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# quantifier

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One of the two symbols ∀ or ∃ used in predicate calculus. ∀ is the universal quantifier and is read “for all”. ∃ is the existential quantifier and is read “there exists” or “for some”. In either case the reference is to possible values of the variable v that the quantifier introduces. ∀v. F means that the formula F is true for all values of v, while ∃v. F means that F is true for at least one value of v. As an example, suppose that P(x,y) is the predicate “x is less than or equal to y”. Then the following expression ∃x . ∀y . P(x,y) says that there exists an x that is less than or equal to all y. This statement is true if values range over the natural numbers, since x can be taken to be 0. It becomes false however if values are allowed to range over negative integers as well. Note also that it would be false even for natural numbers if the predicate were “x is less than y”. Other notations such as (∀v)F in place of ∀v . F are also found.

x . ∀y . P(x,y)

Subjects: Mathematics.

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