Definition:Falsifiable
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Definition
Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Falsifiable Formula
A logical formula $\phi$ of $\LL$ is falsifiable for $\mathscr M$ if and only if:
That is, there exists some structure $\MM$ of $\mathscr M$ such that:
- $\MM \not\models_{\mathscr M} \phi$
Falsifiable Set of Formulas
A collection $\FF$ of logical formulas of $\LL$ is falsifiable for $\mathscr M$ if and only if:
That is, there exists some structure $\MM$ of $\mathscr M$ such that:
- $\MM \not\models_{\mathscr M} \FF$
Falsifiable for Boolean Interpretations
Let $\mathbf A$ be a WFF of propositional logic.
$\mathbf A$ is called falsifiable (for boolean interpretations) if and only if:
- $\map v {\mathbf A} = \F$
for some 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 the boolean interpretation $v$ of $\mathbf A$.