# Definition:Satisfiable

## Definition

Let $\mathcal L$ be a logical language.

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

### Satisfiable Formula

A logical formula $\phi$ of $\mathcal L$ is satisfiable for $\mathscr M$ iff:

$\phi$ is valid in some structure $\mathcal M$ of $\mathscr M$

That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that:

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

### Satisfiable Set of Formulas

A collection $\mathcal F$ of logical formulas of $\mathcal L$ is satisfiable for $\mathscr M$ iff:

There is some $\mathscr M$-model $\mathcal M$ of $\mathcal F$

That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that:

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

### Satisfiable for Boolean Interpretations

Let $\mathbf A$ be a WFF of propositional logic.

$\mathbf A$ is called satisfiable (for boolean interpretations) iff:

$v \left({\mathbf A}\right) = T$

for some boolean interpretation $v$ for $\mathbf A$.

In terms of validity, this can be rendered:

$v \models_{\mathrm{BI}} \mathbf A$

that is, $\mathbf A$ is valid in the boolean interpretation $v$ of $\mathbf A$.

## Also known as

In each of the above cases, satisfiable is also seen referred to as semantically consistent.