One way in which we might approach the challenge posed by the Sorites Paradox is considering that Sorites-susceptible predicates have several candidate extensions, or several ways in which these expressions can be made precise. For example, a candidate extension for the predicate ‘is a baby’ is the set of humans of less than two years, but also the set of those less than two years and one second, and of those less than two years and two seconds. In this chapter we present and discuss two theories for vague predicates based on this idea: super-valuationism and subvaluationism. The chapter is structured in three parts. The first presents the super- and subvaluationist theories: their similarities and differences in the semantics, the resulting logics with their most characteristic features. The second reviews the super- and subvaluationist solutions to the Sorites Paradox and provides discussion on several controversies surrounding these theories. The third part introduces proof procedures for s’valuationist logics.

The epistemicist solution to the Sorites Paradox consists of two key components. First, vague terms, just as non-vague ones, have ordinary classical semantic-values and do not require any revision of classical logic or semantics. In the case of ‘tall’, each person is either tall or not tall, and there is a specific number k such that someone is tall if and only if they are over k metres in height, thus ensuring that one of the premises of the paradox is just plainly false. Second, although words like ‘tall’ have such sharp boundaries, we do not and cannot know what these boundaries are, and this explains why we might be tempted to (falsely) conclude that such boundaries do not exist.

Epistemicm faces four principle challenges: addressing the counterintuitiveness of the view; accounting for how the sharp meanings of vague terms are determined; explaining why we are principally ignorant of the sharp-boundaries of vague terms; and explaining what makes vagueness into a distinctive phenomenon, different than other kinds of ignorance. The chapter focuses on discussion of these four challenges, presents the most prominent defence of Epistemicism - Timothy Williamson’s- and discusses a range of objections from recent literature to these responses.

Non-transitivism solves the Sorites Paradox by denying the transitivity of logical consequence. After introducing the non-transitivist solution, the chapter presents the main reasons in its favour: its fit with the intuitive diagnosis of what goes wrong in soritical reasoning, its vindication of the naive theory of vagueness and its preservation of the compelling classical fundamental operational principles. The chapter then examines a rival of non-transitivism – on-contractivism – which might seem equally well supported in those respects, arguing that non-transitivism is variously superior to it. Next, the chapter focuses on a specific family of non-transitive logics – tolerant logics – explaining their basic lattice-theoretic semantics and giving details of one particularly strong logic. Finally, the chapter develops a non-transitivist approach to the Forced-March Paradox, arguing that the ideal behaviour of a non-transitivist’s confidence along the Forced March requires a super-additive and boundedly non-monotonic theory of probability, and showing how, by using the tolerant logic just mentioned, one can go through the Forced March and return a knowledgeable verdict about each case.

This chapter examines some aspects of the influence of the Sorites Paradox in psychology. The first section starts out with a brief discussion of the analysis of slippery slope arguments in the psychology of reasoning, to introduce the relevance of probabilistic considerations in that domain. We then devote most of this chapter to the analysis in psychophysics and in the psychology of concepts of the complex relationship between discrimination and categorisation for items that differ very little. The second section emphasises the centrality of probabilistic modelling to represent the way in which small differences between stimuli affect decisions of membership under a common category. The third section focuses on experimental data concerning unordered transitions between prototypes, then the final section looks at data concerning ordered transitions between prototypes (dynamic Sorites).

Degree-theoretical approaches to vagueness attempt to flesh out the idea that properties referred to by vague predicates come in degrees, and that sentences containing such predicates can be true to a degree in between absolute truth and absolute falsity. This many-valued semantics is wedded either to some fuzzy logic, or to a non-truth-functional logic, or even to classical logic. The first part of the chapter is devoted to surveying these different alternatives. Subsequently, we discuss the standard fuzzy approach (SFA) to the Sorites, based on infinite-valued Łukasiewicz logic. The mainstream objections to the SFA are then dispelled from a perspective that views classical logic as an ambiguous logic. Next, we address the status of the Tolerance principle in the SFA. We provide a semantics for vague predicates within Rational Pavelka Logic (RPL), contending that the conditional premisses in a Sorites are ambiguous between a reading as Łukasiewicz conditionals and a reading as 'tolerance conditionals'. In conclusion, we formalise in RPL a purely structural version of the paradox, where no logical constant is involved. We ascribe this paradox to an equivocation over consequence.

Take any putative ordinary object which is divisible into a finite number of small units and tolerant to the loss of one of them. We can remove these units one at a time, and since our object definitely doesn’t exist when there are zero units, and since we cannot pinpoint which removal brings about this destruction,the Sorites Puzzle threatens common sense. We can rescue ordinary objects from its grip, but since independently motivated linguistic explanations of vagueness depend on there being multiple candidate contents of vague terms, these efforts succeed only if there are multiple candidates that can be meant by ordinary object terms. Thus, many more objects than common sense accepts have to exist. The 'arguments from vagueness' also offer soritical reasons for object proliferation. Problematising the categories composite object and persisting thing (rather than specific object concepts), these show that either there are none of these (except simples) or every candidate for making up one, does. The latter, less revisionary alternative is plenitudinism. I defend Modally Full Plenitude, because it is three-dimensionalist and non-reductivist about de re modality – a distinct persisting object in any region, and for every modal profile satisfied by the matter in that region.

