Lindström quantifier

In mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. They are a generalization of first-order quantifiers, such as the existential quantifier, the universal quantifier, and the counting quantifiers. They were introduced by Per Lindström in 1966. 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 ${\displaystyle {\bar {a}}}$ a tuple of elements of the domain dom(A) of A. In other words, ${\displaystyle \phi ^{A,x,{\bar {a}}}}$ denotes a (monadic) property defined on dom(A). In general, where x is replaced by an n-tuple ${\displaystyle {\bar {x}}}$ of free variables, ${\displaystyle \phi ^{A,{\bar {x}},{\bar {a}}}}$ denotes an n-ary relation defined on dom(A). Each quantifier ${\displaystyle Q_{A}}$ 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. Their truth conditions can be specified as

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