Definition:Formal Semantics/Valid

Definition
Let $\mathcal L$ be a formal language.

Part of specifying a formal semantics $\mathscr M$ for $\mathcal L$ is to define a notion of validity.

Concretely, a precise meaning needs to be assigned to the phrase:


 * "The $\mathcal L$-WFF $\phi$ is valid in the $\mathscr M$-structure $\mathcal M$."

It can be expressed symbolically as:


 * $\mathcal M \models_{\mathscr M} \phi$

Also denoted as
When the formal semantics in use is clear from the context, $\mathcal M \models \phi$ is commonly seen in place of $\mathcal M \models_{\mathscr M} \phi$.

Also see

 * Definition:Formal Semantics


 * Definition:Model (Logic)
 * Definition:Tautology