No Boolean Interpretation Models a WFF and its Negation

Theorem

Let $v$ be a boolean interpretation.

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

Then $v$ can not model both $\mathbf A$ and $\neg \mathbf A$.

Proof

Suppose that $v$ models $\mathbf A$:

$v \models \mathbf A$

Then $v \left({\mathbf A}\right) = T$ by definition of models.

By definition of boolean interpretation, $v \left({\neg \mathbf A}\right) = F$.

In particular, $v (\neg \mathbf A) \ne T$, so that:

$v \not\models \neg \mathbf A$

Hence the result.

$\blacksquare$