# Quantifier

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The general name for a logical operation that constructs from a predicate a statement characterizing the domain of validity of . In mathematical logic, the most widely used quantifiers are the universal quantifier and the existential quantifier . The statement means that the domain of validity of is the same as the domain of values of the variable . The statement means that the domain of validity of is non-empty. If one is interested in the behaviour of the predicate not on the whole domain of values of , but only on the part singled out by a predicate , then one often uses the restricted quantifiers and . In this case, the statement means the same as , while has the same meaning as , where is the conjunction sign and is the implication sign.
More generally, the model-theoretic interpretation of an arbitrary "quantifier" (with the same syntactic behaviour as and ) can (according to A. Mostowski) be given by a mapping associating with each model a class of subsets of . Then one stipulates as a truth-definition for that, e.g., a sentence holds in if and only if the set is in . Thus, with the existential quantifier is associated the class of non-empty subsets of and with the universal quantifier is associated the class . However, there are many more possible quantifiers, e.g. given by , (the Chang quantifier), , etc. This set-up can be generalized to "quantifiers" binding more than one variable occurring in more than one formula (example: the equi-cardinality quantifier binding two variables and in two formulas and , yielding the formula , which is interpreted by ). Even more general is the Lindström quantifier. And each quantifier has its own logic.