Editing Plural quantification (section)

==Background and motivation==

=== Multigrade (variably polyadic) predicates and relations ===
Sentences like
: Alice and Bob cooperate.
: Alice, Bob and Carol cooperate.
are said to involve a ''multigrade'' (also known as ''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 [[Willard Van Orman Quine|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===
{{main|Nominalism}}
Broadly speaking, nominalism denies the [[problem of universals|existence of universals]] ([[Abstract entity|abstract entities]]), like sets, classes, relations, properties, etc. Thus the plural logics 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.

Boolos argued that [[second-order logic|second-order]] [[monadic logic|monadic]] quantification may be systematically interpreted in terms of plural quantification, and that, therefore, second-order monadic quantification is "ontologically innocent".<ref>{{citation|title=A Companion to W. V. O. Quine|series=Blackwell Companions to Philosophy|first1=Gilbert|last1=Harman|first2=Ernest|last2=Lepore|publisher=John Wiley & Sons|year=2013|isbn=9781118608029|page=390|url=https://books.google.com/books?id=xZtNAgAAQBAJ&pg=PA390}}.</ref>

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 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{{who|date=November 2012}} have suggested that plural logic opens the prospect of simplifying the [[foundations of mathematics]], avoiding the [[paradox]]es of set theory, and simplifying the complex and unintuitive axiom sets needed in order to avoid them.{{clarify|date=November 2012}}

Recently, Linnebo & Nicolas (2008) have suggested that natural languages often contain [[superplural variable]]s (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".