus three do not, and so on. In the end it would appear that no amount of wheat can make a heap. We are faced with paradox since from apparently true premises by seemingly uncontroversial reasoning we arrive at an apparently false conclusion. (shrink)

The term “vagueness” describes a property of natural concepts, which normally have fuzzy boundaries, admit borderline cases, and are susceptible to Zeno's sorites paradox. We will discuss the psychology of vagueness, especially experiments investigating the judgment of borderline cases and contradictions. In the theoretical part, we will propose a probabilistic model that describes the quantitative characteristics of the experimental finding and extends Alxatib's and Pelletier's () theoretical analysis. The model is based on a Hopfield network for predicting truth values. Powerful (...) as this classical perspective is, we show that it falls short of providing an adequate coverage of the relevant empirical results. In the final part, we will argue that a substantial modification of the analysis put forward by Alxatib and Pelletier and its probabilistic pendant is needed. The proposed modification replaces the standard notion of probabilities by quantum probabilities. The crucial phenomenon of borderline contradictions can be explained then as a quantum interference phenomenon. (shrink)

This paper presents and defends a definition of vagueness, compares it favourably with alternative definitions, and draws out some consequences of accepting this definition for the project of offering a substantive theory of vagueness. The definition is roughly this: a predicate 'F' is vague just in case for any objects a and b, if a and b are very close in respects relevant to the possession of F, then 'Fa' and 'Fb' are very close in respect of truth. The definition (...) is extended to cover vagueness of many-place predicates, of properties and relations, and of objects. Some of the most important advantages of the definition are that it captures the intuitions which motivate the thought that vague predicates are tolerant, without leading to contradiction, and that it yields a clear understanding of the relationships between higher-order vagueness, sorites susceptibility, blurred boundaries, and borderline cases. The most notable consequence of the definition is that the correct theory of vagueness must countenance degrees of truth. (shrink)

This article bears on four topics: observational predicates and phenomenal properties, vagueness, strict finitism as a philosophy of mathematics, and the analysis of feasible computability. It is argued that reactions to strict finitism point towards a semantics for vague predicates in the form of nonstandard models of weak arithmetical theories of the sort originally introduced to characterize the notion of feasibility as understood in computational complexity theory. The approach described eschews the use of nonclassical logic and related devices like degrees (...) of truth or supervaluation. Like epistemic approaches to vagueness, it may thus be smoothly integrated with the use of classical model theory as widely employed in natural language semantics. But unlike epistemicism, the described approach fails to imply either the existence of sharp boundaries or the failure of tolerance for soritical predicates. Applications ofmeasurement theory(in the sense of Krantz, Luce, Suppes, & Tversky (1971)) to vagueness in the nonstandard setting are also explored. (shrink)

This paper proposes an experimental investigation of the use of vague predicates in dynamic sorites. We present the results of two studies in which subjects had to categorize colored squares at the borderline between two color categories (Green vs. Blue, Yellow vs. Orange). Our main aim was to probe for hysteresis in the ordered transitions between the respective colors, namely for the longer persistence of the initial category. Our main finding is a reverse phenomenon of enhanced contrast (i.e. negative hysteresis), (...) present in two different tasks, a comparative task involving two color names, and a yes/no task involving a single color name, but not found in a corresponding color matching task. We propose an optimality-theoretic explanation of this effect in terms of the strict-tolerant framework of Cobreros et al. (J Philos Log 1–39, 2012), in which borderline cases are characterized in a dual manner in terms of overlap between tolerant extensions, and underlap between strict extensions. (shrink)

This paper examines the mind and language of an omniscient being from a supervaluationist perspective. Two questions hall receive special attention. How ought the supervaluationist explicate the concept of omniscience? And what ought the supervaluationist expect an omniscient speaker to say about a Sorites series?

Draft (Version 1.1, October 2007): (PDF file) A reply to Jason Stanley’s Analysis criticism of my interest-relative view on vagueness.

