# User:IssaRice/Computability and logic/Models symbol

The "models" symbol, $\models$, is used for several purposes in mathematical logic. Roughly, there are two basic purposes:

1. When the symbol that comes before " $\models$" is a structure/interpretation, then it says something about truth in that structure/interpretation.
2. When the symbol that comes before " $\models$" is a sentence or set of sentences, then it says something about semantic consequence (also called logical consequence, logical implication, semantic implication). In this case, we are talking about all possible structures/interpretations.

In either of the above two purposes, we are talking about the semantics (rather than syntax) of a logical system.

• If $\mathfrak A$ is a structure/interpretation and $\Gamma$ is a set of sentences, then $\mathfrak A \models \Gamma$ means ...
• If $T$ is a theory and $\phi$ is a sentence, then $T \models \phi$ means ...
• If $T$ is a theory and $\Gamma$ is a set of sentences, then $T \models \Gamma$ means ...
• If $\Sigma$ is a set of axioms for a theory $T$, and $\phi$ is a sentence, then $\Sigma \models \phi$ means ...
• If $\Sigma$ is a set of axioms for a theory $T$, and $\Gamma$ is a set of sentences, then ...
• if $\phi$ is a formula (or wff), then ...
• also the variant without anything in front, e.g., $\models \phi$
Part before " $\models$" $\models$ Part after " $\models$" Possible pronunciations Meaning
A structure/interpretation $\mathfrak A$ $\models$ A sentence or formula $\phi$ The structure $\mathfrak A$ satisfies the formula $\phi$.
The formula $\phi$ is true in $\mathfrak A$.
The structure $\mathfrak A$ makes true the formula $\phi$.
A set of sentences or formulas $\Gamma$ $\models$ A sentence or formula $\phi$ $\phi$ is a logical consequence of $\Gamma$. $\Gamma$ logically implies $\phi$. $\phi$ is a semantic consequence of $\Gamma$. $\phi$ is true in every model of $\Gamma$.
Nothing $\models$ A sentence or formula $\phi$ $\phi$ is valid. $\phi$ is a tautology (especially in the case of propositional logic).