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.
Contents
- History
- Multigrade variably polyadic predicates and relations
- Nominalism
- Formal definition
- Syntax
- Model theory
- Criticism
- References
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.
History
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.
The general idea can be traced back to Leibniz. (Levey 2011, pp. 129–133)
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.
Multigrade (variably polyadic) predicates and relations
Sentences like
Alice and Bob cooperate.Alice, Bob and Carol cooperate.are said to involve a multigrade (aka variably polyadic, also anadic) predicate or relation ("cooperate" in this example), meaning that they stand for the same concept even though they don't have a fixed arity (cf. Linnebo & Nicolas 2008). The notion of multigrade relation/predicate has appeared as early as the 1940s and has been notably used by Quine (cf. Morton 1975). Plural quantification deals with formalizing the quantification over the variable-length arguments of such predicates, e.g. "xx cooperate" where xx is a plural variable. Note that in this example it makes no sense, semantically, to instantiate xx with the name of a single person.
Nominalism
Broadly speaking, nominalism denies the existence of universals (abstract entities), like sets, classes, relations, properties, etc. Thus the plural logic(s) were developed as an attempt to formalize reasoning about plurals, such as those involved in multigrade predicates, apparently without resorting to notions that nominalists deny, e.g. sets.
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 shipmatesThey are meeting togetherThey lifted a pianoThey are surrounding a buildingThey admire only one anotheralso cannot be interpreted in monadic second 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. 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" (e.g. 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".
Formal definition
This section presents a simple formulation of plural logic/quantification approximately the same as given by Boolos in Nominalist Platonism (Boolos 1985).
Syntax
Sub-sentential units are defined as
Full sentences are defined as
The last two lines are the only essentially new component to the syntax for plural logic. Other logical symbols definable in terms of these can be used freely as notational shorthands.
This logic turns out to be equi-interpretable with monadic second order logic.
Model theory
Plural logic's model theory/semantics is where the logic's lack of sets is cashed out. A model is defined as a tuple
Satisfaction is given as
Where for singular variable symbols,
As in the syntax, only the last two are truly new in plural logic. Boolos observes that by using assignment relations
This can be taken as the simplest, and most obvious argument that plural logic as Boolos defined it is ontologically innocent.
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 anotherThe 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 anotherBut 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.