Tautology iff Negation is Unsatisfiable

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Theorem

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

Then $\mathbf A$ is a tautology if and only if 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 $\map v {\mathbf A} = \T$.

Hence, by definition of the boolean interpretation of negation:

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

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

Hence, by definition of the boolean interpretation of negation:

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

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

$\blacksquare$

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5$: Theorem $2.39$