Plural quantification
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, plural quantification is the theory that an individual variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 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 storeys.

The point of the theory is to give first-order logic the power of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, 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
George Boolos
George Stephen Boolos was a philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.- Life :...

, though it is older (see notably Simons 1982), and is related to the view of classes defended by John Stuart Mill
John Stuart Mill
John Stuart Mill was a British philosopher, economist and civil servant. An influential contributor to social theory, political theory, and political economy, his conception of liberty justified the freedom of the individual in opposition to unlimited state control. He was a proponent of...

 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
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

 in chapter VI of Russell (1903), but later dropped in favour of a "no-classes" theory. See also Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...

 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
Remko Scha
Remko J. H. Scha is a professor of computational linguistics at the faculty of humanities and Institute for Logic, Language and Computation at the University of Amsterdam. He is also an internationally renowned composer and performer of algorithmic art....

, Godehard Link
Godehard Link
Godehard Link is a professor of logic and philosophy of science at the University of Munich.-External links:*...

, Fred Landman
Fred Landman
Fred Landman is a professor of semantics at Tel Aviv University and the author of a number of books about linguistics. Prof. Landman is known for his influential work on progressives, polarity phenomena, groups, and other topics in semantics and pragmatics....

, Roger Schwarzschild
Roger Schwarzschild
Roger Schwarzschild is a professor of linguistics in the department of linguistics at Rutgers University. His primary research is in the fields of semantics and pragmatics.-External links:*...

, Peter Lasersohn
Peter Lasersohn
Peter Lasersohn is a professor of linguistics at the University of Illinois at Urbana-Champaign.-Education:*Ph.D. in Linguistics: Ohio State University, 1988*M.A. in Linguistics: Ohio State University, 1985*B.A...

 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
∀x(if Alice admires x, then x = Bob or x = Carol)
∀x(if Bob admires x, then x = Alice or x = Carol)
∀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, Oliver & Smiley (2001), Rayo (2002), Yi (2005) and McKay (2006) 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).

So one can propose a unified account of plural terms that allows for both distributive and non-distributive satisfaction of predicates, while defending this position 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. But it should not be assumed that they must then automatically also refer to one thing (a set or a sum).

Several writers have suggested that plural logic opens the prospect of simplifying the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...

, avoiding the paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...

es of set theory, and "simplifying the complex and unintuitive axiom sets needed in order to avoid them.

Recently, Linnebo & Nicolas (2008) have suggested that languages like English contain superplural quantifiers, while Nicolas (2008) has argued that plural logic should be used to account for the semantics of mass nouns, like "wine" and "furniture".

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.

External links

Web pages of some people important in the field:
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