Tautology iff Negation is Unsatisfiable

Theorem

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

Then $\mathbf A$ is a tautology iff its negation $\neg \mathbf A$ is unsatisfiable.

Proof

Necessary Condition

Let $\mathbf A$ be a tautology.

Let $v$ be a boolean interpretation of $\mathbf A$.

Then $v \left({\mathbf A}\right) = T$.

Hence, by definition of the boolean interpretation of negation:

$v \left({\neg \mathbf A}\right) = F$

Since $v$ was arbitrary, it follows that $\neg \mathbf A$ is unsatisfiable.

$\Box$

Sufficient Condition

Let $\neg \mathbf A$ be unsatisfiable.

Let $v$ be a boolean interpretation of $\neg \mathbf A$.

Then $v \left({\neg \mathbf A}\right) = F$.

Hence, by definition of the boolean interpretation of negation:

$v \left({\mathbf A}\right) = T$

Since $v$ was arbitrary, it follows that $\mathbf A$ is a tautology.

$\blacksquare$