Reference to dependencies established in multiple-wh questions

In discourse, a universal statement establishes a dependency between sets of objects, which can support the evaluation of singular pronouns in a subsequent sentence. Two well-known phenomena involving such reference to a dependency are quantificational subordination and telescoping. This paper argues that a multiple-wh question admitting a pair-list answer can support subordination and telescoping, just like universal statements. Accordingly, the relevant phenomena are classified into two kinds of reference to dependencies, called ‘question subordination’ and ‘question telescoping’, which exhibit different properties. A dynamic family-of-questions analysis is developed to account for these phenomena. Briefly, a multiple- wh question generates a set of sub-questions, i.e., a family of questions, and then the set is transformed into a set of possible pair-list answers. Following Dynamic Plural Logic, the family of questions and possible pair-list answers encode different kinds of dependencies. Accessing the dependency encoded in a possible pair-list answer gives rise to question subordination, whereas accessing the dependency encoded in the family of questions gives rise to question telescoping.

On the Innocence and Determinacy of Plural Quantification

Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. Both assumptions are problematic, as is shown by providing a Henkin-style semantics for plural logic that does not resort to sets but takes a plural term to have plural reference. This semantics gives rise to a generalized notion of ontological commitment, which is used to develop some ideas of earlier critics of the alleged ontological innocence of plural logic.

The Semantics of Plurals

Plural logic provides an appealing framework for the regimentation of natural language plurals. However, the choice of a regimenting language leaves wide open the semantic question of how this language should be interpreted. One option is to interpret a plural term as denoting a non-empty set. Another is to embrace plurals in the metalanguage and take a plural term to have plural reference. A detailed comparison of the options reveals that there is no simple solution to the problem of choosing among them.

Taking Plurals at Face Value

Plural logic is a logical system in which plural terms and predicates figure as primitive expressions alongside the singular resources of ordinary first-order logic. The philosophical significance of this system depends on two of its alleged features: being pure logic and providing more expressive power than first-order logic. This chapter first introduces the language and axioms of plural logic and then analyzes this logic’s main philosophical applications in metaphysics, philosophy of mathematics, and semantics.

The Refutation of Singularism?

Traditional analyses of plurals tended to eliminate plural expressions in favor of singular ones. These “singularist” analyses have recently faced many objections, which are intended to provide indirect support for the alternative analysis provided by plural logic. This chapter evaluates four such objections and concludes that they are less compelling than is often assumed. This conclusion is borne out by a close examination of various plural versions of Cantor’s theorem.

The Many and the One

Plural logic has become a well-established subject, especially in philosophical logic. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between this logic and other theoretical frameworks such as set theory, mereology, higher-order logic, and modal logic. The applications of plural logic rely on two assumptions, namely that this logic is ontologically innocent and has great expressive power. These assumptions are shown to be problematic. The result is a more nuanced picture of plural logic’s applications than has been given so far. Questions about the correct logic of plurals play a central role in the last part of the book, where traditional plural logic is rejected in favor of a “critical” alternative. The most striking feature of this alternative is that there is no universal plurality. This leads to a novel approach to the relation between the many and the one. In particular, critical plural logic paves the way for an account of sets capable of solving the set-theoretic paradoxes.

Absolute Generality and Singularization

Is it possible to assert something of absolutely everything there is? While such absolute generality appears possible, it faces serious theoretical challenges. This chapter examines one such challenge, based on the possibility of one-to-one mappings from pluralities to objects. The challenge gives rise to a trilemma. First, generality relativists deny the possibility of absolute generality, at the cost of being unable to express various important insights. Second, traditional generality absolutists deny the possibility of the mentioned mappings, but are pushed up a hierarchy of logics of higher and higher order, resulting in an expressibility deficit akin to that of the relativists. This motivates taking a closer look at the third option, which restricts traditional plural logic.

Plurals and Mereology

This chapter provides a systematic comparison of plural logic and an atomistic version of classical mereology. Since these two systems are mutually interpretable, it is formally possible to eliminate one in favor of the other. However, reasons are offered to retain both systems. In particular, mereology is a useful tool for the analysis of plurals in natural language.

Critical Plural Logic

This chapter develops and motivates an alternative, more critical plural logic, thus exploring the third horn of the trilemma from the previous chapter. First, a liberal view of mathematical definitions is defended, according to which any objects can be used to define a set. This entails that the traditional plural comprehension scheme needs to be restricted. Some successor principles are then formulated on the basis of the idea that any plurality needs to be circumscribed. Finally, the resulting critical plural logic is shown to give rise to a natural and elegant approach to set theory.

Introduction

English and other natural languages contain plural expressions. Plural logic is based on the idea of taking these expressions at face value rather than eliminating them in favor of singular resources. This logic has great potential significance in logic, philosophy, linguistics, and beyond, but it also raises some hard questions. Three such questions will figure centrally in the book as a whole. Are primitive plural resources legitimate? If so, what is their relation to singular resources, and what is their philosophical significance?