Contextualism about vagueness (hereafter ‘Contextualism’) is the view that vagueness consists in a particular species of context-sensitivity and that properly accommodating this fact into our semantic theory will yield a plausible solution to the sorites paradox.[1],[2] But Contextualism, as many commentators have noted, faces the following immediate objection: if we hold the context fixed, vagueness still remains, therefore vagueness is not a species of context-sensitivity. Call this ‘the simple objection’.[3] Absent a convincing reply to the simple objection, Contextualism is in (...) very bad shape. Oddly enough, defenders of Contextualism have said very little in reply. Proponents of the objection have tended to assume that this is because no reply is in the offing—the simple objection is taken to be unassailable. In this short paper, we sketch two replies to the simple objection which result in two very different kinds of Contextualism: Epistemicist Contextualism and Radical Contextualism. With these two theories in hand, the simple objection loses its force. (shrink)

Are moral terms semantically plastic—that is, would very slight changes in our patterns of use have shifted their meanings? This is a delicate question for moral realists. A 'yes' answer seems to conflict with the sorts of intuitions that support realism; but a 'no' answer seems to require a semantics that involves hefty metaphysical commitments. This tension can be illustrated by thinking about how standard accounts of vagueness can be applied to the case of moral terms, and also by considering (...) how realists should respond to the Moral Twin Earth problem. After presenting the puzzle, I will argue that moral realists can accept the semantic plasticity of moral expressions while accounting for contrary intuitions in a way that is nearly cost-free. (shrink)

This book proposes a new solution to the problem of vagueness. There are several different ways of addressing this problem and no clear agreement on which one is correct. The author proposes that it should be understood as the problem of explaining vague predicates in a way that systematizes six intuitions about the phenomenon and satisfies three criteria of adequacy for an ideal theory of vagueness. The third criterion, which is called the “criterion of precisification”, is the most controversial one. (...) It is based on the intuition that a predicate is vague only if it is imprecise. The author considers some different definitions of linguistic imprecision, proposing that a predicate is imprecise if and only if there is no sharp boundary between objects to which its application yields some particular truth-value and objects to which its application does not yield that truth-value. The volume critically reviews the current theories of vagueness and proposes a new one, the Theory of Vagueness as Arbitrariness, which defines a vague predicate as an arbitrary predicate that must be precisified in order to contribute to a sentence that has truth-conditions. The main advantages of this theory over the current alternatives are that it satisfies all three criteria and systematizes the relevant intuitions. (shrink)

The tolerance principle, the idea that vague predicates are insensitive to sufficiently small changes, remains the main bone of contention between theories of vagueness. In this paper I examine three sources behind our ordinary belief in the tolerance principle, to establish whether any of them might give us a good reason to revise classical logic. First, I compare our understanding of tolerance in the case of precise predicates and in the case of vague predicates. While tolerance in the case of (...) precise predicates results from approximation, tolerance in the case of vague predicates appears to originate from two more specific sources: semantic indeterminacy on the one hand, and epistemic indiscriminability on the other. Both give us good and coherent grounds to revise classical logic. Epistemic indiscriminability, it is argued, may be more fundamental than semantic indeterminacy to justify the intuition that vague predicates are tolerant. (shrink)

The principle of tolerance characteristic of vague predicates is sometimes presented as a soft rule, namely as a default which we can use in ordinary reasoning, but which requires care in order to avoid paradoxes. We focus on two ways in which the tolerance principle can be modeled in that spirit, using special consequence relations. The first approach relates tolerant reasoning to nontransitive reasoning; the second relates tolerant reasoning to nonmonotonic reasoning. We compare the two approaches and examine three specific (...) consequence relations in relation to those, which we call: strict-to-tolerant entailment, pragmatic-to-tolerant entailment, and pragmatic-to-pragmatic entailment. The first two are nontransitive, whereas the latter two are nonmonotonic. (shrink)

accounts in general, contrary to what he seems to think. Stanley’s discussion concerns the dynamic or ‘forced march’ version of the sorites, viz. the version framed in terms of the judgments that would be made by a competent speaker who proceeds step by step along a sorites series for a vague predicate ‘F’. According to Stanley, the contextualist treatment of the paradox is based on the idea that the speaker shifts the content of the predicate whenever necessary to make it (...) the case that each successive pair of adjacent items are category-identical – in other words, either both items satisfy ‘F’ or neither does. These adjustments allow the speaker to progress from a clear case for ‘F’ to a clear case for ‘not-F’ without breaking the seeming continuity of the series. (shrink)

