Category:Semantic Consequences

This category contains results about Semantic Consequences.
Definitions specific to this category can be found in Definitions/Semantic Consequences.

Let $\mathscr M$ be a formal semantics for a formal language $\LL$.

Let $\FF$ be a collection of WFFs of $\LL$.

Let $\map {\mathscr M} \FF$ be the formal semantics obtained from $\mathscr M$ by retaining only the structures of $\mathscr M$ that are models of $\FF$.

Let $\phi$ be a tautology for $\map {\mathscr M} \FF$.

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

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

That is to say, $\phi$ is a semantic consequence of $\FF$ if and only if, for each $\mathscr M$-structure $\MM$:

$\MM \models_{\mathscr M} \FF$ implies $\MM \models_{\mathscr M} \phi$

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

Note in particular that for $\FF = \O$, the notation agrees with the notation for a $\mathscr M$-tautology:

$\models_{\mathscr M} \phi$

The concept naturally generalises to sets of formulas $\GG$ on the right hand side:

$\FF \models_{\mathscr M} \GG$

if and only if $\FF \models_{\mathscr M} \phi$ for every $\phi \in \GG$.