This chapter seeks keywords and concepts that will enable us to grasp the contradictory and conflictive globality of the current moment and sharpen our analysis of equally contradictory and conflictive global pasts. In a plea to move beyond equating the global with openness, connection, and integration, I address the role of closure, boundaries, and limits in global history in a wider sense. For this purpose, I explore in an experimental and deliberately open-ended fashion how thinking about global spherescan be utilised fruitfully for the current practice of history writing. The first part explores the radically inclusive yet claustrophobic vision of the globe as a closed sphere from which there is no escape. Building on earlier closed-world and one-world discourses, this thinking gained prominence after the Second World War in the face of the threat of nuclear destruction and environmental degradation. I then move to think about the globe as composed of many bounded spheres – geopolitical but also social. Here, I take central examples from the realm of communication and language and discusses the public sphere as an exclusionary rather than inclusionary figure of thought.

In this short “exploration” chapter we consider an alternative semantics for modal logic using topological spaces. No background whatsoever in topology is assumed. We motivate the fundamental mathematical definition of a topological space from an epistemic perspective, and connect it to the modal logics studied previously, both intuitively and formally. We also show how to transform reflexive and transitive frames into topological spaces in a truth-preserving way, and use this to establish completeness.

This chapter argues that labeling is the key linguistically bespoke operation. I trace the recursive property of Gs to the fact that they employ labels. The chapter argues that labels are the device for taking an expression in the range of Merge and putting it into the domain of Merge. Thus labels close Merge in the domain of the lexical atoms. So closing the operation effectively delivers a recursive system of unbounded hierarchy. The chapter also critically reviews some arguments for the simplicity of Merge based on the role it is intended to play in explaining the evolution of language. I dispute this and argue that what needs explanation is something quite different: What is the source of the power to form inductive definitions? Recursion is a consequence of closure afforded by labels.

The view of evidence, defeat, and suspension put forth here delivers the result that paradigmatic scepticism about knowledge and justification is an instance of resistance to evidence. This chapter argues that this result is correct. In order to do that, I look at extant neo-Moorean responses to purported instances of failure of knowledge closure (Pryor 2004, Williamson 2007) and warrant transmission and argue that they are either too weak – in that they concede too much to the sceptic – or too strong – in that they cannot accommodate the intuition of reasonableness surrounding sceptical arguments. I propose a novel neo-Moorean explanation of the data, relying on my preferred account of defeat and permissible suspension, on which the sceptic is in impermissible suspension but in fulfilment of their contrary-to-duty epistemic obligations.

Since Edgar Allan Poe’s assertion that the short story must be read in a “single sitting,” short story theory has focused on the importance of endings as a hallmark of the form. This crystallized, in the 1980s and 1990s, in the rise of closure studies, a critical field that sought to taxonomize the ways stories end and its effects on the reader. This essay examines a feminist countertradition of short story writing that uses grammar as a tool to disrupt the form’s inbuilt narrative teleology. By interrogating the short story’s narrative temporality, writers such as Gertrude Stein, Lydia Davis, and Lorrie Moore use grammar to situate themselves, in distinctly gendered ways, in and against broader systems of time. Through a close examination of these writers, the essay explores how grammar offers a way of assessing not only the short story’s closures but also its various expansions and radical possibilities.

Chapter 3 challenges the critical commonplace in accounts of the lyric that Schubert’s thematic groups compose closed, self-contained entities that yield static forms.In particular, it re-examines the perceived equivalence of the term ‘closed’ with cadential closure and the Perfect Authentic Cadence (PAC). It argues that Schubert’s lyric forms compose ’closed’ ABA structures while simultaneously undermining or downplaying the role of cadence as a marker of finality. The first section clears the theoretical path by examining the primary and secondary parameters of closure and their non-congruence. Three analytical case studies follow and address, respectively, the concept of functional retrogression in Schubert’s earliest quartets, the composer’s particular ways of articulating or downplaying the medial caesura (MC), and his use of the elided PAC MC, explored through the formally unorthodox Quartettsatz. Each analytical case study demonstrates the destabilisation of moments of punctuation in Schubert’s lyric forms as well as the charged conflict between tonal and rhetorical parameters, which often creates a critical tension between punctuation and continuation.

This chapter discusses three stylistic hallmarks of Clara Schumann’s songs: (1) a fascination with expansive themes; (2) a tendency to modify cadences so that themes pushe onward to the next and decisive closure arrives only at the end of songs; and (3) an inventive use of the piano. I explore each of these hallmarks via brief analyses of representative songs, in preparation for the more detailed music analyses in later chapters.

This chapter considers the reciprocal productivity between Beckett’s Endgame and Shakespeare’s romance The Tempest. It examines the settings of the two plays, their dialectics of making and unmaking, their dynamics of confinement and release, the materiality of air and earth, and the notion of ending. It looks at the insular dominions of Prospero’s island and the space inhabited by Beckett’s characters in Endgame, and argues that the imperfections and shortcomings of a medium are not an end in themselves but become the grounds on which plays such as The Tempest and Endgame transcend the finitude of their art and reflect back on it, asserting its very finitude as a condition of possibility.In both plays, the game of chess figures as a structural and thematic component and reflects on the art of the playwright. The chapter analyses the brief scene in which Prospero discovers Ferdinand and Miranda playing chess. The scene functions as a mise-en-abîme, as a play-within-the-play, and becomes a metaphor for the play itself. In the many references to chess in Beckett’s works, above all in Endgame, chess, as this chapter argues, presents a matrix of multiplicity that remains tied to form.

Is humanitarianism a network, hierarchy, or market? This chapter argues that it combines principles of hierarchy and networks in the form of a club. It develops a sociologically inspired version of club governance to understand the rise and resilience of the Humanitarian Club. This sociological explanation illuminates how clubs, like many groups, are: distinguished by collective interests, identities, and values that create a common mentality and a sense of we-ness; and often generate a distinction from and feeling of superiority to outsiders. The chapter examines the structures of inequality and patterns of inclusion and exclusion, and traces the rise of the humanitarian elite and the creation of a Humanitarian Club that is produced and sustained by four kinds of capital and that create sharp distinctions between (Western) insiders who can deliver the goods and (Southern) outsiders who are viewed as inferior. Although the humanitarian field has attempted on countless occasions to create more inclusion and diversity, the chapter argues that these forms of capital guard the doors of the Club and maintain a humanitarian field in which Western aid organizations dominate Southern aid agencies.

The idea that the universe is infinite in size is central to the Epicurean system. Infinity, however, is also a concept that, in the history of philosophical, scientific and artistic discussion before and after Lucretius, has defied explanation, engendered paradox, and stimulated the romantic sensibility. This chapter looks at the strategies, philosophical and literary, deployed by Lucretius to achieve closure on this inherently open topic. Co-ordinate with the relationships of analogy and complementarity between the text of the DRN and the nature of the universe it describes, the poem’s poetics of closure and enclosure, on the one hand, and of non-closure or ‘false closure’, on the other, express and enact the infinity of the universe conceived both as all-encompassing and as open-ended. These rival conceptions of infinity are modeled throughout the poem, and especially in Epicurus’ triumph of the mind (1.62–79) and Lucretius’ reworking of a thought-experiment attributed to Archytas of Tarentum (1.951–83). Taken together, they bring out the tension, or complementarity, in Lucretius between the totalizing scientist who prescribes an intellectual panacea and the sublime poet who reaches into the beyond.

This chapter introduces the received view in epistemology that inferential knowledge requires essential premises to be known. I formulate the principle of Knowledge Counter-Closure, which expresses the received view, and describe a schema for possible counterexamples to this principle.