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)

When I take off my glasses, the world looks blurred. When I put them back on, it looks sharpedged. I do not think that the world really was blurred; I know that what changed was my relation to the distant physical objects ahead, not those objects themselves. I am more inclined to believe that the world really is and was sharp-edged. Is that belief any more reasonable than the belief that the world really is and was blurred? I see more (...) accurately with my glasses on than off, so visual appearances when they are on have some cognitive priority over visual appearances when they are off. If I must choose which kind of visual appearance to take at face value, I will choose the sharp-edged look. But what should I think when I see a mist, which looks very blurred however well I am seeing? Indeed, why choose to take any of the looks at face value? Why not regard all the choices as illegitimate projections of ways of seeing the world onto the world itself? (shrink)

This paper deals with the truth-Conditions and the logic for vague languages. The use of supervaluations and of classical logic is defended; and other approaches are criticized. The truth-Conditions are extended to a language that contains a definitely-Operator and that is subject to higher order vagueness.

The conceptual spaces approach has recently emerged as a novel account of concepts. Its guiding idea is that concepts can be represented geometrically, by means of metrical spaces. While it is generally recognized that many of our concepts are vague, the question of how to model vagueness in the conceptual spaces approach has not been addressed so far, even though the answer is far from straightforward. The present paper aims to fill this lacuna.

In VAGUENESS AND DEGREES OF TRUTH, Nicholas Smith develops a new theory of vagueness: fuzzy plurivaluationism. -/- A predicate is said to be VAGUE if there is no sharply defined boundary between the things to which it applies and the things to which it does not apply. For example, 'heavy' is vague in a way that 'weighs over 20 kilograms' is not. A great many predicates -- both in everyday talk, and in a wide array of theoretical vocabularies, (...) from law to psychology to engineering -- are vague. -/- Smith argues, based on a detailed account of the defining features of vagueness, that an accurate theory of vagueness must involve the idea that truth comes in degrees. The core idea of degrees of truth is that while some sentences are true and some are false, others possess intermediate truth values: they are truer than the false sentences, but not as true as the true ones. Degree-theoretic treatments of vagueness have been proposed in the past, but all have encountered significant objections. In light of these, Smith develops a new type of degree theory. Its innovations include a definition of logical consequence that allows the derivation of a classical consequence relation from the degree-theoretic semantics, a unified account of degrees of belief and their relationships with degrees of truth and subjective probabilities, and the incorporation of semantic indeterminacy -- the view that vague statements need not have unique meanings -- into the degree-theoretic framework. -/- As well as being essential reading for those working on vagueness, Smith's book provides an excellent entry-point for newcomers to the area -- both from elsewhere in philosophy, and from computer science, logic and engineering. It contains a thorough introduction to existing theories of vagueness and to the requisite logical background. (shrink)

Stewart Shapiro's ambition in Vagueness in Context is to develop a comprehensive account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary according to their context: a person can be tall with respect to male accountants and not tall (even short) with respect to professional basketball players. The key feature of Shapiro's account is that the extensions of vague (...) terms also vary in the course of conversations and that, in some cases, a competent speaker can go either way without sinning against the meaning of the words or the non-linguistic facts. As Shapiro sees it, vagueness is a linguistic phenomenon, due to the kinds of languages that humans speak; but vagueness is also due to the world we find ourselves in, as we try to communicate features of it to each other. (shrink)

The aim of this essay is to argue that, if a robust form of moral realism is true, then moral vagueness is ontic vagueness. The argument is by elimination: I show that neither semantic nor epistemic approaches to moral vagueness are satisfactory.

