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This chapter presents a second topic neutral system of plural logic, full plural logic. It is obtained from the mid-plural system of the previous chapter by allowing plural variables to be bound by quantifiers and also by an operator symbolizing plurally unique description. Naturally, there are great gains in expressive power. The price is the loss of axiomatizability. A partial axiomatization is presented which combines the axioms for mid-plural logic with plural versions of those for singular logic. A plural version of comprehension is derived from the resulting axioms, and versions of plural choice are formalized.

This introductory chapter surveys plural phenomena and argues for their logical significance. Terms may be classified as singular or plural, according to the number of things they are capable of denoting. Predicates and function signs may take plural terms as arguments, and function signs may express multivalued functions and produce plural functional terms (the wives of Henry VIII). Received philosophical and formal logic, however, makes no place for these plural phenomena. Strategies for a logic of plurals can therefore be broadly classified as singularist or pluralist. The singularist forces plurals into the old singular mould, whereas the pluralist develops a new plural logic that directly accommodates plural terms, plural predicates, multivalued functions, and plural quantification. In Chapters 3 and 4, general singularist strategies are eliminated. In this chapter, Michael Dummett’s more piecemeal singularist attempts to deal with plural phenomena are criticized.

This paper presents analyses of natural language quantifiers and determiners. In doing so, the paper pays special attention to plural determiners, determiners that can combine with plural noun phrases (e.g., all, some, any, the, most), and argues that Generalized Quantifier Theory gives clearly incorrect accounts of those determiners by assuming the traditional bias against plural constructions. The paper gives a sketch of a recent approach to plural constructions, the pluralist approach, that regards them as peers of their singular cousins that have autonomous semantic functions, and presents analyses of plural determiners that result from taking the approach. The paper also presents analyses of Bach-Peters sentences that use polyadic quantifiers or determiners, and gives a sketch of a treatment of donkey anaphora based on the view that indefinite noun phrases by themselves have no quantificational component.

The book concludes with a Postscript describing some items of unfinished business. It begins with three tricky topics: first, pseudo-singular terms—syntactically singular but semantically plural noun phrases—and then two extensions of full plural logic in very different directions—higher-level plural logic and higher-order plural logic. It adds a bunch of other items the authors have hardly explored

This chapter presents a second topic neutral system of plural logic, full plural logic. It is obtained from the mid-plural system of the previous chapter by allowing plural variables to be bound by quantifiers and also by an operator symbolizing plurally unique description. Naturally, there are great gains in expressive power. The price is the loss of axiomatizability. A partial axiomatization is presented which combines the axioms for mid-plural logic with plural versions of those for singular logic. A plural version of comprehension is derived from the resulting axioms, and versions of plural choice are discussed.

This introductory chapter surveys plural phenomena and argues for their logical significance. Terms may be classified as singular or plural, according to the number of things they are capable of denoting. Predicates and function signs may take plural terms as arguments, and function signs may express multivalued functions and produce plural functional terms (the wives of Henry VIII). Received philosophical and formal logic, however, makes no place for these plural phenomena. Strategies for a logic of plurals can therefore be broadly classified as singularist or pluralist. The singularist forces plurals into the old singular mould, whereas the pluralist develops a new plural logic that directly accommodates plural terms, plural predicates, multivalued functions and plural quantification. In Chapters 3 and 4, general singularist strategies are eliminated. In this chapter, Michael Dummett's more piecemeal singularist attempts to deal with plural phenomena are criticised.

A plurality or plural object is a single object that is also many, and pluralitism is the thesis that there is such an object. This chapter argues that pluralitism and closely related theses (e.g. the many–one identity thesis and the composition as identity thesis) violate logic. To do so, it formulates an approach to the logic and semantics of plural constructions that results in plural logic, and relates treatments of plural constructions to accounts of natural number. It gives a critical examination of Frege’s views of numbers and plural constructions, and sketches the view of plural constructions as devices for talking about the many (as such), and an account of natural numbers as properties of a special kind, plural properties.

This chapter applies plural logic to set theory. Set theory may be axiomatized with full plural logic as the underlying logic and with the function sign { }—read ‘set of’—as its only nonlogical primitive. This style of axiomatization is inspired by Cantor, both with regard to the idea of a set as a collection of many members and the plural language used to express it. Cantor’s idea of sets as collections does not accommodate empty or singleton sets. So a system of Cantorian set theory is developed which excludes them. It is argued that this is no loss. But for those who prefer to retain the anomalous sets, the chapter concludes by explaining how a conventional iterative set theory can be based on full plural logic.

This chapter offers a detailed account of Jacques Derrida's quasi-transcendental thinking. It tries to undo the ties of the interpretative straitjacket that binds Derrida's thinking into an aneconomic freeplay of differences, which sees “deconstruction” as merely the hysterical dismantling of any construction. It also lays a basis for grasping Derrida's deconstructive readings of Sigmund Freud. The discussion begins by criticizing Richard Rorty's early misreadings, which provide excellent material for an attempt to counter the one-sidedness of readings that make of Derrida's philosophical strategy a freeplay relativism. To counter such misreadings, this chapter offers an account of différance in accordance with the “plural logic of the aporia”, aligning “différance as temporization” with the economic aporia and “différance as spacing” with the aneconomic aporia.

This chapter considers whether mereology should rightly be thought of as a first-order theory with parthood as a binary predicate. It considers extensions of classical mereology aimed at overcoming the expressive limits of standard first-order languages, focusing on second-order and plural formulations. Relatedly, Lewis’s megethology and applications to the philosophy of mathematics are discussed. Then, several ways of modifying the framework to make room for mereological considerations involving time and modality are presented, such as the possibility that an object may have different parts at different times, or that it could have had different parts from the ones it actually has. Finally, a number of theories are expounded that can be developed in order to deal with the phenomenon of mereological indeterminacy, i.e., the fact that in some cases the very question of whether something is part of something else does not appear to have a definite answer.

“There are no gaps in logical space,” writes Lewis (1986), giving voice to sentiment shared by many philosophers. But different natural ways of trying to make this sentiment precise turn out to conflict with one another. One is a pattern idea: “Any pattern of instantiation is metaphysically possible.” Another is a cut and paste idea: “For any objects in any worlds, there exists a world that contains any number of duplicates of all of those objects.” Jumping off from discussions from Forrest and Armstrong (1984) and Nolan (1996), the authors use resources from model theory to show the inconsistency of certain packages of combinatorial principles and the consistency of others.