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 $xx,yy,zz$, etc., and a relation $\mathcal{⊰}$ holding between singular variables and plural variables where $x\mathcal{⊰}yy$ is read as ‘$x$ is one of the $y$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:

• • There exist more than one individuals of which $Px$ is true.

can be captured by the sentence:

• • $\exists x\exists y\left(Px\wedge Py\wedge x\ne y\right)$

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:

• • Some critics admire only one another.

Using plural quantification, the sentence may be expressed by the formula:

• • $\exists xx\left(\forall y\left(y\mathcal{⊰}xx\to \text{Critic}y\right)\wedge \forall y\forall z\left(\left(y\mathcal{⊰}xx\wedge \text{Admire}yz\right)\to \left(z\mathcal{⊰}xx\wedge y\ne z\right)\right)\right)$

where the predicates $\text{Critic}$ and $\text{Admire}$ are suitably interpreted.