logical consequence

语义蕴含

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

Other names / variants: semantic consequence

Translation evidence: Literature usage

Meaning in context

For a sentence set Γ, a sentence φ is a consequence when every structure satisfying Γ satisfies φ.

对于语句集 Γ，若每个满足 Γ 的结构都满足 φ，则称 Γ 语义蕴含 φ。

Example

{P(c),∀x(P(x)→Q(x))}⊨Q(c)\{P(c),\forall x(P(x)\to Q(x))\}\models Q(c).

Usage and distinctions

⊨ compares truth in structures; ⊢ asserts existence of a derivation.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Avigad, Hua, Lewis and van Doorn — Logic and Proof§10.2 Truth in a Model; §10.4 Validity and Logical Consequence; §10.5 Soundness and Completeness · Checked 2026-10-14

Translation sources

  1. 杨睿之（复旦大学）— 数理逻辑，第15讲PDF p.6 定义「语义蕴含」；同页用「逻辑蕴涵」 · Checked 2026-10-14
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