This paper investigates the way that linguistic expressions influence vagueness, focusing on the interpretation of the positive (unmarked) form of gradable adjectives. I begin by developing a semantic analysis of the positive form of ‘relative’ gradable adjectives, expanding on previous proposals by further motivating a semantic basis for vagueness and by precisely identifying and characterizing the division of labor between the compositional and contextual aspects of its interpretation. I then introduce a challenge to the analysis from the class (...) of ‘absolute’ gradable adjectives: adjectives that are demonstrably gradable, but which have positive forms that relate objects to maximal or minimal degrees, and do not give rise to vagueness. I argue that the truth conditional difference between relative and absolute adjectives in the positive form stems from the interaction of lexical semantic properties of gradable adjectives—the structure of the scales they use—and a general constraint on interpretive economy that requires truth conditions to be computed on the basis of conventional meaning to the extent possible, allowing for context dependent truth conditions only as a last resort. (shrink)

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)

How does vagueness interact with metaphysical modality and with restrictions of it, such as nomological modality? In particular, how do definiteness, necessity (understood as restricted in some way or not), and actuality interact? This paper proposes a model-theoretic framework for investigating the logic and semantics of that interaction. The framework is put forward in an ecumenical spirit: it is intended to be applicable to all theories of vagueness that express vagueness using a definiteness (or: determinacy) operator. We (...) will show how epistemicists, supervaluationists, and theorists of metaphysical vagueness like Barnes and Williams (2010) can interpret the framework.  We will also present a complete axiomatization of the logic we recommend to both epistemicists and local supervaluationists. . (shrink)

Vagueness is the study of concepts that admit borderline cases. The epistemology of vagueness concerns attitudes we should have towards propositions we know to be borderline. On this basis Andrew Bacon develops a new theory of vagueness in which vagueness is fundamentally a property of propositions, explicated in terms of its role in thought.

Vagueness is currently the subject of vigorous debate in the philosophy of logic and language. Vague terms -- such as 'tall', 'red', 'bald', and 'tadpole' -- have borderline cases ; and they lack well-defined extensions. The phenomenon of vagueness poses a fundamental challenge to classical logic and semantics, which assumes that propositions are either true or false and that extensions are determinate.This anthology collects for the first time the most important papers in the field. After a substantial introduction (...) that surveys the field, the essays form four groups, starting with some historically notable pieces. The 1970s saw an explosion of interest in vagueness, and the second group of essays reprints classic papers from this period. The following group of papers represent the best recent work on the logic and semantics of vagueness. The essays in the final group are contributions to the continuing debate about vague objects and vague identity. (shrink)

You are morally permitted to save your friend at the expense of a few strangers, but not at the expense of very many. However, there seems no number of strangers that marks a precise upper bound here. Consequently, there are borderline cases of groups at the expense of which you are permitted to save your friend. This essay discusses the question of what explains ethical vagueness like this, arguing that there are interesting metaethical consequences of various explanations.

Words, languages, symphonies, fictional characters, games, and recipes are plausibly abstract artifacts— entities that have no spatial location and that are deliberately brought into existence as a result of creative acts. Many accept that composition is unrestricted: for every plurality of material objects, there is a material object that is the sum of those objects. These two views may seem entirely unrelated. I will argue that the most influential argument against restricted composition—the vagueness argument—doubles as an argument that there (...) can be no abstract artifacts. There is no way to resist the vagueness argument against abstract artifacts that does not also undermine the vagueness argument against restricted composition. (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)

Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professionalbasketball players. The main feature (...) of Shapiro's account is that the extensions of vague terms also vary in the course of a conversation, even after the external contextual features, such as the comparison class, are fixed. A central thesis is that in some cases, a competent speaker ofthe language can go either way in the borderline area of a vague predicate without sinning against the meaning of the words and the non-linguistic facts. Shapiro calls this open texture, borrowing the term from Friedrich Waismann.The formal model theory has a similar structure to the supervaluationist approach, employing the notion of a sharpening of a base interpretation. In line with the philosophical account, however, the notion of super-truth does not play a central role in the development of validity. The ultimate goal of the technical aspects of the work is to delimit a plausible notion of logical consequence, and to explore what happens with the sorites paradox.Later chapters deal with what passes for higher-order vagueness - vagueness in the notions of 'determinacy' and 'borderline' - and with vague singular terms, or objects. In each case, the philosophical picture is developed by extending and modifying the original account. This is followed with modifications to the model theory and the central meta-theorems.As Shapiro sees it, vagueness is a linguistic phenomenon, due to the kinds of languages that humans speak. But vagueness is also due to the world we find ourselves in, as we try to communicate features of it to each other. Vagueness is also due to the kinds of beings we are. There is no need to blame the phenomenon on any one of those aspects. (shrink)

