existential quantifier

存在量词

Foundations & Logic · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

∃x φ\exists x\,\varphi holds when some domain value for xx makes φ\varphi true under the remaining assignment.

保持其他变元赋值不变，若论域中至少一个元素作为 x 的值使 φ 为真，则 ∃x φ 为真。

Example

∃n∈Z  (n2=9)\exists n\in\mathbb Z\;(n^2=9), witnessed by 33.

Usage and distinctions

Existence does not assert uniqueness; ∃! is a different assertion.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Avigad, Hua, Lewis and van Doorn — Logic and Proof§7.2 The Universal Quantifier; §7.3 The Existential Quantifier · Checked 2026-10-14

Translation sources

  1. 杨睿之（复旦大学）— 数理逻辑，2021年秋季PDF pp.17–20: 一阶逻辑的语言；自由出现与约束出现 · Checked 2026-10-14
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