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 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.