This chapter introduces paradoxes from three areas. First, the famousantinomies from Frege–Cantor set theory are presented asconsequences of the naive set concept; the standard solution,Zermelo–Fraenkel set theory and the idea of iterative sets,is discussed and found to be problematic. Second, the soritesparadox for vague predicates/properties is presented, and somestandard solutions are discussed, putting the focus on“cutoff points.” A paraconsistent “glut”approach is recommended as the best way to accept the existence ofcutoff points, as inconsistent and not unique. Third, a puzzle aboutboundaries in space is presented, standard solutions discussed andfound wanting, and a paraconsistent response outlined. It issuggested that all three paradoxes are interrelated by the conceptof “revenge” and should be met with dialetheism.

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.

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).

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.

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.

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.