Plural logic

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.