This paper criticizes George Boolos's famous use of plural quantification to argue that monadic second-order logic is pure logic. I deny that plural quantification qualifies as pure logic and express serious misgivings about its alleged ontological innocence. My argument is based on an examination of what is involved in our understanding of the impredicative plural comprehension schema.

In this sequel to "The logic and meaning of plurals. Part I", I continue to present an account of logic and language that acknowledges limitations of singular constructions of natural languages and recognizes plural constructions as their peers. To this end, I present a non-reductive account of plural constructions that results from the conception of plurals as devices for talking about the many. In this paper, I give an informal semantics of plurals, formulate a formal characterization of truth for the (...) regimented languages that results from augmenting elementary languages with refinements of basic plural constructions of natural languages, and account for the logic of plural constructions by characterizing the logic of those regimented languages. (shrink)

This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle (...) which pluralities form sets, the paper concludes by affirming a naturalistic approach to the philosophy of set theory. (shrink)

English has plural terms as well as singular terms. But our standard formal languages, e.g., the predicate calculus, feature only singular terms. How can the plural idiom be formalized?‘Changing the subject’ is by far the most common plurals strategy among both philosophers and linguists: a plural term is replaced by a singular term standing for some complex object that ‘contains’ the individuals to which the plural term alludes. For example, one might simply replace ‘A, B imply C’ with ‘{A, B} (...) implies [singular] C’. We uncover a surprising variety of ways to change the subject, of ever‐increasing complexity and ingenuity. Our question is whether any can made to work. (shrink)

Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

NOWADAYS, even schoolchildren babble about "null sets" and "singletons" and "one-one correspondences," as if they knew what they were talking about. But if they understand even less than their teachers, which seems likely, they must be using the technical jargon with only an illusion of understanding. Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should (...) be child's play. (shrink)

Atomism denies that complexes exist. Common-sense metaphysics may posit masses, composite individuals and sets, but atomism says there are only simples. In a singularist logic, it is difficult to make a plausible case for atomism. But we should accept plural logic, and then atomism can paraphrase away apparent reference to complexes. The paraphrases require unfamiliar plural universals, but these are of independent interest; for example, we can identify numbers and sets with plural universals. The atomist paraphrases would fail if plurals (...) presuppose complexes: but an Appendix shows that reference to complexes is not required in the semantics of plurals. (shrink)

This is the first systematic survey of modern nominalistic reconstructions of mathematics, and for this reason alone it should be read by everyone interested in the philosophy of mathematics and, more generally, in questions concerning abstract entities. In the bulk of the book, the authors sketch a common formal framework for nominalistic reconstructions, outline three major strategies such reconstructions can follow, and locate proposals in the literature with respect to these strategies. The discussion is presented with admirable precision and clarity, (...) and should be accessible even to readers with only minimal background in logic and mathematics. There will be many who will turn directly to these pages and use them as a brief manual on the state of the art of nominalism in mathematics. But the most intriguing parts of this elegant book—at least in my view—are the introduction and the conclusion, where the authors examine the significance of reconstructive nominalism. (shrink)

Contemporary accounts of logic and language cannot give proper treatments of plural constructions of natural languages. They assume that plural constructions are redundant devices used to abbreviate singular constructions. This paper and its sequel, "The logic and meaning of plurals, II", aim to develop an account of logic and language that acknowledges limitations of singular constructions and recognizes plural constructions as their peers. To do so, the papers present natural accounts of the logic and meaning of plural constructions that result (...) from the view that plural constructions are, by and large, devices for talking about many things (as such). The account of logic presented in the papers surpasses contemporary Fregean accounts in its scope. This extension of the scope of logic results from extending the range of languages that logic can directly relate to. Underlying the view of language that makes room for this is a perspective on reality that locates in the world what plural constructions can relate to. The papers suggest that reflections on plural constructions point to a broader framework for understanding logic, language, and reality that can replace the contemporary Fregean framework as this has replaced its Aristotelian ancestor. (shrink)

The paper explores the view that in mathematics, in particular where the infinite is involved, the application of classical logic to statements involving the infinite cannot be taken for granted. L. E. J. Brouwer’s well-known rejection of classical logic is sketched, and the views of David Hilbert and especially Hermann Weyl, both of whom used classical logic in their mathematical practice, are explored. We inquire whether arguments for a critical view can be found that are independent of constructivist premises and (...) consider the entanglement of logic and mathematics. This offers a convincing case regarding second-order logic, but for first-order logic, it is not so clear. Still, we ask whether we understand the application of logic to the higher infinite better than we understand the higher infinite itself. (shrink)

Barry Schein proposes combining a second-order treatment of plurals with DonaldDavidson's suggestion that there are positions for reference to events in ordinary predicates inorder to account for several of the more puzzling features of ...

Plural predication is a pervasive part of ordinary language. We can say that some people are fifty in number, are surrounding a building, come from many countries, and are classmates. These predicates can be true of some people without being true of any one of them; they are non-distributive predications. However, the apparatus of modern logic does not allow a place for them. Thomas McKay here explores the enrichment of logic with non-distributive plural predication and quantification. His book will be (...) of great interest to philosophers of language, linguists, metaphysicians, and logicians. (shrink)

This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here for (...) the first time. (shrink)

The aim of this essay is to show that the subject-matter of ontology is richer than one might have thought. Our route will be indirect. We will argue that there are circumstances under which standard first-order regimentation is unacceptable, and that more appropriate varieties of regimentation lead to unexpected kinds of ontological commitment.

Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-style semantics for plural logic (...) that does not resort to sets or set-like objects to interpret plural variables, but adopts the view that a plural variable has many objects as its values. Using this semantics, we also articulate a generalized notion of ontological commitment which enables us to develop some ideas of earlier critics of the alleged ontological innocence of plural logic. (shrink)

The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers.