universal quantifier

全称量词

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Translation evidence: Literature usage

Meaning in context

∀x φ\forall x\,\varphi holds under an assignment in a structure precisely when φ\varphi holds for every domain value assigned to xx.

在给定结构和赋值下，∀x φ 为真当且仅当把 x 赋为论域中的任意元素时 φ 都为真。

Example

∀n∈Z  (n2≥0)\forall n\in\mathbb Z\;(n^2\geq0).

Usage and distinctions

Restricted universal quantification uses implication: ∀x∈A P means ∀x(x∈A→P).

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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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