They were later studied for their applications in logic in computer science and database query languages.
In order to facilitate discussion, some notational conventions need explaining.
The expression for A an L-structure (or L-model) in a language L, φ an L-formula, and
a tuple of elements of the domain dom(A) of A.
denotes a (monadic) property defined on dom(A).
denotes an n-ary relation defined on dom(A).
is relativized to a structure, since each quantifier is viewed as a family of relations (between relations) on that structure.
For a concrete example, take the universal and existential quantifiers ∀ and ∃, respectively.
is the singleton whose sole member is dom(A), and
In other words, each quantifier is a family of properties on dom(A), so each is called a monadic quantifier.
Any quantifier defined as an n > 0-ary relation between properties on dom(A) is called monadic.
Before we go on to Lindström's generalization, notice that any family of properties on dom(A) can be regarded as a monadic generalized quantifier.
For example, the quantifier "there are exactly n things such that..." is a family of subsets of the domain of a structure, each of which has a cardinality of size n. Then, "there are exactly 2 things such that φ" is true in A iff the set of things that are such that φ is a member of the set of all subsets of dom(A) of size 2.
Lindström quantifiers are classified according to the number structure of their parameters.
The first result in this direction was obtained by Lindström (1966) who showed that a type (1,1) quantifier was not definable in terms of a type (1) quantifier.
After Lauri Hella (1989) developed a general technique for proving the relative expressiveness of quantifiers, the resulting hierarchy turned out to be lexicographically ordered by quantifier type: For every type t, there is a quantifier of that type that is not definable in first-order logic extended with quantifiers that are of types less than t. Although Lindström had only partially developed the hierarchy of quantifiers which now bear his name, it was enough for him to observe that some nice properties of first-order logic are lost when it is extended with certain generalized quantifiers.
For example, adding a "there exist finitely many" quantifier results in a loss of compactness, whereas adding a "there exist uncountably many" quantifier to first-order logic results in a logic no longer satisfying the Löwenheim–Skolem theorem.
In 1969 Lindström proved a much stronger result now known as Lindström's theorem, which intuitively states that first-order logic is the "strongest" logic having both properties.