I argue for an account of vagueness according to which the root of vagueness lies not in the type of semantic-value that is best associated with an expression, but in the type of linguistic practice that renders the expression meaningful. I suggest, in particular, that conventions about how to use sentences involving attributions of vague predicates to borderline cases prevail to a lesser degree than conventions about how to use sentences involving attributions of vague predicates to clear cases.

Philosophers disagree about whether vagueness requires us to admit truth-value gaps, about whether there is a gap between the objects of which a given vague predicate is true and those of which it is false on an appropriately constructed sorites series for the predicate—a series involving small increments of change in a relevant respect between adjacent elements, but a large increment of change in that respect between the endpoints. There appears, however, to be widespread agreement that there is some sense (...) in which vague predicates are gappy which may be expressed neutrally by saying that on any appropriately constructed sorites series for a given vague predicate there will be a gap between the objects of which the predicate is definitely true and those of which it is definitely false. Taking as primitive the operator ‘it is definitely the case that’, abbreviated as ‘D’, we may stipulate that a predicate F is definitely true (or definitely false) of an object just in case ‘DF (a)’, where a is a name for the object, is true (or false) simpliciter.1 This yields the following conditional formulation of a ‘gap principle’: (DΦ(x) ∧ D¬Φ(y)) → ¬R(x, y). Here ‘Φ’ is to be replaced with a vague predicate, while ‘R’ is to stand for a sorites relation for that predicate: a relation that can be used to construct a sorites series for the predicate—such as the relation of being just one millimetre shorter than for the predicate ‘is tall’. Disagreements about the sense in which it is correct to say that vague predicates are gappy can then be recast as disagreements about how to understand the definitely operator. One might give it, for example, a pragmatic construal such as ‘it would not be misleading to assert that’; or an epistemic construal such as ‘it is known that’ or ‘it is knowable that’; or a semantic construal such as ‘it is true that’. (shrink)

In this paper I argue that there is a significant but often overlooked metaphysical distinction to be made between contextualism and contrastivism. The orthodox view is that contrastivism is merely a form of contextualism. This is a mistake. The contextualist view is incompatible with certain naturalist claims about the metaphysical nature of concepts within whichever domain is being investigated, while the contrastivist view is compatible with these claims. So, choosing one view over the other will involve choosing to affirm or (...) deny a significant metaphysical claim. As such, a demarcation ought to be put in place between contextualism and contrastivism. (shrink)

Stewart Shapiro has objected to the epistemicist theory of vagueness on grounds that it gives counterintuitive predictions about cases involving conditional obligation. This paper details a response on the epistemicist’s behalf. I first argue that Shapiro’s own presentation of the objection is unsuccessful as an argument against epistemicism. I then reconstruct and offer two alternative arguments inspired by Shapiro’s considerations, and argue that these fail too, given the information-sensitive nature of conditional obligations.

This volume analyzes and studies how vagueness occurs and matters as a specific problem in the context of theories that are primarily about something else.