Elizabeth Barnes and Robert Williams have developed a theory of metaphysical indeterminacy, via which they defend the theoretical legitimacy of vague objects. In this paper, we argue that while the Barnes–Williams theory supplies a viable account of genuine metaphysical vagueness, it cannot underwrite an account of genuinely vague objects. First we clarify the distinction between these two key theses. Then we argue that the Barnes–Williams theory of metaphysical vagueness not only fails to deliver genuinely vague objects, it in (...) fact provides grounds for rejecting them. (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)

Vague expressions are omnipresent in natural language. As such, their use in legal texts is virtually inevitable. If a law contains vague terms, the question whether it applies to a particular case often lacks a clear answer. One of the fundamental pillars of the rule of law is legal certainty. The determinacy of the law enables people to use it as a guide and places judges in the position to decide impartially. Vagueness poses a threat to these ideals. In (...) borderline cases, the law seems to be indeterminate and thus incapable of serving its core rule of law value. -/- In the philosophy of language, vagueness has become one of the hottest topics of the last two decades. Linguists and philosophers have investigated what distinguishes "soritical " vagueness from other kinds of linguistic indeterminacy, such as ambiguity, generality, open texture, and family resemblance concepts. There is a vast literature that discusses the logical, semantic, pragmatic, and epistemic aspects of these phenomena. Legal theory has hitherto paid little attention to the differences between the various kinds of linguistic indeterminacy that are grouped under the heading of "vagueness ", let alone to the various theories that try to account for these phenomena. -/- The paper is an introduction to a book of the same title. Bringing together leading scholars working on the topic of vagueness in philosophy and in law, the book fosters a dialogue between philosophers and legal scholars by examining how philosophers conceive legal ambiguity from their theoretical perspective and how legal theorists make use of philosophical theories of vagueness. (shrink)

In this paper I explore the implications of moral vagueness (viz., the vagueness of moral predicates) for non-naturalist metaethical theories like those recently championed by Shafer-Landau, Parfit, and others. I characterise non-naturalism in terms of its commitment to 7 theses: Cognitivism, Correspondence, Atomism, Objectivism, Supervenience, Non-reductivism, and Rationalism. I start by offering a number of reasons for thinking that moral predicates are vague in the same way in which ‘red’, ‘tall’, and ‘heap’ are said to be. I then (...) argue that the moral non-naturalist seeking to countenance moral vagueness faces a dilemma: are moral properties vague, or perfectly sharp? On either horn of the dilemma, serious problems arise for some of the central tenets of non-naturalism: vague properties seem to threaten Objectivism, Supervenience, and Non-reductivism; on the other hand, sharp properties raise problems for Supervenience and Rationalism. The difficulties on each horn of the dilemma are real, and while they may not be insuperable, they do, at the very least, drastically limit the things non-naturalists can consistently say about moral properties, facts, and reasons. (shrink)

Vagueness in law can lead to indeterminacies in legal rights and obligations. This book responds to the challenges that those indeterminacies pose to theories of law and adjudication.

This article discusses two arguments in favor of perdurance. The first is Sider’s argument from vagueness, “one of the most powerful” in favor of perdurantism. I make the observation that endurantists have principled grounds to claim that the argument is unsound, at least if endurance is formulated in locative rather than mereological terms. Having made this observation, I use it to emphasize a somewhat neglected difference between endurantists and perdurantists with respect to their views on material objects. These views, (...) in the case of endurantists, lead to a further, less than conclusive but nevertheless interesting argument against endurantism—the anti-fundamentality argument—which I discuss and tentatively endorse. That argument posits that endurantists must take location to be a fundamental relation, and that this has as a consequence the metaphysical possibility of some rather unwelcome scenarios. Perdurantists may avoid this consequence by denying that location is fundamental, perhaps by embracing supersubstantivalism. (shrink)

