Plural quantification

In mathematics and logic, plural quantification is the theory that an individual variable x may take on plural, as well as singular values. As well as substituting individual objects such as Alice, the number 1, the tallest building in London etc. for x, we may substitute both Alice and Bob, or all the numbers between 0 and 10, or all the buildings in London over 20 stories.

The point of the theory is to give first-order logic the power of set theory, but without any "existential commitment" to such objects as sets. The classic expositions are Boolos 1984, and Lewis 1991.

Background
The view is commonly associated with George Boolos, though it is older, and is related to the view of classes defended by John Stuart Mill and other nominalist philosophers. Mill argued that universals or "classes" are not a peculiar kind of thing, having an objective existence distinct from the individual objects that fall under them, but "is neither more nor less than the individual things in the class". (Mill 1904, II. ii. 2,also I. iv. 3).

A similar position was also discussed by Bertrand Russell in chapter VI of Russell (1903), but later dropped in favour of a "no-classes" theory. See also Gottlob Frege 1895 for a critique of an earlier view defended by Ernst Schroeder.

Interest revived in plurals with work in linguistics in the 1970s by Remko Scha, Godehard Link, Fred Landman, Roger Schwarzschild, Peter Lasersohn and others, who developed ideas for a semantics of plurals.

Plural quantification
Standard first order logic has difficulties in representing some sentences with plurals. Most well-known is the Geach-Kaplan sentence "some critics admire only one another". Kaplan proves that it is nonfirstorderizable by showing its second-order translation to be true in every nonstandard model of arithmetic but false in every standard one. (This requires formalizing it in the usual language of arithmetic.) Hence its paraphrase into a formal language commits us to quantification over (i.e. the existence of) sets. But it seems implausible that a commitment to sets is essential in explaining these sentences. "Alice, Bob and Carol admire only one another" does not involve sets and is equivalent to the conjunction of the following


 * &forall;x(if Alice admires x, then x = Bob or x = Carol)
 * &forall;x(if Bob admires x, then x = Alice or x = Carol)
 * &forall;x(if Carol admires x, then x = Alice or x = Bob)

where x ranges over Bob, Alice, and Carol. But this seems to be an instance of "some people admire only one another", which is nonfirstorderizable.

Boolos argued that 2nd-order monadic existential quantification may be systematically interpreted in terms of plural existential quantification, and that, therefore, 2nd-order monadic existential quantification is "ontologically innocent".

Later Tom McKay (2003) and others argued that sentences such as


 * They are shipmates
 * They are meeting together
 * They lifted a piano
 * They are surrounding a building
 * They admire only one another

also cannot be interpreted, in standard first order logic. This is because predicates such as "are shipmates", "are meeting together", "are surrounding a building" are not distributive. A predicate F is distributive if, whenever some things are F, each one of them is F. But in standard logic, every predicate is distributive. Yet such sentences also seem innocent of any existential assumptions. If true, they are about individuals who are shipmates, who meet together, lift pianos &c, and nothing else (not sets, or abstract Platonic entities).

McKay has argued for a unified account of plural terms that allows for both distributive and non-distributive satisfaction of predicates, which he has defended against the "singularist" assumption that such predicates are predicates of sets of individuals (or of mereological sums)


 * Such views make the singularist assumption, that every plural predication is at its root a singular predication, thus making it possible to apply the framework of standard first-order logic. Fundamentally, the problem with such approaches is that they have not taken plurality seriously. No set ever surrounds a building, though its members may. The fact that some individuals are surrounding a building does not automatically imply that some single individual (of any kind) surrounds the building.


 * Plural noun phrases can refer to several things. We will not assume that they must then automatically also refer to one thing.

McKay, as well as other writers such as Cameron (1999), have suggested that plural logic opens the prospect of simplifying the foundations of mathematics, avoiding the paradoxes of set theory, and "simplifying the complex and unintuitive axiom sets needed in order to avoid them.

Criticism
Philippe de Rouilhan (2000) has argued that Boolos relied on the assumption, never defended in detail, that plural expressions in ordinary language are "manifestly and obviously" free of existential commitment. But when I utter "there are critics who admire only one another" is it manifest and obvious that I am only committing myself with respect to critics? Or is Boolos victim of a "grammatical illusion" (p. 10)? Consider


 * There is at least one critic who admires only himself.
 * There are critics who admire only one another

The first case is clearly "innocent". But what about the second? There is an obvious logical difference, since in the first case the plural is distributive, in the second, it is collective, and irreducibly so. How is it obvious that this difference is innocent? Also, the second is equivalent to


 * Some group (or collection) of critics is such that they admire only one another

But what is a "group" or "collection" in this sense? "That is the whole problem". Perhaps Boolos has accorded a kind of innocence to [the second] that would actually belong only to the first.