Quantifiers, generalized

Generalized quantifiers are logical tools with a wide range of uses. As the term indicates, they generalize the ordinary universal and existential quantifiers from first-order logic, ‘∀x’ and ‘∃x’, which apply to a formula A(x), binding its free occurrences of x. ∀xA(x) says that A(x) holds for all objects in the universe and ∃xA(x) says that A(x) holds for some objects in the universe, that is, in each case, that a certain condition on A(x) is satisfied. It is natural then to consider other conditions, such as ‘for at least five’, ‘at most ten’, ‘infinitely many’ and ‘most’. So a quantifier Q stands for a condition on A(x), or, more precisely, for a property of the set denoted by that formula, such as the property of being non-empty, being infinite, or containing more than half of the elements of the universe. The addition of such quantifiers to a logical language may increase its expressive power.

A further generalization allows Q to apply to more than one formula, so that, for example, Qx(A(x),B(x)) states that a relation holds between the sets denoted by A(x) and B(x), say, the relation of having the same number of elements, or of having a non-empty intersection. One also considers quantifiers binding more than one variable in a formula. Qxy,zu(R(x,y),S(z,u)) could express, for example, that the relation (denoted by) R(x,y) contains twice as many pairs as S(z,u), or that R(x,y) and S(z,u) are isomorphic graphs.

In general, then, a quantifier (the attribute ‘generalized’ is often dropped) is syntactically a variable-binding operator, which stands semantically for a relation between relations (on individuals), that is, a second-order relation. Quantifiers are studied in mathematical logic, and have also been applied in other areas, notably in the semantics of natural languages. This entry first presents some of the main logical facts about generalized quantifiers, and then explains their application to semantics.