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).

Notes

Other tricky things:

• Some books only use the models symbol for one of the two use cases. E.g. Boolos/Burgess/Jeffrey only uses the symbol for truth in an interpretation. Therefore you might be really confused when you start reading other books and you start seeing stuff like $\Gamma \models \phi$.
• In the Boolos/Burgess/Jeffrey book, if a set of sentences comes after $\models$, then the sentences are taken disjunctively rather than conjunctively. (see p. 168)
• https://math.stackexchange.com/a/2506938/35525