A type of quantification that corresponds to a natural language quantifier such as ‘some things are…’ (rather than ‘some thing is…’) that is strictly stronger than the first-order existential and universal quantifiers. Formal accounts of plural quantification includes plural variables , etc., and a relation holding between singular variables and plural variables where is read as ‘ is one of the s’.
As plural quantifiers bind variables corresponding to individuals rather than properties, plural quantification is not obviously equivalent to second-order quantification over properties, and arguably remains first-order in spirit.
One might think that classical logic is capable of expressing similar constructions, e.g., the sentence:
can be captured by the sentence:
However, there are natural sentences expressible with plural quantifiers not equivalent to any sentence of first-order logic. The most well-known example is the Geach-Kaplan sentence:
Using plural quantification, the sentence may be expressed by the formula:
where the predicates and are suitably interpreted.