Book synopsis: Vagueness is currently the subject of vigorous debate in the philosophy of logic and language. Vague terms-such as "tall", "red", "bald", and "tadpole"—have borderline cases ; and they lack well-defined extensions. The phenomenon of vagueness poses a fundamental challenge to classical logic and semantics, which assumes that propositions are either true or false and that extensions are determinate. Another striking problem to which vagueness gives rise is the sorites paradox. If you remove one grain from (...) a heap of sand, surely you must be left with a heap. Yet apply this principle repeatedly as you remove grains one by one, and you end up, absurdly, with a solitary grain that counts as a heap. This anthology collects papers in the field. After an introduction that surveys the field, the essays form four groups, starting with some historically notable pieces. The 1970s saw an explosion of interest in vagueness, and the second group of essays reprints classic papers from this period. The following group of papers represent current work on the logic and semantics of vagueness. The essays in the final group are contributions to the continuing debate about vague objects and vague identity. (shrink)

It is natural to think of precise probabilities as being special cases of imprecise probabilities, the special case being when one’s lower and upper probabilities are equal. I argue, however, that it is better to think of the two models as representing two different aspects of our credences, which are often vague to some degree. I show that by combining the two models into one model, and understanding that model as a model of vague credence, a natural interpretation arises that (...) suggests a hypothesis concerning how we can improve the accuracy of aggregate credences. I present empirical results in support of this hypothesis. I also discuss how this modeling interpretation of imprecise probabilities bears upon a philosophical objection that has been raised against them, the so-called inductive learning problem. (shrink)

The paper is divided into two parts, each with subsections. In the first part, I shall discuss some matters that have been extensively examined by Kim, namely what the basic types of supervenience are and how they are pairwise logically related; in the course of this discussion, I shall distinguish a weak from a strong notion of global supervenience. In the second part, I shall examine supervenience in a context in which Kim has not: I shall attempt to solve a (...) puzzle that arises when we consider supervenience relations involving vague properties and/or predicates. This is, of course, not a special case of property/predicate supervenience, but the typical one: for virtually all properties/predicates are vague. The two parts of the paper stand independently of each other. Of the discussion in the first part, essentially all that is presupposed in the second is the notion of “world strong supervenience” defined in section 1 of part I. (shrink)

Abstract: I argue that practically all vague language is context-sensitive in a covert and unfamiliar way. I first outline a novel puzzle concerning the interaction of conditionals and vagueness. I then argue that the best way of resolving the puzzle is through positing context-sensitive penumbral connections between sundry parts of language. I argue that these penumbral connections shift through a distinct form of Lewisian accommodation. The upshot is that meaning is a far shiftier thing than has typically been thought.

Vague existence can seem like the worst kind of vagueness in the world, or seem to be an entirely unintelligible notion. This bad reputation is based upon the rumour that if there is vague existence then there are non-existent objects. But the rumour is false: the modest brand of vague existence entailed by certain metaphysical theories of composition does not deserve its bad reputation.

Some comparisons are hard. How should we think about such comparisons? According to John Broome, we should think about them in terms of vagueness. But the vagueness account has remained unpopular thus far. Here I try to bolster it by clarifying the notion of comparative vagueness that lies at its heart.

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)

In his note "can there be vague objects?" ("analysis", 1978), Gareth evans presents a purported proof that there can be no vague identity statements. Some readers think that evans endorses the proof and its false conclusion. Not so. His point is that those who put vagueness in the world, Rather than in language, Will have no way to fault the proof and no way to escape the false conclusion.

John MacFarlane has formulated a version of truth-relativism, and argued for its application in some cases – future contingents, knowledge attributions and epistemic modals among them. Mark Richard also defends a version of relativism, which he applies to vagueness-inducing features of the semantics of gradable adjectives. On MacFarlane’s and Richard’s characterization, truth-relativist claims posit a distinctive kind of context-dependence, the dependence of the evaluation of an assertion as true or otherwise on aspects of the context of the evaluation itself (...) – in contrast with the context of the assertion. The paper follows Evans in distinguishing two forms of truth-relativism, a moderate one concerning the evaluation of contents or propositions, an a radical one concerning the evaluation of acts; it argues against Richard’s truth-relativist proposals for gradable adjective, which is understood to be of the second kind, while accepting a form of moderate content-relativism for those cases. (shrink)

