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 $\mathrm{\Gamma }$ is a set of sentences, then $\mathfrak{A}\models \mathrm{\Gamma }$ means ...
• If $T$ is a theory and $\varphi$ is a sentence, then $T\models \varphi$ means ...
• If $T$ is a theory and $\mathrm{\Gamma }$ is a set of sentences, then $T\models \mathrm{\Gamma }$ means ...
• If $\mathrm{\Sigma }$ is a set of axioms for a theory $T$, and $\varphi$ is a sentence, then $\mathrm{\Sigma }\models \varphi$ means ...
• If $\mathrm{\Sigma }$ is a set of axioms for a theory $T$, and $\mathrm{\Gamma }$ is a set of sentences, then ...
• if $\varphi$ is a formula (or wff), then ...
• also the variant without anything in front, e.g., $\models \varphi$

Part before "$\models$"	$\models$	Part after "$\models$"	Possible pronunciations	Meaning
A structure/interpretation $\mathfrak{A}$	$\models$	A sentence or formula $\varphi$	The structure $\mathfrak{A}$ satisfies the formula $\varphi$. The formula $\varphi$ is true in $\mathfrak{A}$.
A set of sentences or formulas $\mathrm{\Gamma }$	$\models$	A sentence or formula $\varphi$	$\varphi$ is a logical consequence of $\mathrm{\Gamma }$. $\mathrm{\Gamma }$ logically implies $\varphi$. $\varphi$ is a semantic consequence of $\mathrm{\Gamma }$. $\varphi$ is true in every model of $\mathrm{\Gamma }$.
Nothing	$\models$	A sentence or formula $\varphi$	$\varphi$ is valid. $\varphi$ 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 $\mathrm{\Gamma }\models \varphi$.
• 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)