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 storeys.

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 (see notably Simons 1982), 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 proved that it is nonfirstorderizable (the proof can be found in that article). Hence its paraphrase into a formal language commits us to quantification over (i.e. the existence of) sets. But some find it implausible that a commitment to sets is essential in explaining these sentences.

Note that an individual instance of the sentence, such as "Alice, Bob and Carol admire only one another", need not involve sets and is equivalent to the conjunction of the following first-order sentences:

∀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 all critics [it being taken as read that critics cannot admire themselves]. But this seems to be an instance of "some people admire only one another", which is nonfirstorderizable.

Boolos argued that 2nd-order monadic quantification may be systematically interpreted in terms of plural quantification, and that, therefore, 2nd-order monadic 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 (monadic) predicate is distributive (in standard logic, these "predicates" would be represented by relations). Yet such sentences also seem innocent of any existential assumptions, and do not involve quantification.

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).

Several writers 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.

Recently, Linnebo & Nicolas (2008) have suggested that natural languages often contain superplural variables (and associated quantifiers) such as "these people, those people, and these other people compete against each other" (eg as teams in an online game), 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.

References

• George Boolos, 1984, "To be is to be the value of a variable (or to be some values of some variables)," Journal of Philosophy 81: 430–449. In Boolos 1998, 54–72.
• --------, 1985, "Nominalist platonism." Philosophical Review 94: 327–344. In Boolos 1998, 73–87.
• --------, 1998. Logic, Logic, and Logic. Harvard University Press.
• Burgess, J.P., "From Frege to Friedman: A Dream Come True?"
• --------, 2004, “E Pluribus Unum: Plural Logic and Set Theory,” Philosophia Mathematica 12(3): 193–221.
• Cameron, J. R., 1999, "Plural Reference," Ratio.
• Cocchiarella, Nino (2002). "On the Logic of Classes as Many". Studia Logica 70: 303–338. doi:10.1023/A:1015190829525. 
• De Rouilhan, P., 2002, "On What There Are," Proceedings of the Aristotelian Society: 183–200.
• Gottlob Frege, 1895, "A critical elucidation of some points in E. Schroeder's Vorlesungen Ueber Die Algebra der Logik," Archiv fur systematische Philosophie: 433–456.
• Landman, F., 2000. Events and Plurality. Kluwer.
• David K. Lewis, 1991. Parts of Classes. London: Blackwell.
• Linnebo, Øystein; Nicolas, David. "Superplurals in English". Analysis 68 (3): 186–97. http://d.a.nicolas.free.fr/research/Linnebo-Nicolas-Superplurals.pdf. 
• McKay, Thomas J. (2006), Plural Predication, New York: Oxford University Press, ISBN 0199278148, 9780199278145 
• John Stuart Mill, 1904, A System of Logic, 8th ed. London: .
• Nicolas, David (2008). "Mass nouns and plural logic". Linguistics and Philosophy 31 (2): 211–244. doi:10.1007/s10988-008-9033-2. http://d.a.nicolas.free.fr/Nicolas-Mass-nouns-and-plural-logic-Revised-2.pdf. 
• Oliver, Alex; Smiley, Timothy (2001). "Strategies for a Logic of Plurals". Philosophical Quarterly 51 (204): 289–306. doi:10.1111/j.0031-8094.2001.00231.x. 
• Oliver, Alex (2004). "Multigrade Predicates". Mind 113: 609–681. doi:10.1093/mind/113.452.609. 
• Rayo, Agustín (2002). "Word and Objects". Noûs 36: 436–64. 
• --------, 2006, “Beyond Plurals,” in Rayo and Uzquiano (2006).
• --------, 2007, “Plurals,” forthcoming in Philosophy Compass.
• --------, and Gabriel Uzquiano, eds., 2006. Absolute Generality Oxford University Press.
• Bertrand Russell, B., 1903. The Principles of Mathematics. Oxford Univ. Press.
• Peter Simons, 1982, “Plural Reference and Set Theory,” in Barry Smith, ed., Parts and Moments: Studies in Logic and Formal Ontology. Munich: Philosophia Verlag.
• --------, 1987. Parts. Oxford University Press.
• Uzquiano, Gabriel (2003). "Plural Quantification and Classes". Philosophia Mathematica 11 (1): 67–81. 
• Yi, Byeong-Uk (1999). "Is two a property?". Journal of Philosophy 95: 163–190. 
• --------, 2005, “The Logic and Meaning of Plurals, Part I,” Journal of Philosophical Logic 34: 459–506.

External links

• Plural quantification entry by Øystein Linnebo in the Stanford Encyclopedia of Philosophy

Web pages of some people important in the field:

• John P. Burgess.
• Øystein Linnebo.
• Tom McKay. Go for "staff", then "McKay."
• David Nicholas.