Ted Sider has famously argued that existence, in the unrestricted sense of ontology, cannot be vague, as long as vagueness is modeled by means of precisifications. The first section of Chapter 9 exposes some controversial assumptions underlying Sider’s alleged reductio of vague existence. The upshot of the discussion is that, although existence cannot be vague, it can be super-vague, i.e. higher-order vague, for all orders. The second section develops and defends a novel framework, dubbed negative supervaluationary semantics, which makes (...) room for the possibility of super-vague existence. (shrink)

We say that a sentence A is a permissive consequence of a set X of premises whenever, if all the premises of X hold up to some standard, then A holds to some weaker standard. In this paper, we focus on a three-valued version of this notion, which we call strict-to-tolerant consequence, and discuss its fruitfulness toward a unified treatment of the paradoxes of vagueness and self-referential truth. For vagueness, st-consequence supports the principle of tolerance; for truth, it (...) supports the requisite of transparency. Permissive consequence is non-transitive, however, but this feature is argued to be an essential component to the understanding of paradoxical reasoning in cases involving vagueness or self-reference. (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 integration of normative analysis with empirical data often remains unclear despite the availability of many empirical bioethics methodologies. This paper sought bioethics scholars’ experiences and reflections of doing empirical bioethics research to feed these practical insights into the debate on methods. We interviewed twenty-six participants who revealed their process of integrating the normative and the empirical. From the analysis of the data, we first used the themes to identify the methodological content. That is, we show participants’ use of familiar (...) methods explained as “back-and-forth” methods (reflective equilibrium), followed by dialogical methods where collaboration was seen as a better way of doing integration. Thereafter, we highlight methods that were deemed as inherent integration approaches, where the normative and the empirical were intertwined from the start of the research project. Second, we used the themes to express not only how we interpreted what was said but also how things were said. In this, we describe an air of uncertainty and overall vagueness that surrounded the above methods. We conclude that the indeterminacy of integration methods is a double-edged sword. It allows for flexibility but also risks obscuring a lack of understanding of the theoretical-methodological underpinnings of empirical bioethics research methods. (shrink)

This paper explores the idea that vague predicates like “tall”, “loud” or “expensive” are applied based on a process of analog magnitude representation, whereby magnitudes are represented with noise. I present a probabilistic account of vague judgment, inspired by early remarks from E. Borel on vagueness, and use it to model judgments about borderline cases. The model involves two main components: probabilistic magnitude representation on the one hand, and a notion of subjective criterion. The framework is used to represent (...) judgments of the form “x is clearly tall” versus “x is tall”, as involving a shift of one’s criterion, and then to derive observed patterns of acceptance for borderline contradictions, namely sentences of the form “x is tall and not tall”, relative to the acceptance of their conjuncts. (shrink)

Consider an object or property a and the predicate F. Then a is vague if there are questions of the form: Is a F? that have no yes-or-no answers. In brief, vague properties and kinds have borderline instances and composite objects have borderline constituents. I'll use the expression "borderline cases" as a covering term for both. ;Having borderline cases is compatible with precision so long as every case is either borderline F, determinately F or determinately not F. Thus, in addition (...) to borderline cases, vagueness requires "fuzzy borders". A has fuzzy borders if there is no sharp line between a's borderline cases, a's determinate cases and a's determinate non-cases. To illustrate, loudness is vague because in a continuum of gradually increasing volume, there is no fact about where the last definite loud noise is and where the first borderline noise is. ;Actual borderline cases are not required for vagueness. Suppose actual noises are either ear-shatteringly loud or silent. Even so, loudness is vague since it could have borderline instances. Just so for objects. Suppose a table is cleanly bounded such that there is a sharp break between the table and its surroundings. Still, that table is vague since there are worlds at which it is indeterminate whether some wood molecule a is, or is not, part of that table. ;There are three views about which things can be vague. On the linguistic view, vagueness is solely a feature of expressions. On the nihilistic view, nothing is vague because vagueness is incoherent. Pitted against these views is the ontological conception of vagueness according to which borderline instances and constituents arise due to genuine indeterminacy in nature. ;My dissertation defends the ontological conception. I argue against the adequacy of the theory's rivals, disarm a damaging objection and demonstrate the causal superiority of vague events. Anyone familiar with the literature on vagueness will notice that only a subset of issues are discussed. What I offer is preliminary--I hope only to make the theory of ontological vagueness attractive enough to merit future work. (shrink)

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)