Double turnstile

The double turnstile is a binary relation. It has several different meanings in different contexts:

• To show semantic consequence, with a set of sentences on the left and a single sentence on the right, to denote that if every sentence on the left is true, the sentence on the right must be true, e.g. ${\displaystyle \Gamma \vDash \varphi }$. This usage is closely related to the single-barred turnstile symbol which denotes syntactic consequence.
• To show satisfaction, with a model (or truth-structure) on the left and a set of sentences on the right, to denote that the structure is a model for (or satisfies) the set of sentences, e.g. ${\displaystyle {\mathcal {A}}\models \Gamma }$.
  • In this context, the semantic consequence in the previous list can be stated as "For a given model ${\displaystyle {\mathcal {A}}}$, if ${\displaystyle {\mathcal {A}}\models \Gamma }$ then ${\displaystyle {\mathcal {A}}\vDash \varphi }$".
• To denote a tautology, ${\displaystyle \vDash \varphi }$. which is to say that the expression ${\displaystyle \varphi }$ is a semantic consequence of the empty set.

In TeX, the turnstile symbols ${\displaystyle \vDash }$ and ${\displaystyle \models }$ are obtained from the commands \vDash and \models respectively. In Unicode it is encoded at U+22A8 ⊨ TRUE (HTML ⊨ · ⊨, ⊨)

In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, and is capable of putting labels below or above it, in the correct places. The article A Tool for Logicians is a tutorial on using this package.

• List of logic symbols
• List of mathematical symbols