Definition:Semantic Consequence

Definition
Let $\mathscr M$ be a formal semantics for a formal language $\mathcal L$.

Let $\mathcal F$ be a collection of WFFs of $\mathcal L$.

Let $\mathscr M \left({\mathcal F}\right)$ be the formal semantics obtained from $\mathscr M$ by retaining only the structures of $\mathscr M$ that are models of $\mathcal F$.

Let $\phi$ be a tautology for $\mathscr M \left({\mathcal F}\right)$.

Then $\phi$ is called a semantic consequence of $\mathcal F$, and this is denoted as:


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

That is to say, $\phi$ is a semantic consequence of $\mathcal F$ iff, for each $\mathscr M$-structure $\mathcal M$:


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

where $\models_{\mathscr M}$ is the models relation.

Note in particular that for $\mathcal F = \varnothing$, the notation agrees with the notation for a $\mathscr M$-tautology:


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

The concept naturally generalises to sets of formulas $\mathcal G$ on the RHS:


 * $\mathcal F \models_{\mathscr M} \mathcal G$

iff $\mathcal F \models_{\mathscr M} \phi$ for every $\phi \in \mathcal G$.

Also known as
One also says that $\mathcal F$ semantically entails $\phi$, in particular if $\mathcal F$ comprises just one WFF.

Another common term used is logical consequence (and, correspondingly, logical entailment).

However, the adjective "logical" is heavily used and prone to ambiguity, so these terms should not be used on.

Also see

 * Definition:Tautology (Formal Semantics)


 * Definition:Provable Consequence