Plural Quantification - Plural Quantification

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 . 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" (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".