Double Turnstile

Definition

Double Turnstile

The symbol $⊨$ has no fixed meaning on its own. It depends on what sits on the left:

Notation Left side Meaning
$M, ℓ ⊨ φ$ structure + assignment Satisfaction — $φ$ is true under this interpretation
$M ⊨ φ$ structure Sentence satisfaction — $φ$ has no free variables, assignment is irrelevant
$F_{1}, \dots, F_{n} ⊨ G$ formulas Entailment — $G$ is true in every interpretation that makes all $F_{i}$ true
$⊨ φ$ nothing $φ$ is valid — true in all interpretations

Notation	Left side	Meaning
$M, ℓ ⊨ φ$	structure + assignment	Satisfaction — $φ$ is true under this interpretation
$M ⊨ φ$	structure	Sentence satisfaction — $φ$ has no free variables, assignment is irrelevant
$F_{1}, \dots, F_{n} ⊨ G$	formulas	Entailment — $G$ is true in every interpretation that makes all $F_{i}$ true
$⊨ φ$	nothing	$φ$ is valid — true in all interpretations