Definition:Unsatisfiable/Boolean Interpretations

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

$\mathbf A$ is called unsatisfiable (for boolean interpretations) if and only if:

$\map v {\mathbf A} = \F$

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

In terms of validity, this can be rendered:

$v \not \models_{\mathrm {BI} } \mathbf A$

that is, $\mathbf A$ is invalid in every boolean interpretation of $\mathbf A$.

Also known as

Unsatisfiable formulae are more commonly referred to as contradictions.

To avoid ambiguity with inconsistent formulae, the latter term is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$ when specifically referring to unsatisfiability.

Also see

  • Results about contradictions can be found here.