Call a term ‘pseudo-singular’ if it is syntactically singular but semantically plural. ‘The pair who wrote Principia’ is a good example, standing as it does for the two individuals, Whitehead and Russell. In this journal (2021), Eric Snyder and Stewart Shapiro launched an attack on the idea, calling it ‘linguistically and logically untenable.’ In this reply we rebut every one of their criticisms.

What is the relation between some things and the set of these things? Mathematical practice does not provide a univocal answer. On the one hand, it relies on ordinary plural talk, which is implicitly committed to a traditional form of plural logic. On the other hand, mathematical practice favors a liberal view of definitions which entails that traditional plural logic must be restricted. We explore this predicament and develop a “critical” alternative to traditional plural logic.

AbstractAlmost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to the theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals.

This book tackles the logic of plural terms (‘Whitehead and Russell’, ‘the men who wrote Principia Mathematica’, ‘Henry VIII's wives’, ‘the real numbers’, ‘√—1’, ‘they’); plural predicates (‘surrounded the fort’, ‘are prime’, ‘are consistent’, ‘imply’); and plural quantification (‘some things’, ‘any things’). Current logic is singularist: it only allows terms to stand for at most one thing. By contrast, the foundational thesis of this book is that a particular term may legitimately stand for several things at once, in other words, there is such a thing as genuinely plural denotation. Plural logic is logic based on plural denotation. The book begins by making the case for taking plural phenomena seriously, and argues, by eliminating rival singularist strategies, that the only viable response is to adopt a plural logic. The subsequent development of the conceptual ground includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists. A formal system of plural logic is then presented in three stages, before being applied to Cantorian set theory as an illustration. A system of higher-level plural logic is then outlined. It bears a striking similarlty to the set theory.

Extract This is just what it says it is. It does not presume to set out a research programme for others to follow, nor is it a mea culpa. It simply describes a miscellany of topics which, if we were not keen to call a halt to our work, we should have carried further. Second-order plural logic This extension of full plural logic is modelled on the familiar second-order extension of the classical predicate calculus. But the second-order extension now envisaged will be built on a base of plural logic, and will thus permit direct representation of e.g. ‘Whitehead and Russell did something that Wittgenstein and Ramsey did not’. It also allows one to give full expression to principles such as plural versions of choice, which can only be partially captured using schemes in full plural logic (see §13.6). Third- and higher-order extensions are likewise possible. We have already made informal use of higher-order resources in the semantic metalanguages for singular, mid-plural, and full plural logic. As a second example, we now present two plural versions of Cantor’s theorem, and use them to explain why the strategy of changing the subject must fail.

Plural phenomena are significant and inescapable. Granted, the plural idiom is sometimes reducible to the singular, e.g. ‘2 and 3 are prime is equivalent to ‘2 is prime and 3 is prime’. ‘Are prime’, however, belongs to the special class of predicates known as distributives. No such reductions are possible for the general case of collective (nondistributive) predicates, and they are to be found everywhere, from the everyday (‘Whitehead and Russell wrote Principia Mathematica’) to the heart of logic itself (‘The axioms are consistent’, ‘Those premises imply this conclusion’). It is no good dismissing grammatical number as a logically irrelevant complication like person or gender, since plural expressions are crucially involved in valid patterns of argument. To take an elementary example, ‘The Brontë sisters supported one another; the Brontë sisters were Anne, Charlotte and Emily; so Anne, Charlotte and Emily supported one another’. There can be no warrant for ignoring such patterns while attending to their singular counterparts. And some arguments do not even have a singular counterpart. For example, ‘Some numbers are prime. So some numbers are such that they are prime and a number is prime only if it is one of them.’ Logicians wedded to the singular logic of the predicate calculus typically try to dodge the issue of plurals by invoking sets, but we shall see that this is untenable. Socrates exploited the difference between distributive and collective predicates in Hippias Major, but little of interest happened subsequently until Russell put plurals at the centre of his project for providing a foundation for mathematics, through his idea of the ‘class as many’ in The Principles of Mathematics. After another fallow period, the subject revived in the 1970s and 80s with the work of Black, Morton, Sharvy, Simons and Boolos. It would be premature to attempt a comprehensive survey. This entry offers a nontechnical outline of plural predicate logic, including the major differences between it and singular logic and some matters still to be resolved.

Having previously dispatched singularism, this chapter turns to plural logic. First comes philosophical logic, beginning with the notion of a term. Singular and plural terms are different species of a common genus. As against a narrower Russellian conception, terms include definite descriptions and functional terms alongside proper names and demonstratives. Terms of any of these kinds may denote some thing(s) or may be empty. One aim of this book is to counterbalance the recent preoccupation with proper names and descriptions by placing functions and functional terms centre stage. Russell and Frege are criticized for failing to do justice to functions, despite their signal interest in mathematics. Of special interest are partial functions which map something to nothing, co-partial functions which map nothing to something, as well as functions which take several arguments at a given place, and multivalued functions which produce several values for a given choice of arguments.

The most common singularist strategy is changing the subject, which replaces a plural term apparently denoting several things by a singular term standing for a single thing, a set or aggregate or group. This chapter argues that no version of changing the subject works. Naive versions can be quickly dismissed. More sophisticated variants, in which the predicate as well as the subject is changed, fall foul of an analogue of Russell’s paradox. The appendix to this chapter criticizes extensions of Donald Davidson’s event-analysis of singular verbs of action to plural predication in general.