The vagueness-intuitionist credits epistemicism with a crucial insight, namely that vagueness is a cognitive, rather than a semantic phenomenon: the vagueness of the distinction between, say, yellow and orange consists in our inability to, so to speak, bring those two concepts up against one another to mark a sharp and stable boundary. The chapter proposes a knowledge-theoretic semantics for a first-order logic of vagueness that respects this conception of the nature of the phenomenon. The semantics diverges from traditional intuitionist-style semantics (Heyting, Beth, Kripke) in the treatment of negation, taking the idea of incompatibility between atomic vague predications as primitive, and setting the negation of atomic A as acceptable just when some statement is acceptable which is incompatible with A. It is shown how a semantics based on this idea vindicates an intuitionist-style repudiation of the principle of Bivalence and thereby the law of excluded middle, and thus enables the reasoning of the Sorites Paradox to go through as a demonstration of the negation of its major premisewithout thereby incurring the unwelcome implication, sustained by classical logic, of the existence of a sharp boundary to the extension of the relevant predicate.

The Sorites Paradox is one of the most venerable and complex paradoxes in the territory of philosophy of logic. Together with the Sorites, the semantic paradoxes also occupy a very prominent place in research in this area. In this chapter we examine the relation between the Sorites and the best-known of the semantic paradoxes: the Liar Paradox. Traditionally, the Sorites and the Liar have been considered to be unrelated. Nevertheless, there have been several attempts to uniformly cope with them. This chapter begins by examining when and why in general, a uniform solution to more than one paradox should be expected and, in particular, why a uniform solution to the Liar and the Sorites should be expected. Subsequently the chpater focuses on the work of Paul Horwich, who has used epistemicist ideas that were first applied to solve the Sorites in order to attempt to give a solution to the Liar. It shows in some detail, as a particular example of the influence that the Sorites has had over the semantic paradoxes, whether the epistemicist approach Horwich presents in order to face the Sorites can be successfully applied to his theory of truth.

The first part of the chapter surveys some of the main ways in which the Sorites Paradox has figured in arguments in practical philosophy in recent decades, with special attention to arguments where the paradox is used as a basis for criticism. Not coincidentally, the relevant arguments all involve the transitivity of value in some way. The second part of the chapter is more probative, focusing on two main themes. First, it further addresses the relationship between the Sorites Paradox and the main arguments discussed in the first part, by elucidating in what sense they rely on (something like) tolerance principles. Second, it briefly discusses the prospect of rejecting the respective principles, aiming to show that we can do so for some of the arguments but not for others. The reason is that in the latter cases the principles do not function as independent premises in the reasoning but, rather, follow from certain fundamental features of the relevant scenarios. It is also argued that not even adopting what is arguably the most radical way to block the Sorites Paradox – that of weakening the consequence relation – suffices to invalidate these arguments.

This chapter outlines the impact of the Sorites Paradox in linguistics, with particular focus on its relation to semantic and pragmatic analyses of gradability and comparison. Section 2 describes the importance of philosophical work on vagueness and the Sorites Paradox for early attempts in linguistics to provide compositional analyses of the relation between positive and comparative adjectives. Section 3 then discusses subsequent linguistic analyses of these phenomena, and the extent to which they succeed or fail in providing isnights on vagueness and the Sorites Paradox. Section 4 explores the ways in which the Sorites Paradox has been used to uncover grammatically significant distinctions between classes of gradable predicates, and Section 5 concludes with a discussion of the connection between the Sorites Paradox and new lines of research geared towards understanding communication under semantic uncertainty.

The Sorites Paradox has a very long history, though one less rich than that of its close relative, the Liar. This chapter is divided into three sections. Section 1 presents a very brief and schematic overview of the pre-analytic history of the Sorites Paradox, from the Ancient Greeks to Kant and Hegel. Section 2 focuses on Chrysippus (Stoic from the third century BC), the ancient philosopher who worked most on the Sorites. Chrysippus held an epistemic view of vagueness and proposed a strategy to respond to the paradox in the context of a dialectical questioning (similar to what is now commonly called the Forced March) that deserves to be thoroughly examined. Section 3 focuses on Leibniz, who produced, in the author's view, the most interesting contribution in modern pre-analytic times on the paradox. Leibniz started by accepting that vague notions have sharp boundaries, but then changed his mind and moved to something like a semantic view of vagueness.

Incoherentism about vagueness is the view that vague expressions/concepts are incoherent due to their vagueness. This chapter elaborates on what incoherentism is, and defends a particular incoherentist view. It presents an overview of important arguments for and against incoherentism. Among arguments for the view are claims that it provides an attractive account of the nature of vagueness, and of the way in which vagueness is associated with indeterminacy. Among arguments against the view are claims that it presupposes a mistaken view on semantic/conceptual competence, and that the view sits ill with how ubiquitous vagueness is. The specific view defended is compared to the views of Michael Dummett, Terence Horgan and Peter Unger.

The chapter explains and defends a dialetheic account of vagueness and its solution to the Sorites Paradox. According to this, statements in the middle of a Sorites progression are both true and false. After an explanation of an appropriate paraconsistent logic, detailed models of Sorites transitions are provided. Crucial to any supposed solution to the Sorites Paradox is how it handles the matter of cut-offs. Much of the chapter concerns how a dialetheic solution handles this.

This Introduction provides the tools necessary for understanding the Sorites Paradox as well as a first orientation concerning its solutions and influence. After presenting the aims and the structure of the book and characterising the notion of vagueness, the Introduction presents the Sorites Paradox in its two main versions: the lack-of-boundaries version and the tolerance version. The Introduction proceeds with an overview of the solutions to the Sorites Paradox that will be developed in the volume: the ones that preserve classical logic (epistemicism, supervaluationism, contextualism, incoherentism) and the ones that do not (intuitionism, rejection of excluded middle, dialetheism, Degree theory and non-transitivism). Finally, after presenting and briefly discussing the Forced-March Paradox, the Introduction ends with a survey of the main areas in which the influence of the Sorites Paradox has been important, which are discussed in the second part of the volume.