# Definition:Semantic Consequence

*This page is about semantic consequence in the context of formal semantics. For other uses, see consequence.*

## Definition

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 $\phi \in \GG$.

Let $\FF$ be a collection of WFFs of propositional logic.

Then a WFF $\mathbf A$ is a **semantic consequence** of $\FF$ if and only if:

- $v \models_{\mathrm{BI}} \FF$ implies $v \models_{\mathrm{BI}} \mathbf A$

where $\models_{\mathrm{BI}}$ is the models relation.

Let $\FF$ be a collection of WFFs of predicate logic.

Then a WFF $\mathbf A$ is a **semantic consequence** of $\FF$ if and only if:

- $\AA \models_{\mathrm{PL} } \FF$ implies $\AA \models_{\mathrm{PL} } \mathbf A$

for all structures $\AA$, where $\models_{\mathrm{PL} }$ is the models relation.

## Also known as

For **semantic consequence**, one also says that **$\FF$ semantically entails $\phi$, in particular if $\FF$ comprises just one WFF.**

The term **formal consequence** is also sometimes encountered.

Another common term used is **logical consequence**, but this is mainly used more generally in the context of **logical implication**.

Correspondingly, the term **logical entailment** can also be found.

However, the adjective "logical" is heavily used and prone to ambiguity, so these terms should not be used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Results about
**semantic consequences**can be found**here**.

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.7$: Tableaus - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**consequence**:**2.**(**formal consequence**) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**consequence**:**2.**(**formal consequence**)