On some accounts of vagueness, predicates like “is a heap” are tolerant. That is, their correct application tolerates sufficiently small changes in the objects to which they are applied. Of course, such views face the sorites paradox, and various solutions have been proposed. One proposed solution involves banning repeated appeals to tolerance, while affirming tolerance in any individual case. In effect, this solution rejects the reasoning of the sorites argument. This paper discusses a thorny problem afflicting this approach to vagueness. (...) In particular, it is shown that, on the foregoing view, whether an object is a heap will sometimes depend on factors extrinsic to that object, such as whether its components came from other heaps. More generally, the paper raises the issue of how to count heaps in a tolerance-friendly framework. (shrink)

Of the many families of words that are thought to be vague, so-called observational predicates may be both the most fascinating and the most confounding. Roughly, observational predicates are terms that apply to objects on the basis of how those objects appear to us perceptually speaking. ‘Red’, ‘loud’, ‘sweet’, ‘acrid’, and ‘smooth’ are good examples. Delia Graff explains that a “predicate is observational just in case its applicability to an object (given a fixed context of evaluation) depends only on the (...) way that object appears” (2001, 3). By the same token observational predicates are, as Crispin Wright observes, terms “whose senses are taught entirely by ostension” (1976). Like other vague predicates, observational words appear to generate sorites paradoxes. Consider for example a series of 20 colored patches progressing from a clearly red one to a clearly orange one, so ordered that each patch is just noticeably different in hue from the one before. The following argument then seems forced upon us: (1) Patch #1 is red. (2) Any patch that differs only slightly in hue from a red patch is itself red. (3) Therefore patch #20 is red. Premise (2) expresses what Wright has called the tolerance of ‘red’: the application of the predicate tolerates small changes in a decisive parameter (here, hue). Of course, most vague predicates, hence most versions of the sorites, are not observational. For instance, given a series of. (shrink)

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it (...) is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy). (shrink)

The author argues that vagueness in law is typically extravagant, in the sense that it is possible for two competent users of the language, who understand the facts of each case, to take such different views that there is not even any overlap between the cases that each disputant would identify as borderline. Extravagant vagueness is a necessary feature of legal systems. Some philosophers of law and philosophers of language claim that bivalence is a property of statements in the domains (...) that concern them (the domain of law in the former case, the whole domain of meaningful discourse in the latter). The author argues that the bivalence claim should be rejected. In philosophy of law, the motivation underlying the bivalence claim is an urge to assert the principle that the law must be capable of standing against arbitrary use of political power. The challenge – once the bivalence claim is rejected – is to articulate the law's opposition to arbitrariness in a way that is compatible with the possibility of indeterminacy in the application of vague laws. (shrink)

Intuitively, vagueness involves some sort of indeterminacy: if Plato is a borderline case of baldness, then there is no fact of the matter about whether or not he’s bald—he’s neither bald nor not bald. The leading formal treatments of such indeterminacy—three valued logic, supervaluationism, etc.—either fail to validate the classical theorems, or require that various classically valid inference rules be restricted. Here we show how a fully classical, yet indeterminist account of vagueness can be given within natural semantics, an alternative (...) semantics for classical proof theory. The key features of the account are: there is a single notion of truth—definite truth—and a single notion of validity; sentences can be true, false, or undetermined; all classical theorems and all classical inference rule are valid; the sorites argument is unsound; ‘definitely’ is treated as a meta-language predicate; higher-order vagueness is handled via semantic ascent. (shrink)

The presentation of the “dual picture of vagueness” in my earlier work is supplemented here with a number of additional considerations. I emphasize how the picture lends itself naturally to treatments of the contribution of a typical degree adjective to propositional content and to truth conditions. A number of reasonable refinements of the picture are presented, especially concerning occasions of use of a degree adjective in which a class containing a sorites series is somehow involved in content fixing, but in (...) such a way that an extension ultimately gets fixed for the predicate. I also develop the observation that standard theories of the logical and semantic paradoxes rely on the idea that the content-fixing mechanisms for the paradoxical expressions work well in general but fail in some localized cases, and that while standard theories of the sorites paradox are clearly not based on this idea at all, the dual theory is fully based on the idea. Next, I offer a few remarks on some recent theories of vagueness and the sorites that are broadly in the spirit of the dual picture, emphasizing the aspects where the latter compares favorably against those recent theories. Finally, the question whether the dual theory is statable with truth according to the theory itself is considered in more detail than in my earlier presentation. (shrink)

