Definition:Satisfiable

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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.


Also see


Sources