Definition:Falsifiable

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:

$\phi$ is not valid in some structure $\MM$ of $\mathscr M$

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:

Some $\mathscr M$-structure $\MM$ is not a model of $\FF$

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 $\mathbf A$.