As far as ‘modern’ logical theories of vagueness are concerned, a main distinction can be drawn between ‘semantical’ ones and ‘pragmatical’ ones. The latter are defended here, because they tend to retake into account important contextual dimensions of the problem abandoned by the former. Their inchoate condition seems not alarming, since they are of surprisingly recent date. This, however, could very well be an accidental explanation. That is, the true reason for it might sooner or later turn out to be (...) bearing exactly on the fundamental human limitations, when it comes to theorizing, that these approaches are urging us to appreciate. (shrink)

Several philosophers offer explanations of linguistic vagueness by appealing to the referential context-dependence of vague terms. Timothy Williamson argues pre-emptively that any such approach must fail, on the grounds that context-dependence is neither necessary nor sufficient for vagueness. He supports this claim, in turn, by example. This paper argues that his examples fail to show that context-dependence is either unnecessary or insufficient for vagueness, and hence that he has failed by his own lights to show that it cannot explain vagueness.

One feature of vague predicates is that, as far as appearances go, they lack sharp application boundaries. I argue that we would not be able to locate boundaries even if vague predicates had sharp boundaries. I do so by developing an idealized cognitive model of a categorization faculty which has mobile and dynamic sortals (`classes', `concepts' or `categories') and formally prove that the degree of precision with which boundaries of such sortals can be located is inversely constrained by their flexibility. (...) Given the literature, it is plausible that we are appropriately like the model. Hence, an inability to locate sharp boundaries is not necessarily because there are none; boundaries could be sharp and it is plausible that we would nevertheless be unable to locate them. (shrink)

Most words in natural language are vague, that is to say, they lack sharp boundaries and, hence, they have (actual or potential) borderline cases, where the word in question neither definitely applies nor definitely fails to apply. Vagueness gives rise to paradoxes, the best known of which is the sorites (concerned with how many grains of sand are needed to make a heap). Besides offering a solution to such paradoxes, a theory of vagueness should systematically describe how the truth conditions (...) of sentences with vague terms are determined; and it should also define the right logical principles for reasoning with such sentences. This article offers an introduction to the main theories of vagueness and to the problems they have to face. (shrink)

In natural language we rarely use relation-words with more than three argument places. This paper studies one systematic device, rooted in natural language, by which relations of greater adicity can be expressed. It is based on a higher-order relation between 1-place, 2-place, and 4-place relations (and so on) of which the relation between the positive and comparative degrees of a predicate is a special case. Two formal languages are presented in this connection, one of which represents the language of communication (...) and the other the contextual information against which the first language is interpreted. A semantical theory is described, which treats the two languages in an interdependent way. Logical consequence is non-compact. Connections with issues about vagueness are made. (shrink)

Colin Howson argues that (1) my sociologistic reliabilism sheds no light on the objectivity of epistemic content, and that (2) sorites does not threaten the reliability of modus ponens . I reply that argument (1) misrepresents my position, and that argument (2) is beside the point.

In the literature on vagueness one finds two very different kinds of degree theory. The dominant kind of account of gradable adjectives in formal semantics and linguistics is built on an underlying framework involving bivalence and classical logic: its degrees are not degrees of truth. On the other hand, fuzzy logic based theories of vagueness—largely absent from the formal semantics literature but playing a significant role in both the philosophical literature on vagueness and in the contemporary logic literature—are logically nonclassical (...) and give a central role to the idea of degrees of truth. Each kind of degree theory has a strength: the classical kind allows for rich and subtle analyses of the comparative form of gradable adjectives and of various types of gradable precise adjectives, while the fuzzy kind yields a compelling solution to the sorites paradox. This paper argues that the fuzzy kind of theory can match the benefits of the classical kind and hence that the burden is on the latter to match the advantages of the former. In particular, we develop a new version of the fuzzy logic approach that—unlike existing fuzzy theories—yields a compelling analysis of the comparative as well as an adequate account of gradable precise predicates, while still retaining the advantage of genuinely solving the sorites paradox. (shrink)