Existential quantifier

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A logical operation used in forming statements with the expression "for a certain x" ( "an x exists for which", "there exists an x such that" ). In formalized languages, existential quantifiers are denoted by $\exists x$, $(\exists x)$, $\cup_x$, $\vee_x$, $\Sigma_x$.


The unique existential quantifier forms the assertion that "there exists exactly one x such that" and is denoted $\exists!x$.


[a1] A. Grzegorczyk, "An outline of mathematical logic" , Reidel (1974)
[b1] S.C. Kleene, "Mathematical Logic" repr. Dover (2013) ISBN 0486317072
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Existential quantifier. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article