Definition:Satisfiable
Definition
Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Satisfiable Formula
A logical formula $\phi$ of $\LL$ is satisfiable for $\mathscr M$ if and only if:
That is, there exists some structure $\MM$ of $\mathscr M$ such that:
- $\MM \models_{\mathscr M} \phi$
Satisfiable Set of Formulas
A collection $\FF$ of logical formulas of $\LL$ is satisfiable for $\mathscr M$ if and only if:
- There is some $\mathscr M$-model $\MM$ of $\FF$
That is, there exists some structure $\MM$ of $\mathscr M$ such that:
- $\MM \models_{\mathscr M} \FF$
Satisfiable for Boolean Interpretations
Let $\mathbf A$ be a WFF of propositional logic.
$\mathbf A$ is called satisfiable (for boolean interpretations) if and only if:
- $\map v {\mathbf A} = \T$
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
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Axiom systems