Double Turnstile

Definition

Double Turnstile

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

Notation	Left side	Meaning
$\mathcal{M},\ell \models \phi$	structure + assignment	Satisfaction — $\phi$ is true under this interpretation
$\mathcal{M}\models \phi$	structure	Sentence satisfaction — $\phi$ has no free variables, assignment is irrelevant
$F_1,\dots ,F_n\models G$	formulas	Entailment — $G$ is true in every interpretation that makes all $F_i$ true
$\models \phi$	nothing	$\phi$ is valid — true in all interpretations